Talks

Transformers have been a major breakthrough in Natural Language Processing (NLP) due to their ability to capture long-range dependencies through self-attention. However, the (self-)attention mechanism suffers from massive memory consumption, especially for tasks with large context windows and high resolution data, such as climate data.

In this talk, I will present a transformer model that uses multiple icosahedral grids to enable large (physical) context windows and high resolutions for various climate-related modelling tasks.

As the model is still under development, the presentation will focus on its technical foundations and properties such as resolution independence and multi-scale output in the context of climate data.

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The goal of parallel-in-time methods is to employ parallelism in the time direction in addition to the space direction, in the hope of obtaining further parallel speedups at the limits of what is possible due to spatial parallelism with domain decomposition alone. Recently diagonalisation techniques have emerged as a way of solving the coupled system for the solution of a differential equation at several timesteps simultaneously. One approach, sometimes referred to as “ParaDiag II” involves preconditioning this “all-at-once” system obtained from time discretisation of a linear constant coefficient ODE (perhaps obtained as the space discretisation of a time dependent PDE) with a nearby system that can be diagonalised in time, allowing the solution of independent blocks in parallel. For nonlinear PDEs this approach can form the basis of a preconditioner within a Newton-Krylov method for the all-at-once system after time averaging the (now generally time dependent) Jacobian system. After some preliminary description of the ParaDiag II approach, I will present results from our investigation of ParaDiag II applied to some testcases from the hierarchy of models used in the development of dry dynamical cores for atmosphere models, including performance benchmarks. Using these results I will identify the key challenges in obtaining further speedups and identify some directions to address these.

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We present a deep learning-based object tracking framework designed to accurately extract particle trajectories in diverse experimental settings. This framework, which leverages the state-of-the-art object detection model YOLO and the Hungarian Algorithm, is particularly effective for scenarios where objects remain within the scene without coalescence. Our simple approach, requiring minimal initial human input, enables efficient, fast, and accurate extraction of observables of interest across various experimental configurations. The result is high-fidelity data ideally suited for data-driven modeling applications..
The framework is applied to walking droplets experiments, where a liquid droplet, known as a walker, propels itself laterally on the free surface of a vibrating bath of the same liquid. Walking droplets are of significant scientific interest as they are the only known example of quantum-like behaviors at a macroscopic scale Our methodology can track individual walker(s) in real-time across a broad spectrum of experimental settings without suffering from identity-switch issues.

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In this talk I will present some recent results on polynomial decay rates for C0-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Unlike many of the recent developments in the literature our results do not require the semigroup to be uniformly bounded. The talk is based on joint work with Chenxi Deng and Jan Rozendaal.

Entropy stability is the foundation of numerical methods for hyperbolic conservation laws, thereby ensuring the stability and reliability of the resulting numerical solutions. Summation-by-parts (SBP) operators provide a general framework to systematically develop entropy-stable schemes by mimicking continuous properties on a discrete level. They have proven to be a powerful tool to provide stable and high-order accurate numerical solutions. Classically, they are developed in order to differentiate polynomials up to a certain degree exactly. However, in many cases alternative function spaces are more appropriate to approximate the underlying solution space. Especially in multidimensional problems with potentially complex domains radial basis functions are known to possess very good approximation properties. The theory of radial basis function approximation provides us with stability and convergence results for scattered data approximation in a meshfree setting.
This talk discusses properties and efficient construction algorithms for multidimensional function space SBP (MFSBP) operators based on scattered data. I focus on radial basis function spaces and show some preliminary results for using MFSBP operators to solve conservation laws. I give an outlook on how convergence results of radial basis functions can be used to prove long-time error behavior of SBP discretizations for linear advection problems.

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Inhomogeneous random graphs are a prominent tool for modeling real-world complex networks as they manage to capture key concepts such as the scale-free property. In this talk we will focus on two particular inhomogeneous random graph models, the Norros-Reittu model and the random connection model. The Norros-Reittu model uses a deterministic vertex set and can be seen as a generalisation of the famous Erdős–Rényi graph. The random connection model on the other hand yields a spatial random graph, which leads to natural clustering effects. Our main goal is to determine the asymptotic behaviour of the size of the largest component as the number of vertices or the size of the observation window, respectively, goes to infinity. For the Norros-Reittu model we also study asymptotics of other counting statistics.

This talk gives an overview of the results obtained in my PhD under the supervision of Prof. Dr. Matthias Schulte.

Lower bounds for variances are often needed to derive central limit theorems. In this talk, a generalised reverse Poincaré inequality is established, which provides a lower variance bound for Poisson functionals that depends on the difference operator of some fixed order.
Poisson functionals, i.e. random variables that depend on a Poisson process, have many applications in stochastic geometry. In this talk we apply the introduced lower variance bound to statistics of spatial random graphs, $L^p$ surface areas of random polytopes and geometric functionals of excursion sets of Poisson shot noise processes.

This talk gives an overview of the results obtained in my PhD under the supervision of Prof. Dr. Matthias Schulte.

Accurate flow simulations remain a challenging task. In this talk we discuss the use of deep neural networks for augmenting classical finite element simulations in fluid-dynamics.
We first investigate the classical DFG-benchmark in 3D. We extend these settings to higher Reynolds numbers. At high Reynolds numbers, accurate simulations in 3D settings become increasingly difficult, and the classical methods reach their limits. To address this issue, we discuss approaches to connect the finite element method with neural networks. We propose the Deep Neural Network Multigrid Solver, which combines a geometric multigrid solver with a deep neural network to overcome limitations of classical methods. This approach uses classical simulation techniques where their strengths are eminent, such as the efficient representation of a coarse, large-scale flow field. Neural networks are used when a full resolution of the effects does not seem possible or efficient. While our method is tightly embedded in a geometric multigrid framework, it remains flexible, allowing for the coarse grid problem to be addressed with various solvers and enabling local enrichment without a global fine grid.
We demonstrate the efficiency, generalizability, and scalability of our proposed approach using 3D simulations. Our focus is on issues of stability, generalizability.
In the second part of the talk we give an outlook on future developments towards an extension to space-time multigrid methods within the framework of space-time finite element methods. We discuss the efficient implementation of space-time multigrid methods using the matrix-free framework provided by the deal.ii finite element library and demonstrate its effectiveness for the heat and acoustic wave equation.

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Radial basis function finite difference (RBF-FD) discretization has recently emerged as an al-
ternative to classical finite difference or finite element discretization of (systems) of partial
differential equations. We focus on the construction of preconditioners for the iterative solution
of the resulting linear systems of equa- tions. In RBF-FD, a higher discretization accuracy may
be obtained by increasing the stencil size. This, however, leads to a less sparse and often also
worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We
propose to construct preconditioners based on stiffness matrices resulting from RBF-FD dis-
cretization with smaller stencil sizes compared to the one for the actual system to be solved. In
our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS)
with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners
in the solution of the three-dimensional convection-diffusion equation.

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I will be introducing myself formally and presenting my master's thesis, which concerns the study of numerical schemes for solving singularly perturbed differential equations, focusing on the Haar wavelet method.

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This talk focuses on the mathematical models underlying reinforcement learning in artificial intelligence, particularly the reward functions in Markov Decision Processes. I argue that ethical principles related to well-being, safety, and equality are inherently reflected in these mathematical models. Building on this foundation, I then demonstrate how ethics can inform computational mathematics, while also addressing the challenges one encounters in this domain. Specifically, I discuss how the mathematical models behind reinforcement learning may rely on a distorted representation of ethics with respect to the determinacy and commensurability of ethical values.

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In Numerical Weather Prediction and Climate modelling, computational efficiency and numerical accuracy are paramount. This work aims to implement time-parallel Spectral Deferred Correction (SDC) methods in LFRic-Atmosphere, the Met Office’s next-generation atmospheric model, designed to exploit the new supercomputers with improved scalability; the use of a quasi-uniform cubed-sphere mesh is integral to this, as is the underlying lowest-order compatible finite element spatial discretisation. LFRic-Atmosphere has an iterative semi-implicit time stepping structure with a Method of Lines finite-transport scheme using an explicit Runge-Kutta time discretisation. Time parallel SDC offers increased temporal accuracy with small computation cost, this could be utilised over the whole time discretisation, or to target a specific time discretised component.
I will present two approaches in this talk. The first approach is using serial SDC as the time discretisation of LFRic-Atmosphere’s finite-volume transport scheme. The second approach is using a serial IMEX SDC time stepper to compare to the semi-implicit time stepping structure in LFRic-Atmosphere. My initial work has explored both using the shallow water equations, I will present results from the standard shallow water test-cases.

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The Maxey-Riley Equation (MRE) models the motion of a finite-sized, spherical particle moving in a fluid. Applications using the MRE are, for example, the study of the spread of Coronavirus particles in a room, the formation of clouds and the so-called marine snow. The MRE is a second-order, implicit integro-differential equation with a singular kernel at initial time. For over 35 years, researchers used approximations and numerical schemes with high storage requirements or ignored the integral term, although its impact can be relevant. A major break-through was reached in 2019, when Prasath et al. mapped the MRE to a time-dependent Robin-type boundary condition of the 1D Heat equation, thus removing the requirement to store the full history. They provided an implicit integral form of the solution by using the so-called Fokas method that could be later solved with a numerical scheme and a nonlinear solver. While Prasath et al.’s method can deliver numerical solutions accurately, the need to evaluate nested integrals makes it computationally costly and it becomes impractical for computing trajectories of a large number of particles. In this talk, we present a new fourth order finite differences scheme and compare its accuracy and performance with Prasath et al’s method as well as other existing schemes. We then apply our method for the calculation of Lagrangian Coherent Structures, a large scale fluid structure, and point out for which cases, the approximations on the MRE have a considerable influence on these structures and the use of the full MRE models is relevant.

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This presentation explores the intricacies of training large language models (LLM) on High-Performance Computing (HPC) systems, unveiling the key components, challenges, and optimizations involved in handling the computational demands of state-of-the-art language models. Delving into the nuances of model architecture, data preprocessing, and hyperparameter tuning, a comprehensive understanding of parallelization strategies, scalability challenges, and resource allocation will be given. Additionally, the talk touches on the implications for research, highlighting potential progress and future applications of LLMs.

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With exascale-capable supercomputers already on the horizon, reducing communication operations in orthogonalization kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods, first introduced in Swirydowicz et al. (Numer. Lin. Alg. Appl. 28(2):e2343, 2020), have been shown to improve the scalability of the Arnoldi method in high-performance, distributed computing. Block versions of low-synchronization Gram-Schmidt show further potential for speeding up algorithms, as column-batching allows for maximizing cache usage with matrix-matrix operations. We will examine how low-synchronization block Gram-Schmidt variants can be transformed into block Arnoldi variants for use in standard Krylov subspace methods like block generalized minimal residual methods (BGMRES). We also demonstrate how an adaptive restarting heuristic can handle instabilities that arise with the increasing condition number of the Krylov basis. The performance, accuracy, and stability of these methods are assessed via a flexible comparison tool written in MATLAB.

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Neural networks trained with machine learning techniques are currently attracting great attention as nonlinear approximation methods to solve forward and inverse problems involving high-dimensional partial differential equations (PDEs). In a recent paper, Neural Galerkin scheme has been proposed to solve PDEs by means of deep learning. In this approach, the deep learning process generates the training data samples with an active learning process for the numerical approximation. We apply this approach in this talk to tackle a parameter estimation problem and propose an algorithm based on Neural Galerkin scheme to estimate a scalar coefficient involved in a non-linear PDE problem. We provide numerical results with Korteweg-de Vries (KdV) equation in one dimension.

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We discuss approximation methods for linear equations in sequence spaces. When cutting out a finite matrix from an infinite dimensional operator, a choice of boundary conditions has to be made. Choosing zero boundary conditions leads to the classical finite section method, for which conditions for the applicability are known. We derive similar conditions for the applicability for the choice of periodic boundary conditions.
As an important tool, we demonstrate a way to approximate spectral quantities of an infinite dimensional operator with the help of finitely supported vectors.
Moreover, we investigate discrete Schrödinger operators and find conditions for the applicability of the finite section method.

This talk gives an overview of the results obtained in my PhD under the supervision of Prof. Dr. Marko Lindner.

Zoom link: https://tuhh.zoom.us/j/8757671580?pwd=ZjgyYURxYWxrQmJjaUVtTE5uTnBHUT09

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Almost everywhere convergence is an essential part of classical measure theory. However, when passing to the quantum setting of noncommutative -spaces, the absence of an explicit measure space makes it very difficult to give expression to notions like almost everywhere convergence. There is a rich literature devoted to different ways of circumventing this challenge, positing various notions of “measure theoretic” convergence in the noncommutative case. However, not many of these seem to be suited to dealing with Haagerup -spaces. In this talk we review several noncommutative notions of convergence before proposing versions of these notions which have been recast in terms of spectral projections. The harmony of exisiting notions with these revised notions is then investigated in the semifinite setting, at which point we also demonstrate the efficacy of the “new” approach by establishing a matching noncommutative monotone convergence theorem. On the basis of the theory achieved in the semifinite setting, we then show how this “reshaped” theory may be lifted to the setting of Haagerup -spaces. In closing we show that even here a monotone convergence theorem based on these notions is valid. This is joint work with L. Labuschagne and C. Steyn.

Evolution equations are a class of partial differential equations arising frequently in physical applications, such as heat or wave equations. To account for unknown material parameters or measurement inaccuracies, they can be considered with random coefficients or a noise term. However, analytical solutions are often out of reach and a numerical solution is required. Several questions arise regarding the influence of the random terms on the discretisation. In this talk, I will give an overview of convergence rates that can be obtained in the random setting.

First, evolution equations with random coefficients are investigated. Solving them numerically requires a discretisation in space, in time, and of random coefficients, which, individually, are well-known. We present conditions under which they can be combined to obtain a joint convergence rate for the full discretisation. In the second part, temporal discretisation of semi-linear stochastic evolution equations is investigated with a focus on hyperbolic problems. Optimal bounds for the pathwise uniform strong error are shown. This extends and improves previous results from exponential Euler to general contractive time discretisation schemes, such as implicit Euler, and from the group to the semigroup case.

This talk gives an overview of the results obtained in my PhD under the supervision of Dr. habil. Christian Seifert.

Time optimal controllability of abstract differential equations refers to reaching a desired target state within a minimal transition time. Further imposing a bound on control functions representing the energy available for control leads to the interesting Bang-Bang property, i.e. the time-optimal control function attains full norm on the transition time interval. Building upon investigations in Fattorini (SIAM J. Control Ser. A, 2(1): 54-59, 1964) and later Wang & Zhang (SIAM J. Control Optim., 55(3): 1862-1886, 2017), we generalize results on existence, Bang-Bang property and uniqueness of time optimal controls to reflexive Banach spaces. An example in heat diffusion will illuminate the relation of the Bang-Bang property with observability inequalities.

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Finite element methods (FEM) for displacement problems in elasticity lead to systems of nonlinear equations. These equations are usually solved with Newton's method or a related method. Based on a benchmark problem in high-order FEM, we explore traditional solution techniques for the nonlinear equation system such as step width selection and Quasi-Newton methods. We also consider algorithms specifically designed for displacement problems in nonlinear structural analysis like load step and arc-length methods. We extend traditional load step methods to a new approach exploiting the hierarchical structure of the problem and saving about 50% of computation time (vs. benchmark). In an outlook, we discuss new developments in nonlinear preconditioning and their applicability to displacement problems in nonlinear FEM.

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We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the k-th eigenvalue of the Dirichlet Laplacian there exist at least k+2 eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains. The proof relies on a novel variational principle.

Time-domain simulation of large-scale problems becomes computationally prohibitive if space-parallelization saturates. This is particularly challenging if long time periods are considered, e.g., if the start-up of an electrical machine until steady state is simulated. In this contribution, several parallel-in-time methods are discussed for initial-boundary-value problems and for time-periodic boundary value problems. All those methods are based on a subdivision of the time interval into as many subintervals as computing cores are available. For example, the well-known parareal method works similarly to multiple shooting methods; it solves two types of problems iteratively until convergence is reached: a cheap problem defined on coarse grids is solved sequentially on the whole time-interval to propagate initial conditions (and approximate derivatives) and secondly, high-fidelity problems are solved on the subintervals in parallel. We also discuss Paraexp and Waveform Relaxation methods in the context of real world engineering problems from electrical engineering.

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Physics Informed Neural Networks (PINNs) are becoming increasingly important in solving initial and boundary value problems. In contrast to conventional neural networks, they do not require labelled data for training and can thus be assigned to the field of unsupervised learning [3]. In this work, a PINN is to be trained to learn the equation of motion of a charged particle in an electromagnetic field. It turns out that networks trained using the L-BFGS opimisation algorithm show better convergence behaviour than those trained using the Adam optimisation algorithm commonly used in deep learning. In addition, it turns out that pre-training neural networks on the solution of a numerical method such as the Crank-Nicolson method can significantly speed up the training of PINNS.

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Parallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our reasearch proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal's single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.

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The availability of reliable, high-resolution climate and weather data is important to inform long-term decisions on climate adaptation and mitigation and to guide rapid responses to extreme events. Forecasting models are limited by computational costs and, therefore, often generate coarse-resolution predictions. Two common ways to decrease computational efforts with DL are downscaling, the increase of the resolution directly on the predicted climate variables, and emulation, the replacement of model parts to achieve faster runs initially. Here, we look at several downscaling tasks and an aerosol emulation problem. While deep learning shows promising results it may not obey simple physical constraints, such as mass conservation or mass positivity. We tackle this by investigating both soft and hard constraining methodologies in different setups, showing that incorporating hard constraints can be beneficial for both downscaling and emulation problems.

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The theory of evolutionary equations, afforded by Rainer Picard (Dresden) et al., provides a well-posedness theorem applicable to a vast amount of linear PDEs including heat, wave and Maxwell's equations as well as equations including fractional derivatives and integrals. In this talk, I will discuss this well-posedness theorem in the autonomous case. I will show how to impose initial and boundary conditions on such evolutionary equations, and I will present a possible evolutionary approach to control theory for PDEs.

Graphics Processing Units (GPUs), initially designed as accelerators for graphics applications, have revolutionized the computing landscape with their unparalleled computational prowess. Today, GPU-accelerated systems are present everywhere – for example, in our smartphones, cars, and supercomputers. GPU-accelerated systems are transforming the world in many ways, and several exciting possibilities, such as digital twins and precision medicine are on the horizon. While GPU-accelerated systems are desirable, their optimal utilization is crucial; otherwise, they can be very expensive in terms of power and energy consumption, which is not good as we aspire to reduce our carbon footprint. A single GPU can draw up to 700 watts, while GPU-powered supercomputers scale to the energy-hungry range of 1 to 10 megawatts.
In this presentation, I will talk about the performance, power, and energy efficiency of GPUs. I will present a GPU power simulator that we developed to estimate the power and energy efficiency of GPUs and show how we can use the simulator to investigate bottlenecks that cause low performance and low energy efficiency, highlighting the wide gap between the achieved energy efficiency of GPUs and the energy-efficiency aim of exascale computing.
Finally, I will briefly highlight two ongoing projects aimed at harnessing GPUs effectively within High-Performance Computing (HPC) clusters.
In the first project, we are developing techniques to predict the scalability of applications on HPC clusters. The project aims to automatically choose the best number of nodes for an application depending on its scalability. In the second project, we are developing a tool to enable automatic optimization of HPC applications on NVIDIA Hopper (and the next generation) GPUs. As we navigate the intricate interplay of performance, power, and energy efficiency, we embark on a quest to maximize the transformative potential of GPUs while minimizing their environmental footprint.

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Parareal is the most studied Parallel-in-Time method; by introducing parallelism in the time dimension, it allows to relieve communication bottlenecks that appear when parallelism is used only in the spatial dimension.
An expensive part of Parareal is the sequential solve using the coarse operator. So, for performance reasons, it can be interesting to consider the sequential operator not only on a coarser grid in time but also in space.
In this talk, we will discuss an alternative approach to the Generating Function Method (GFM) for computing Parareal bounds and how it can be used to compute linear and superlinear bounds.
We will then extend the analysis to Parareal with spatial-coarsening (coarsening factor 2 in space and time) and discuss the associated challenges. Finally, numerical results for the heat equation will be provided.

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We present some recent developments for the numerical simulation of
transport-dominated problems such as compressible fluid flows and
nonlinear dispersive wave equations. We begin with a brief review
of modern entropy-stable semidiscretizations of hyperbolic conservation
laws and use the method of lines to obtain efficient, fully discrete
numerical methods. Next, we introduce means to preserve the entropy
structures also under time discretization. Therefore, we present the
relaxation approach, a recent technique developed as small modifications
of standard time integration schemes such as Runge-Kutta or linear
multistep methods, which is designed to preserve the conservation or
dissipation of important functionals of the solution. This can be an
entropy in the case of compressible fluid flows, the energy of
Hamiltonian problems, or another nonlinear invariant.

We applied the Parallel-In-Time algorithm Parareal to the ocean-circulation model FESOM2 to demonstrate its applicability to complex problems in climate research. The talk is intended to give an overview of the technical challenges that can be expected when attempting to parallelize state-of-the-art simulation software in time. With the convergence results presented the talk concludes with a discussion of whether and how an efficient application of Parareal could be achieved.

We study the interplay between moment and tail inequalities in the concentration of measure phenomenon. A motivating example are so-called higher order concentration bounds, where functions are addressed which have unbounded first order derivatives (or differences) but whose derivatives of some higher order are bounded. A variety of different situations is considered like (classical) Euclidean spaces, discrete situations, functions of independent random variables and the Poisson space. A special emphasis is put on pointing out the parallels and common ground throughout all these cases.

Since we now have implemented an efficient solver for the Maxey-Riley equation, we can generate a lot of trajectory data. This data could be used to train a neural network, which can predict the trajectories given a starting postion. Because the dynamics are governed by an integro-differential equation, the future path of a trajectory depends on the whole past. This characteristic could be handled using recurrent neural networks.
I will show how the network performs on different velocity fields. There are cases where the model does a great job in the prediction, but there are still many problems to discuss and it will be interesting to hear some thoughts.
Finally I will show a network architecture that combines a Verlet integrator with a recurrent neural network.

Tomographic imaging is an essential tool in medical diagnostics, allowing diseases to be detected much earlier
than would be possible from external observations alone. The aim is to determine a function representing the inner
of the human body from external measurements only so that the procedure is non-invasive and not harmful.
Determining this function, or in practice an appropriately discretized form, involves solving an
inverse problem, which is often ill-posed and must be solved for noisy measurements. In this talk, an
overview of different image reconstruction challenges and ways to address them algorithmically is given.
We also sketch possibilities that arise and allow for multi-contrast image reconstruction from only single
measurements.

Fluid flow problems can be modelled by the Navier-Stokes, or Oseen equations. Their discretization results in saddle point problems. These systems of equations are typically very large and need to be solved iteratively. Standard (block-) preconditioning techniques for saddle point problems rely on an approximation of the Schur complement. Such an approximation can be obtained by a hierarchical matrix (H-Matrix) LU-decomposition for which the Schur complement is computed explicitly. The computational complexity of this computation depends, among other things, on the hierarchical block structure of the involved matrices. However, widely used techniques do not consider the connection between the discretization grids for the velocity field and the pressure, respectively. Thus, a problem dependent hierarchical block structure for the FEM discretization of the gradient operator is presented. The block structure of the corresponding saddle point matrix block is improved by considering the connection between the two involved grids.Numerical results will show that the improved block structure allows for a faster computation of the Schur complement, the bottleneck for the set-up of the H-Matrix LU-decomposition.

The first part of the presentation provides an introduction to quantum mechanics and quantum algorithms. In the second part, I will present an overview of the research at the new TUHH institute on the topic (see www.tuhh.de/quantum) and explain the mathematical aspects of it.

The numerical solution of saddle-point systems arising in computational fluid dynamics with iterative solvers
requires efficient preconditioners. We focus on preconditioners for the pressure Schur complement. Low-rank
corrections aim to enhance spectral properties of standard preconditioners to accelerate convergence. We discuss
a multiplicative low-rank update that exploits a (randomized) low-rank approximation to the error between the
identity and the preconditioned Schur complement. Relaxing the initial Schur complement preconditioner can
have a significant impact on the convergence behaviour of the iterative solver. We test the presented method
for the linearized incompressible Navier-Stokes equations. Numerical results illustrate the performance of the
relaxed update scheme.

In this talk, we present new explicit bounds on the Gaussian approximation of Poisson functionals, based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derived bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators — thereby extending the results from (Last, Peccati and Schulte, 2016). Our main application is a CLT for the Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also applied our techniques to derive quantitative CLTs for edge functionals of the Gilbert graph, of the k-Nearest Neighbour graph and of the Radial Spanning Tree, both in cases where qualitative CLTs are known and unknown.

This talk will be devoted to several topics in classical and modern control theory. Classical stabilization techniques for linear and nonlinear control systems as well as modern attempts to linearize nonlinear systems will constitute the core for this presentation. In particular, the first part will consist of a brief summary of my Master's Thesis on the foundations of mathematical control theory in finite dimension. During the second part, we will catch a glimpse into modern control theory involving the Koopman operator focussing on advances and difficulties. The third part will be on my current PhD topic "Time-Optimal Control of Linear Systems in Non-Reflexive Banach Spaces". The official introduction to my person will also not come too short.

This talk will start with a high-level overview on how machine learning can be used to improve weather and climate predictions. Afterwards, the talk will provide more detail on recent developments of machine learned weather forecast models and how they compare to conventional models and numerical methods.

We study the propagation of electromagnetic waves in heterogeneous structures. The governing equations for this problem are Maxwell's equations with highly oscillatory parameters. We use an analytic homogenization result, which yields an effective Maxwell system that involves additional dispersive effects.

The Finite Element Heterogeneous Multiscale Method (FE-HMM) is used to discretize in space, and we provide a semi-discrete error estimate. The rigorous error analysis in space is supplemented by a standard time discretization combined with a recursive approximation of the convolution that relies on the assumption that the convolution kernel is an exponential function. Eventually, we present numerical experiments both for the microscopic and the macroscopic scale.

We consider stochastic PDEs driven by an additive or multiplicative Gaussian noise of the form
$$
\begin{cases} \mathrm{d} u &=(A u + F(t,u))\,\mathrm{d} t + G(t,u) \,\mathrm{d} W~~~\text{ on } [0,T],\\ u(0) &= u_0 \in L^p(\Omega;X)
\end{cases}
$$
on a Hilbert space $X$. Here, $A$ is the generator of a contractive $C_0$-semigroup $(S(t))_{t\geq 0}$, $W$ is a cylindrical Brownian motion, $F$ and $G$ are globally Lipschitz and of linear growth, $p \in [2,\infty)$, and $u_0$ is the initial data.
Our aim is to obtain strong convergence rates for a temporal discretisation scheme of the form $U_0 = u_0$,
$$
U_j = R_k U_{j-1} + k R_k F(t_{j-1},U_{j-1})+ R_k G(t_{j-1},U_{j-1}) \Delta W^{j},~j=1,\ldots,N_k
$$
with time step $k>0$, Wiener increments $\Delta W^j$, and contractive time discretisation scheme $R:[0,\infty) \to \mathcal{L}(X)$ approximating $S$ to order $\alpha \in (0,\frac{1}{2}]$ on a subspace $Y\subseteq X$. Among others, this setting covers the splitting scheme, the implicit Euler, and the Crank-Nicholson method.

Assuming additional structure of $F$ and $G$ as well as $Y$, we obtain the following bound for the pathwise uniform strong error
$$
\left(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}} \|u(t_j) - U_j\|_X^p \right)^{1/p}
\le C(1+\|u_0\|_{L^p(\Omega;Y)}) \left(\log\left(\frac{T}{k}\right)\right)k^{\alpha}.
$$
In particular, this implies that the convergence rate of the uniform strong error is given by the order of the scheme up to a logarithmic correction factor. This factor can be avoided for the splitting scheme.

This is joint work with Mark Veraar and Jan van Neerven (TU Delft).

In my introductory talk I will introduce myself and present the results of my master thesis.
The objective of my master thesis was to numerically solve a model, simulating the transportation of a tracer bolus through blood flow in the liver. A good model is important, especially in the field of cancer research, because tumor perfusion and other vascular properties are important parameters of cancer’s response to therapy. Good perfusion imaging allows an accurate model of the tumor’s vascular state and perfusion. With this model, critical determinants in the tumor’s progression and its response to therapy can be derived.

For the implementation I used a weighted essentially non-oscillatory (WENO) solver and tested it for accuracy, especially for its ability to solve the advection equation with space dependent velocity. WENO schemes have gained a lot of influence in numerical solutions of hyperbolic problems. The main advantage of WENO schemes and the reason they are so heavily used is their capability to achieve arbitrarily high-order formal accuracy in smooth regions while still maintaining stable and, most of all, non-oscillatory and sharp discontinuity transitions. The essential idea behind the scheme lies in the stencil choosing procedure.

Lower bounds for variances are often needed to derive central limit theorems. In this talk, we establish a specific lower bound for the variance of a Poisson functional that uses the difference operator of Malliavin calculus.
Poisson functionals, i.e. random variables that depend on a Poisson process, are widely used in stochastic geometry. In this talk, we show how to apply our lower variance bound to statistics of spatial random graphs, the $L^p$ surface area of random polytopes and the total edge length of hyperbolic radial spanning trees. This talk is based on joint work with M. Schulte.

Spectral Deferred Correction (SDC) methods are iterative solvers for collocation discretization of ordinary differential equations, but each iterate can also be interpreted as particular Runge-Kutta (RK) scheme. In contrast to fixed RK schemes, viewing SDC as a fixed point iteration allows combining them with various kinds of deliberate perturbations resulting from mesh adaptivity or algebraic adaptivity in PDEs, lossy compression in parallel-in-time solvers, or inexact computations in scale-separated long time integrations, for improved performance. It also fosters a deeper understanding of SDC approximation error behavior, and the construction of more efficient preconditioners. In the talk, we will touch several of these aspects, and provide a - necessarily incomplete - overview of the astonishing flexibility of SDC methods.

The talk discusses the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows.
Here, gradient forces appearing in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their
$L^2$-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions
of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.

This is joint work with Christian Merdon (WIAS)

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior. Nonlinear approximation methods, such as neural networks, were shown to be very efficient approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. We investigate such approximation schemes for solving molecular Schrödinger equations and provide linear and nonlinear convergence analysis.

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Stochastic optimization techniques have become an essential tool for training of
neural networks. One prominent algorithm is stochastic gradient descent
(SGD). Under smoothness and convexity assumptions one can show
convergence of SGD to a minimizer. However, the analyses of variants of
SGD require different techniques. In this talk, we look at recent
advances in modelling SGD by a continuous-time process defined by a
stochastic differential equation to obtain a unified framework. In
particular, we motivate the connection between the discrete and
continuous process and investigate in what sense they convergence to one
another. Further, we present examples of how the continuous-time model
behaves in practice.

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Describing aspects of physical phenomena by forming abstract mathematical models is a common practice in scientific work: the mathematical formalism allows for permeation of the mathematical model as a means of creating insights and knowledge over the described real-world phenomenon.

In this talk, I will present how the topics of my dissertation contribute to the theory of popular mathematical models ranging from quantum physics to mathematical fluid mechanics.

In particular, you will find out

(I) how to classify periodic potentials of discrete Schrödinger operators with respect to the applicability of the finite section method,
(II) how to prove final-state observability for time-dependent diffusion problems, and
(III) how to improve the regularity of weak solutions to the Navier-Stokes equations on rough domains.

Link to slides:
https://math.fabian-gabel.de/talks/fabian_gabel_dissertation_pres.pdf

Link to video recording:
https://youtu.be/_2W-b-vXeZE

At the Large Hadron Collider, the interaction of subatomic particles with matter lead to severalmillions of collisions every second. For each collision, upto thousands of particles are producedfollowing stochastic processes. The accurate description of these particles require thousands ofvariables, which leads to large data sets with high dimensionality. The interaction of particleswith the detectors (like Compact Muon Solenoid) are best simulated with the GEANT4 software.Alternatively, less precise but faster simulations are sometimes preferred to reach higher statisticalprecision. We present recent progresses of refinement of fast simulations with Machine Learningtechniques to enhance the quality of such fast simulations. We demonstrate the use of adversarialnetworks in the context of jet simulation using the Wasserstein distance metric. The architectureconsists of opposing networks, Refiner and Critic. A Refiner refines the distribution of the energyof the jets obtained with the fast simulation. The Critic is used to effectively differentiate betweenthe distributions of refined energy and the distribution obtained by the GEANT4 simulation. Weapply the technique to jet kinematics, when the response is close to Gaussian, first on toy data setsand then on realistic data sets

We discuss spectral inequalities and observability for the harmonic oscillator and more general Schrödinger operators with confinement potentials on the whole space. It turns out that the (super-)exponential decay of the corresponding eigenfunctions allows to consider sensor sets with a density that exhibits a certain decay. This, in particular, permits sensors with finite measure.

Nuclear fusion reactor design crucially depends on numerical simulation. The plasma can usually be modeled using fluid equations (for mass, momentum and energy). However, the reactor also contains neutral (non-charged) particles (which are important in its operation), of which both the position and velocity distribution is important. This leads to a Boltzmann-type transport equation that needs to be discretised with a Monte Carlo method. In high-collisional regimes, the Monte Carlo simulation describing the evolution of neutral particles becomes prohibitively expensive, because each individual collision needs to be tracked.
In this presentation, we overview a number of approaches that can alleviate the computational burden associated with the high-collisional regime. One option is to avoid simulating each invididual collision. In the limit of infinite collision rate, the law of large numbers dictates the approach of an advection-diffusion like particle behaviour, in which the accumulated effect of an infinite amount of collisions is aggregated in a Brownian motion (diffusion). To maintain accuracy and remove exploding simulation costs in high-collisional regimes, one can define hybridized particles that exhibit both kinetic behaviour and diffusive behaviour depending on the local collisionality [3].
Additionally, we can reduce the number of Monte Carlo particles that needs to be simulated via the multilevel Monte Carlo method[5]. Finally, one can also reduce the variance of the simulation by using an approximate fluid model for the neutral particles, discretized with a finite volume methods. This deterministic simulation can be used as a control variate, allowing the Monte Carlo simulation to focus on solely the deviation of the kinetic model with respect to the approximate fluid model.
References
[1] KukushkinA.S.,PacherH.D.,KotovV.,PacherG.W.,andReiterD.(2011)FinalizingtheITERdivertordesign:thekeyroleofSOLPSmodeling Fusion Eng. Des. 86:2865-2873.
[2] ReiterD.,BaelmansM.,andBörner,P.(2005)TheEIRENEandB2-EIRENEcodes,FusionSci.Technol.47:172-186.
[3] MortierB.,SamaeyG.,BaelmansM.(2019)Kinetic-diffusionasymptotic-preservingMonteCarloalgorithmsforplasmaedgeneutralsimulation.
Contributions to Plasma Physics, in press.
[4] Horsten N., Samaey G., Baelmans M. (2019) Hybrid fluid-kinetic model for neutral particles in the plasma edge. Nuclear Materials and Energy
18:201-207.
[5] Løvbak E., Samaey G., Vandewalle S. (2019) A multilevel Monte Carlo method for asymptotic-preserving particle schemes. Submitted. https://arxiv.org/abs/1907.04610.

Zoomlink: https://tuhh.zoom.us/j/84729171896?pwd=ODArbForaUxMM3Q3VTJsNG1kaVNYQT09

Aufgrund des zunehmenden Wachstums im E-Commerce-Sektor haben robotisierte Lagerhaltungs-
systeme – Robotic mobile fulfillment systems (RMFS) – für die Auftragsabwicklung in letzter Zeit
mehr Aufmerksamkeit erhalten. Dabei handelt es sich um eine neue Art von Lagerhaltungssyste-
men, bei denen nicht mehr Kommissionierer:innen in den Lagerbereich geschickt werden, um die
bestellten Artikel zu suchen und zu kommissionieren, sondern Roboter die Regale mit den bestell-
ten Artikeln aus dem Lagerbereich zu den Kommissionierstationen, auch Packstationen genannt,
tragen. An jeder Packstation steht eine Person – der oder die Kommissionierer:in (Packer:in) – die
die Artikel aus den Regalen nimmt und sie entsprechend der Kundenbestellung in Kartons verpackt.
Ein solches RMFS wirft viele Entscheidungsprobleme auf. Wir konzentrieren uns auf Entscheidun-
gen über die optimale Anzahl von Robotern. Wir modellieren das RMFS als ein Warteschlangen-
netzwerk, analysieren seine Stabilität und bestimmen die minimale Anzahl von Robotern für ein
stabiles System.
Dieser Vortrag basiert auf der gemeinsamen Arbeit [1] mit Sonja Otten, Ruslan Krenzler, Lin Xie
und Hans Daduna.

LITERATUR
[1] Otten, S., Krenzler, R., Xie, L., Daduna, H., und Kruse, K. Analysis of semi-open queueing
networks using lost customers approximation with an application to robotic mobile fulfilment
systems, OR Spectrum, 1–46, 2021. DOI: 10.1007/s00291-021-00662-9.

We discuss recent developments around approaches to Crouzeix's conjecture, the statement that the numerical range is a 2 spectral set for any complex square matrix. This includes spectral constants for specific classes of operators.

Zoomlink:
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The Maxey-Riley Equation (MRE) models the motion of a finite-sized, spherical particle moving in a fluid. Applications using the MRE are, for example, the study of the spread of Coronavirus particles in a room, the formation of clouds and the so-called marine snow. The MRE is a second-order, implicit integro-differential equation with a singular kernel at initial time. For over 35 years, researchers used approximations and numerical schemes with high storage requirements or ignored the integral term, although its impact can be relevant. A major break-through was reached in 2019, when Prasath et al. mapped the MRE to a time-dependent Robin-type boundary condition of the 1D Heat equation, thus removing the requirement to store the full history. They provided an implicit integral form of the solution by using the so-called Fokas method that could be later solved with numerical scheme and a nonlinear solver. While Prasath et al.’s method can deliver numerical solutions of very high accuracy, the need to evaluate nested integrals makes it computationally costly and it becomes impractical for computing trajectories of a large number of particles. In the talk, we will present a finite differences approach that it is not only storage efficient but also much faster. We will compare our approach to both Prasath et al.’s method as well as a to the numerical schemes developed by A. Daitche in 2013 for direct integration of the original Maxey-Riley equation with integral term.

Spectral deferred corrections (SDC) is an iterative method for the numerical solution of ordinary differential equations. It can be interpreted as a Picard iteration for the collocation problem, preconditioned with a low order method. SDC has been studied for first order problems, using explicit, implicit or implicit-explicit Euler as preconditioner. It has been shown that SDC can achieve arbitrary high order of accuracy and possesses good stability properties.

In this talk, we will present an analysis of the convergence and stability properties of the SDC method when applied to second-order ODEs and using velocity-Verlet as preconditioner. While a variant of this method called Boris-SDC for the Lorentz equation has been investigated, no general analysis of its properties for general second order problems exists.

We will show that the order of convergence depends on whether the force on the right hand side of the system depends on velocity (like in the Lorentz equation) or not (like in the undamped harmonic oscillator). Moreover, we also show that the SDC iteration is stable under certain conditions. We compare its stability domain with that of the Picard iteration and validate our theoretical analysis in numerical examples.

In this talk, I would like to present the main results of my PhD thesis.
We study a general method to obtain observability estimates for control systems in infinite dimensional spaces by combining an uncertainty principle and a dissipation estimate. Contrary to previous results obtained in the context of Hilbert spaces, we obtain conditions for observability in Banach spaces, allow for more general asymptotic behavior in the assumptions, and retain explicit estimates on the observability constant.
Our approach has applications, e.g., to control systems on non-reflexive spaces and anomalous diffusion operators.
Further, we derive duality results that connect observability estimates to controllability and stabilizability properties. As an application, we study controllability properties of systems given by fractional powers of elliptic differential operators with constant coefficients in $L_p(\mathbb{R}^d)$ for $p\in [1,\infty)$ and thick control sets.

Zoomlink:
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Advancement in computational speed is nowadays gained by using more processing units rather than faster ones.
Faults in the processing units caused by numerous sources including radiation and aging have been neglected in the past.
However, the increasing size of HPC machines makes them more susceptible and it is important to develop a resilience strategy to avoid losing millions of CPU hours.
Parallel-in-time methods target the very largest of computers and are hence required to come with algorithm-based fault tolerance.
We look here at spectral deferred corrections (SDC), which is a time marching scheme that is at the heart of parallel-in-time methods such as PFASST.
Due to its iterative nature, there is ample opportunity to plug in computationally inexpensive fault tolerance schemes, many of which are also easy to implement.
We experimentally examine the capability of various strategies to recover from single bit flips in time serial SDC, which will later be applied to parallel-in-time methods.

Zoomlink:
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Robots have the potential to disrupt many aspects of our lives, from healthcare to manufacturing. To realize this potential, a key challenge is real-time robot manipulation. Given a task, how can a robot quickly generate a motion plan to successfully complete it? How can the robot react in real-time to potential uncertainties in the real-world as it executes its plan? In this talk, we will overview recent developments at the University of Leeds, to realize real-time robot manipulation. These will include parallel-in-time integration methods that leverage parallel computing to significantly speed-up physics predictions for various robot manipulation tasks. It will also include learning-based and optimal control-based methods for robots to handle real-world uncertainties in object pose estimation and model parameters. We hope these recent advances will help accelerate the next generation of intelligent robots.

Zoomlink: https://tuhh.zoom.us/j/85353626407?pwd=MEIzeTEvY3dRTmtYZjFWUHJaVll4UT09

Meeting ID: 853 5362 6407
Passcode: 045209

The Norros-Reittu model is an inhomogeneous random multigraph that exhibits the so-called scale-free or power-law behaviour, which is observed in real-world complex networks. We study the component sizes of the Norros-Reittu model in the subcritical regime, i.e. in the abscence of a giant component, and show convergence of the point process of the component sizes to a Poisson process. It is planned to derive similar results for other models such as the random connection model.

The thesis deals with boundary layer enrichment of convection dominated flow problems using the Hybrid Discontinuous Galerkin Method. It aims to introduce an appropriate and computationally efficient Hybrid Discontinuous Galerkin formulation for the most important model problems of incompressible fluid flow, namely the convection-diffusion equation.The main contribution is the derivation, discussion and analysis of the Enriched Finite elementSpace using non-polynomial spaces (specifically boundary layer functions) for both the Discontinuous Galerkin Methods and the Hybrid Discontinuous Galerkin Method. We evaluate the robustness (i.e linear stability as well as reasonable linear systems) and accuracy of this method using various analytical and realistic problems and compare the results to those obtained using the standard (H)DG method. Numerical results are provided to contrast the Enriched (H)DG methods with standard (H)DG approaches.

In the early 1980s, Erd\H{o}s and S\'os initiated the study of the classical Tur\'an problem with a uniformity condition: the uniform Tur\'an density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Tur\'an densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349--366] and [J. Eur. Math. Soc. 20 (2018), 1139--1159], while the latter still remains open for almost 40 years.
In addition to $K_4^{(3)-}$, the only $3$-uniform hypergraphs whose uniform Tur\'an density is known are those with zero uniform Tur\'an density classified by Reiher, R\"odl and Schacht~[J. London Math. Soc. 97 (2018), 77--97] and a specific family with uniform Tur\'an density equal to $1/27$.

In this talk, we give an introduction to the concept of uniform Tur\'an densities, present a way to obtain lower bounds using color schemes, and give a glimpse of the proof for determining the uniform Turán density of the tight $3$-uniform cycle $C_\ell^{(3)}$, $\ell\ge 5$.

We study the observability and null control problem for
the semigroup generated by the harmonic oscillator
and the partial harmonic oscillator.
We identify sensor sets which ensure null controlabillity
improving and unifying previous results for such problems.
In particular, it is possible to observe the Hermite semigroup
from finite volume sensor sets.
This is joint work with A.Dicke and A. Seelmann.

Since I am new to our institute, I will start by introducing myself and presenting the results of my master thesis on second order information in
neural network training.

Traditionally, neural networks are trained using gradient-based optimization methods like Adagrad or Adam. Using second order methods might result in faster convergence (e.g. locally quadratic convergence in Newton's method). Furthermore, curvature information can provide some insight into the optimization process and help to characterize the cost function of a neural network.

For large problems, applying Newton's method and Quasi-Newton-methods to the cost function of a neural net is only possible through implicit Hessian-vector-products. For this reason, Krvlov subspace methods are particularly well suited for solving the linear system with the Hessian that appears in Newton's method. Krylov subspace methods use matrix-vector-products instead of operating on the full matrix.

Two data sets are used: The first one is constructed to allow the exact calculation (except for rounding errors) of the Hessian and its eigenvalues. Here, we observe that the largest eigenvalue can be approximated with a small number of steps of a Krylov subspace method and with high accuracy. The second data set is the famous MNIST data set for handwritten digit classification. For MNIST and the given computational resources, we cannot calculate the full Hessian of the cost function. Instead, the Krylov subspace method is used to approximate eigenvalues from implicitly calculated Hessian-vector-products. On both data sets, the largest eigenvalue can be observed to be coupled to the value of the cost function.

An inexact Quasi-Newton-method and the L-BFGS method are used to train a neural network on both data sets.

Furthermore, I will talk about first ideas for my dissertation on nonlinear finite element methods with applications in ship structural design.

We are interested in constructing cheap and efficient implicit high order
solvers for compressible turbulent
flow problems. These problems arise for
example in the design of next generation jet engines, air frames, wind tur-
bines or star formation. A suitable high order discretization for these prob-
lems are discontinuous Galerkin spectral element methods (DG-SEM). In
this talk we discuss challenges of solvers for DG-SEM discretizations in space
combined with implicit time-stepping methods.
One option to yield implicit DG-SEM solvers is to apply a space-time
DG-SEM discretization, i.e. discretizing space and time simultaneously with
DG-SEM. We present two approaches for the formulation and implementa-
tion of space-time DG-SEM: Either time is treated as an additional coor-
dinate direction and the Galerkin procedure is applied to the entire prob-
lem. Alternatively, the method of lines is used with DG-SEM in space and
the fully implicit Runge-Kutta method Lobatto IIIC in time. The two ap-
proaches are mathematically equivalent in the sense that they lead to the
same discrete solution. However, in practice they differ in several important
respects, including the terminology used to the describe them, the struc-
ture of the resulting software, and the interaction with nonlinear solvers.
We present challenges and merits of the two approaches and show their im-
pact on numerical tests using implementations based on the Distributed and
Unified Numerics Environment (DUNE).
Another option to construct implicit DG-SEM solvers is the classical
method of lines approach. The spatial directions are discretized with DG-
SEM and any implicit time-stepping method can be applied to the resulting
ODE. This yields large nonlinear systems and a solver has to be chosen
carefully. We suggest to use a preconditioned Jacobian-free Newton-Krylov
method. The challenge here is to construct a preconditioner without con-
structing the Jacobian of the spatial discretization. Our idea is to make use
of a simplified replacement operator for the DG operator and a multigrid
method. We discuss the idea of our suggested preconditioner and present
numerical results to show the potential of this preconditioning technique.

Talk (PDF, 73KB)

https://tuhh.zoom.us/j/82516486683?pwd=RnV4ZEcvREhXeDYyZXdiUE1kUmh1QT09

Meeting-ID: 825 1648 6683
Kenncode: 329040

It is a conjecture going back to J. Rauch (1974) that the hottest and coldest spots in an insulated homogeneous medium such as an insulated plate of metal should converge to the boundary, for "most" initial heat distributions, as time tends to infinity. This so-called hot spots conjecture can be phrased alternatively as follows: the eigenfunction(s) corresponding to the first non-zero eigenvalue of the Neumann Laplacian on a Euclidean domain should take its maximum and minimum on the boundary only. This has been proven to be false for certain domains with holes, but it was shown to hold for several classes of simply connected or convex planar domains. One of the most recent advances is the proof for all triangles given by Judge and Mondal (Annals of Math. 2020). The conjecture remains open in general for simply connected or at least convex domains. In this talk we provide a new approach to the conjecture. It is based on a non-standard variational principle for the eigenvalues of the Neumann and Dirichlet Laplacians.

Jonathan Rohleder is an associate professor at Stockholm University, Sweden. His work focusses on spectral theory.

Specific solutions of the nonlinear Schrödinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrödinger equation is obtained for the case of wind-forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrödinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather-type solutions such as the Peregrine breather occur even in strong gusty wind conditions.

Atmospheric dynamics can be described by the Boussinesq approximation which models bouyancy-driven fluid flows. Its simulation involves the repeated solution of the Navier-Stokes equations. This requires numerical solution methods for the dense Schur complement. In this talk, we will be concerned with Schur complement preconditioners. Furthermore, we will discuss a low-rank update for the Schur complement preconditioners. The update method is based on the error between the preconditioned Schur complement and the identity. It will be illustrated with some numerical results.

Hierarchical Matrices are dense but data-sparse matrices that use low-rank factorisations of suitable submatrices to allow for storage with linear-polylogarithmic complexity. Furthermore, efficient approximations of matrix operations like matrix-vector and matrix-matrix multiplication, matrix inversion and LU decomposition are available. There are several approaches for the computation of QR factorisations in the hierarchical matrix format, however, they suffer from numerical drawbacks that limit their use in many applications. In this talk, I will present a new approach based on block Householder transformations that improves upon some of those problems. To prevent unnecessary high ranks in the resulting factors and increase speed as well as accuracy the algorithm meticulously tracks for which intermediate results low-rank factorisations are available.

I will try to keep things as simple as possible and give a short introduction to hierarchical matrices as well. Previous knowledge of them is not necessary to understand the basic ideas and main obstacles of the new algorithm. I will focus on aspects, that I haven't talked about yet in similar talks in the past, mainly on how a cost estimate is possible although the hierarchical structure of the resulting QR decomposition is step-wise created during the algorithm and not defined beforehand.

We present a novel local pressure correction method for incompressible fluid flows. Pressure correction methods
decouple the velocity and pressure components of the time-dependent Navier-Stokes equations and lead to a sequence of elliptic partial differential equations for both components instead of a saddle point problem. In some situations, the equations
for the velocity components are solved explicitly (with time step restrictions) and thus the elliptic pressure problem remains to be the most expensive step. Here, we employ a multiscale procedure for the solution of the Poisson problem related to pressure. The procedure replaces the global Poisson problem by local Poisson problems on subregions.We propose a new Robin-type boundary condition design for the
local Poisson problems, which contains a coarse approximation of the global Poisson problem. Accordingly, no further communication between subregions is necessary and the method is perfectly adapted for parallel computations. Numerical experiments regarding a known analytical solution and flow around cylinder benchmarks show the effectivity of this new local pressure correction method.

An increased use of renewable energy could significantly contribute to decelerate climate change but cannot be realized easily since most renewable energy sources underlie volatile availability. Using of storage devices and scheduling consumers to times when energy is available can increase the amount of renewable energy that is used. For this purpose, adequate models that forecast the energy generation and consumption as well as the behavior of storage devices are essential. We present a computationally efficient modeling approach based on a least-squares problem that is extended by a hybrid model approach based on kmeans clustering and evaluate it on real-world data at the examples of modeling the state of charge of a battery storage and the temperature inside a milk cooling tank. The experiments indicate that the hybrid approach leads to better forecasting results, especially if the devices show a more complicated behavior. Furthermore, we investigate whether the behavior of the models is qualitatively realistic and find that the battery model fulfills this requirement and is thus suitable for the application in a smart energy management system. Even though forecasts for the hybrid milk cooling model have low error values, further steps need to be taken to avoid undesired effects when using this model in such a sophisticated system.

I will introduce myself and present the topic of my master thesis.

Computed Tomography (CT) is a standard procedure in clinical imaging. In dynamic CT, several CT scans are made to make a process inside the patient visible. Therefore, the X-ray exposure to the patient is relatively high during such a survey. Thus, it is desirable to lower the X-ray exposure to the patient.

In my thesis I investigated an approach which requires only sparse angular sampling for every scan. In order to be able to reconstruct the image anyway, I used a shearlet system combined with an $\ell^1$-regularization. I compared different shearlet systems and checked for different parameters the impact on the results. I used both simulated data as well as real CT data for the tests.

Back in 2009, both Stein's method - a probabilistic technique to derive quantitative limit theorems - and Malliavin calculus - a stochastic version of the calculus of variations - had already established themselves as standard tools in their respective domain, even though both were discovered quite recently in 1972 and 1978, respectively. Then they started an innocent liaison in Paris which quickly developed into a very strong bond (despite numerous affairs), leading to fame and success both in- and outside the probabilistic community. This bond is today known as the Malliavin-Stein approach.

I will highlight some exciting parts of this story, also attributing a fair share of time to yet unwritten chapters (i.e. open problems). Mathematically, this will feature non-linear approximation, limit theorems (central and non-central), stochastic processes, chaos, Markov generators, non-commutative probability theory and the first four moments. Catering to the fact that probabilists are in the minority in our department, things will also be presented from a functional analytic point of view.

The talk will mostly be informal and understandable by non-specialists.

Coupling means the joint construction of two or more random variables, processes, or any random objects. The aim of the construction could be to deduce properties of the individual objects, or to gain insight into distributional relations between them, or to simulate a particular object. It has been called The Probabilistic Method since it is not based on methods from other fields of mathematics.

In this talk we shall consider some basic examples such as the Poisson approximation, stochastic domination, Markov chains and Brownian motion, and perfect simulation

Solving evolution equations with random coefficients numerically requires discretizing in space, time and of random parameters. As numerical methods for all three discretisations are well-known, it is natural to ask under which conditions they can be combined. In this talk, we discuss this question with a special emphasis on preservation of strong convergence rates.

A common approach to spatial discretization consists of solving the weak formulation on finite-dimensional approximating spaces. We present a novel quantified version of the Trotter-Kato theorem in this setting, yielding rates of strong convergence under a joint condition on properties of the corresponding form and the approximating spaces.

This is joint work with Christian Seifert.

The trend towards massively parallel high-performance computers requires the development of parallel algorithms to employ their computational power.
The Parareal algorithm computes the solution of time-dependent problems parallel in time, meaning that approximations to the solution at different times are computed simultaneously. In this talk, we will focus on hyperbolic one-dimensional problems, where a combination of Parareal and a discontinuous Galerkin method will be used. The practical use and challenges of this method will be illustrated by means of a Python implementation for shallow water equations and corresponding numerical results.

How will the next generation of Artificial Intelligence (AI) look like? Comparing today's AI algorithms with biological intelligence, one of the most remarkable differences is the ability of the human brain to somehow understand the 'essence' of things: A small child can easily identify any type of object after having seen only a few examples or recognize a song even when played on different instruments or in a different key. In other words: Brains are able to create abstract concepts of real-world entities - and today's algorithms are not.

With today's AI largely being based on neuron models already invented by the mid of last century, I will argue that we should take a new look at the brain to find inspiration for the next generation of machine learning algorithms. Even though there is still only a very limited understanding of how the brain works computationally, I'll explain why there is hope that we can reverse-engineer some of its algorithmic principles and implement them in a computer. I'll explain why a highly interdisciplinary approach is required from neuroscience, computer science, mathematics and physics to make progress in this question.

The talk will be held on Zoom:
https://tuhh.zoom.us/j/86836210324?pwd=ajJURGY2T3pFNWMvUzVQTkduSTNCQT09
Meeting-ID: 868 3621 0324
Kenncode: 521014

For many processes in sciences, the coefficients of the partial differential equation describing a dynamical system as well as the boundary conditions of it may vary with time. In such cases one speaks of non-autonomous (or time-varying) evolution equations. From an operator theoretical point of view one considers families of Banach space operators which depend on the time parameter and studies the associated non-autonomous abstract Cauchy problem. We consider time-dependent Desch-Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on Lp -spaces. This is joint work with Christian Seifert (TUHH).

I will introduce myself and present the topic of my master thesis.

A fundamental principle in developing the Theory of Quantum Mechanics is to take
well-studied concepts from the Theory of Classical Mechanics and to define
analogues in the quantum mechanical setting.
One such important tool in Classical Mechanics is the theory of optimal
transport and in particular the Wasserstein distance.

In my thesis I studied the mathematical objects needed to translate
the concepts of the optimal transport problem to the realm of Quantum
Mechanics. In particular,
one wants to establish a relation between density matrices (trace-class operators
of trace one) and
probability measures. This can be done by the so-called
(generalized) Toeplitz operators and the (generalized) Husimi
transform.

After I give a brief introduction into both the Optimal Transport and Quantum
Mechanics I will introduce both
the Toeplitz operators and the Husimi transform and discuss some of their
properties.

Suppose that $A$ is a bounded linear operator on the Hardy space $H^p$ that satisfies
\[\langle Az^j,z^k \rangle = a_{k-j} \quad (j,k \in \mathbb{N}_0)\]
for some sequence of complex numbers $\{a_n\}_{n \in \mathbb{Z}}$. By the Brown--Halmos theorem, $A$ must be a Toeplitz operator with bounded symbol, that is, $\{a_n\}_{n \in \mathbb{Z}}$ is the Fourier sequence of a bounded function. Likewise, Nehari's theorem shows that if $A$ satisfies $\langle Az^j,z^k \rangle = a_{k+j+1}$ instead, then $A$ is equal to a Hankel operator with bounded symbol. These results were proven in the 50's and 60's and have become classical in the theory of Hardy spaces.

More recently, due to some applications in mathematical physics, there has been a lot of interest in so-called Toeplitz+Hankel operators. Quite simply put, a Toeplitz+Hankel operator is the sum of a Toeplitz operator $T(a)$ and a Hankel operator $H(b)$. Now clearly, if both $T(a)$ and $H(b)$ are bounded, then $A = T(a)+H(b)$ is necessarily bounded as well. It is therefore natural to ask whether the converse is also true or if the ``unboundedness'' of $T(a)$ and $H(b)$ can somehow cancel out. I will elaborate on this question and present a Brown--Halmos type result for Toeplitz+Hankel operators for both the Hardy spaces $H^p$ and the sequence spaces $\ell^p(\mathbb{N}_0)$. A similar characterization for compactness will be obtained as well.

Based on joint work with Torsten Ehrhardt and Jani Virtanen.

Hallo liebe Institutsmitarbeiter*innen,

anbei der offizielle Einladungstext zum Promotionsvortrag von Alexander Haupt:

_________________________________________________________________________

Sehr geehrte Damen und Herren,

im Rahmen seines Promotionsverfahrens wird

Herr M. Sc. Alexander Michael Haupt

einen kombinierten Live-Online-Vortrag mit dem Titel

„New Combinatorial Proofs for Enumeration Problems and Random Anchored Structures“

halten. Der Vortrag findet statt am

Dienstag, dem 21. September 2021 um 11:00 Uhr.

Zu diesem universitätsöffentlichen Vortrag lade ich Sie herzlich ein.

Aufgrund der aktuell geltenden Regelungen können Interessierte nur per Zoom am Vortrag teilnehmen. Bitte benutzen Sie hierzu die folgenden Zugangsdaten:

https://tuhh.zoom.us/j/88353220627?pwd=amZFYjRNOHl2TkdGb2c1Z29MVGNCUT09

Meeting-ID: 883 5322 0627
Kenncode: 747604

Mit freundlichen Grüßen
Prof. Dr. Matthias Schulte

(Vorsitzender des Prüfungsausschusses)

In this talk, we discuss non-local operators like elliptic integrodifferential operators of fractional type
\[
Au := p.v. \int_{\mathbb{R}^d} \frac{u(x) - u(y)}{|x-y|^{d+2\alpha}}dy \quad \quad (1)
\]
or the Stokes operator with bounded measurable coefficients $\mu$, formally given by
\[
Au := -div( \mu \nabla u ) + \nabla \phi, \quad div(u) = 0 \; in \; \mathbb{R}^d. \quad \quad (2)
\]
These operators satisfy $L^{2}$-resolvent estimates of the form
\[
|| \lambda ( \lambda + A )^{-1} f ||_{L^2} \leq C || f ||_{L^2} \quad (f \in L^2(\mathbb{R^d}))
\]
for $\lambda$ in some complex sector $\left\{z \in \mathbb{C} \smallsetminus {0} : | arg(z) | < \theta \right\}$. We describe how analogues of such a resolvent estimate can be established in $L^{p}$ by virtue of certain non-local Caccioppoli inequalities. Such estimates build the foundation for many important functional analytic properties of these operators like maximal $L^{q}$-regularity.

More precisely, we establish resolvent estimates in $L^{p}$ for $p$ satisfying
\[
\left|\frac{1}{p} - \frac{1}{2} \right| < \frac{\alpha}{d}
\]
in the case (1) and
\[
\left|\frac{1}{p} - \frac{1}{2}\right| < \frac{1}{d} \quad \quad (3)
\]
in the case (2). This resembles a well-known situation for elliptic systems in divergence form with $L^{\infty}$-coefficients. Here, important estimates like Gaussian upper bounds for the semigroup cease to exist and the $L^{p}$-extrapolation has be concluded by other means. In particular, for elliptic systems one can establish resolvent bounds for numbers p that satisfy (3) and if $d \geq 3$, Davies constructed examples which show that corresponding resolvent bounds do not hold for numbers $1 < p < \infty$ that satisfy
\[
\left|\frac{1}{p} - \frac{1}{2} \right| > \frac{1}{d}.
\]
These elliptic results give an indication that the result for the Stokes operator with $L^{\infty}$-coefficients is optimal as well.

We study integral input-to-state stability of bilinear systems with
unbounded control operators and derive natural sufficient conditions. The
results are applied to a bilinearly controlled Fokker-Planck equation.

Gauss quadrature rules are nowadays not only a powerful tool to compute integrals in many scientific applications, but also a numerical method that most people in the scientific community at least heard of at some point in there life.
Even if they are not the only tool to compute integral numerically, they provide the possibility to integrate any function multiplied by a given weight function (or measure), by estimating the integral of the product using a weighted sum of the function evaluations at given values (nodes).
Classical measures are well known (e.g Legendre, Chebyshev, Laguerre, Hermite), and their associated quadrature rules are well studied and documented in the literature.
While some measures allow to estimate integrals over infinite intervals (e.g Laguerre or Hermite), others also allow to integrate a function with singularities (e.g Chebyshev of the first, third and fourth kind).
However, the use of non-classical measures for specific applications can also be considered, and even this is not often used in the community, many algorithms exist to compute the nodes and weights of those quadrature rules.
In this talk we will give a quick overview of those algorithms, their efficiency, numerical stability, and some current challenge that still need to be solved.
Furthermore, under some conditions, all Gauss quadrature rules share some common properties, in particular when considering a large number of nodes.
We will give a quick overview of those common asymptotic properties, and show how they can be generalized to other applications (e.g barycentric Lagrange interpolation).
While some of those properties have been proven in particular cases, we will present some situations where they have not been proved theoretically yet, or still need to be verified.

Atmospheric motion covers a broad range of time- and spatial scales. Low and high pressure systems can influence us for days or even weeks and they extend up to hundreds of kilometers. In contrast, sound waves pass by in seconds with wavelengths of centimeters to meters. Implicit-explicit (IMEX) time stepping methods can help to avoid drastic limitations on the time step induced by the variety of scales without requiring computationally expensive fully nonlinear implicit solves. I will introduce Spectral Deferred Correction (SDC) methods as a strong competitor to currently used schemes. They allow an easy construction of high order schemes in contrast to e.g IMEX Runge-Kutta methods which require a growing number of coupling conditions with increasing order.

Discrete Schrödinger operators are used to describe systems in theoretical solid-state physics.
In this talk we consider discrete Schödinger operators with both periodic and aperiodic potentials. We analyse spectral properties of these operators and find conditions for the applicability of the so-called "finite section method" that allows us to approximate solutions of systems involving discrete Schrödinger operators.

Solving evolution equations numerically requires discretizing both in time and in space. However, these two problems can be treated seperately. A common approach to spatial discretization relies on solving the weak formulation on finite-dimensional subspaces. On a semigroup level, this corresponds to approximating a semigroup by semigroups on finite-dimensional subspaces. For practical applications, quantifying the convergence speed is essential. This can be achieved by the quantified version of the Trotter-Kato theorem presented in this talk. Rates of strong convergence are obtained on dense subspaces under a joint condition on properties of both the form and the approximating spaces. An outlook to evolution equations with random coefficients and their polynomial chaos approximation will be given as well as a generalization allowing to treat the Dirichlet-to-Neumann operator.

Zoom Link folgt.

In many problems, like the discretized Stokes or Navier-Stokes equation, linear systems of saddle point type arise. Since the condition number for such problems can grow unbounded, as the number of unknowns grows, good preconditioners are key for solving such problems fast.

In this talk I will introduce some general preconditioning techniques for saddle point problems, and how to apply them to the discretized Stokes and Navier-Stokes equation

In this talk, we examine linearized kinetic BGK equation in 1D velocity dimension. It is closely related to the Maxwell-Boltzmann equation for gas dynamics. The equation that we are interested is obtained by linearization of the equation around Maxwellian. We discuss the kinetic and macroscopic equations and the boundary and coupling conditions for this equation.

Furthermore, we will drive coupling conditions for macroscopic equations on different network and compare the solutions with Maxwell and half-moment approximations. Moreover, the macroscopic equations on the network with the different Knudsen numbers are numerically compared with each other.

Positive definite kernel functions are powerful tools, which can be used to solve a variety of mathematical problems. One possible application of kernel-based methods is the reconstruction of images from scattered Radon data, which is described in [1]. More precisely, the authors introduced weighted kernel functions to solve the reconstruction problem via generalized interpolation. Although the reconstruction method was quite competitive in comparison to standard Fourier-based methods, a detailed discussion on well-posedness and stability was mainly missing.

In this talk, I will explain the basics of kernel-based generalized interpolation and discuss the well-posedness of the proposed reconstruction method. Like most kernel-based methods, the reconstruction method also suffers from bad condition numbers. I will show how to apply well-known stabilization methods from standard Lagrangian interpolation to the generalized case to improve the stability significantly.


[1] S. De Marchi, A. Iske, G. Santin. Image reconstruction from scattered Radon data by weighted kernel functions. Calcolo 55, 2018.

In my inaugural talk I would like to introduce myself and present the topic of my master thesis. To this end, I will first provide a brief introduction to empirical processes and long-range dependence. Afterwards, we consider the problem of testing the equality of two non-parametric regression functions. Finally, we provide a goodness of fit test for the error distribution.

The characterisation of the dynamics of a small inertial particle in a viscous fluid is a problem that dates to Stokes[1], back in 1851. Since his first attempt, many have tried and several formulas have been obtained for different types of flows, as well
as more general cases; however, the scientific community did not agree in a general formula until 1983, when M. Maxey and J. Riley[2] obtained a formula from first principles. This formula includes an integro-differential term, called the Basset History term, which
requires information for the whole history of the particle dynamics and creates difficulties in the numerical implementation due to fast increasing storage requeriments.
In the last decade, the Maxey-Riley formula has drawn the interest of many mathematicians and so, local and global existence and uniqueness of mild solutions have been proved ([3] & [4]). Nevertheless, a method to bypass the history term and obtain the trajectory
of the particle remained unknown until the publication of an accurate solution method by S.Ganga Prasath et al (2019) [5].
In this presentation I will analyse the Maxey Riley equation and will identify the core ideas within S. Ganga Prasath's method to solve the Maxey Riley equation as well as its implementation for certain fluid flows.

[1] Stokes, G. G. (1851). On the effect of the internal friction of fluids on the motion of pendulums.
[2] Maxey, M. R., & Riley, J. J. (1983). Equation of motion for a small rigid sphere in a nonuniform flow. The Physics of Fluids, 26(4), 883-889.
[3] Farazmand, M., & Haller, G. (2015). The Maxey–Riley equation: Existence, uniqueness and regularity of solutions. Nonlinear Analysis: Real World Applications, 22, 98-106.
[4] Langlois, G. P., Farazmand, M., & Haller, G. (2015). Asymptotic dynamics of inertial particles with memory. Journal of nonlinear science, 25(6), 1225-1255.
[5] Prasath, S. G., Vasan, V., & Govindarajan, R. (2019). Accurate solution method for the Maxey–Riley equation, and the effects of Basset history. Journal of Fluid Mechanics, 868, 428-460.

In this talk, I would like to introduce myself and the topic of my master thesis "Malliavin calculus and Malliavin-Stein method".
As indicated by its name, this talk provides a basic overview of the Malliavin calculus and its operators in the case where the underlying process is an isonormal Gaussian process. After this introduction, it is shown how the Malliavin calculus can be combined with Stein's method for the purpose of one-dimensional normal approximation and, particularly, for the derivation of generalized central limit theorems.

This presentation introduces the semi-implicit Smoothed Particle Hydrodynamics (SPH)
scheme [1] for the shallow water equations following the semi-implicit finite volume and finite
difference approach of Casulli [2]. In standard explicit numerical methods, there is often a severe
limitation on the time step due to the stability restriction imposed by the CFL condition. To this
effect, a semi-implicit SPH scheme is derived, which leads to an unconditionally stable method.
The discrete momentum equation is substituted into the discrete continuity equation to obtain
a symmetric positive definite linear system for the free surface elevation. The resulting system
can be easily solved by a matrix-free conjugate gradient method. Once the new free surface
location is known, the velocity at the new time level can be directly computed and the particle
positions can subsequently be updated. We further discuss a nonlinear algorithm for treating
wetting/drying problems. We derive a mildly nonlinear system for the discrete free surface
elevation from the shallow water equations by taking into consideration a correct mass balance
in wet regions and in transition regions, i.e. the regions from wet particles to dry particles
and those from dry particles to wet particles. The scheme is validated on a two dimensional
inviscid hydrostatic free surface flows for the two dimensional shallow water equations and
wetting/drying test problem.

References
[1] A.O. Bankole, A. Iske, T. Rung, M. Dumbser, A meshfree semi-implicit Smoothed Particle
Hydrodynamics method for free surface flow. Meshfree Methods for Partial Differential
Equations VIII, M. Griebel and M.A. Schweitzer (eds.), Springer LNCSE, Vol. 115, pp.
35-52 (2017).
[2] V. Casulli, Semi-Implicit Finite Difference Methods for the Two-Dimensional Shallow
Water Equations. Jour. of Comp. Phys., Vol 86. pp. 56-74 (1990).

Meeting-ID: 820 3979 6993
Passwort: 694649

Solving differential equations rigorously is a main and vigorous topic in the
field of verified computation. Here, solving rigorously means that a computer
program supplies an approximate solution along with error bounds that respect
all numerical as well as all rounding errors that occurred during the computation.
An exact solution is proved to be enclosed within these rigorous bounds.
In this context so-called Taylor models have been used successfully for solving
ordinary differential equations (ODEs) rigorously. Implementations are COSY INFINITY [1], FLOW [2], ODEIntegretor [3], and RIOT [4]. Here, COSY INFINITY
developed by Berz and Makino and their group is the most advanced
implementation. Recently, we implemented the Taylor model approach in MATLAB/
INTLAB [5].

We give a short introduction to Taylor models, their rigorous arithmetic,
and the Taylor model method for enclosing solutions of ordinary differential
equations in a verified manner. We only treat initial value problems
$y_0 = f(t,y)$, $y(t_0) = y_0$
where the initial value $y_0$ may be an interval vector. For specific ODEs we demonstrate
how to use and call our verified ODE solver. This is designed to be very
similar to calling MATLAB's non-verified ODE solvers like ode45. Finally, results
and run times are compared to those of COSY INFINITY, RIOT and Lohner's
classical AWA.

[1] M. Berz, K. Makino, COSY INFINITY: www.bt.pa.msu.edu/index_cosy.htm
[2] X. Chen, Reachability analysis of non-linear hybrid systems using Taylor models,
Dissertation RWTH Aachen, 2015. FLOW: https://flowstar.org/dowloads/
[3] T. Dzetkulic, Rigorous integration of non-linear ordinary differential equations in
Chebyshev basis, Numer. Algor. 69, 183-205, 2015.
ODEintegrator: https://sourceforge.net/projects/odeintegrator
[4] I. Eble, Über Taylor-Modelle, Dissertation at Karlsruhe Inst. of Technology, 2007.
RIOT: www.math.kit.edu/ianm1/~ingo.eble/de
[5] S.M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing
(ed. by Tibor Csendes), Kluwer Academic Publishers, 77-104, 1999.
INTLAB: http://www.ti3.tu-harburg.de/intlab/

Talk (PDF, 100KB)

https://tuhh.zoom.us/j/87535538628?pwd=RTJ0ZGp0ZWc1NVk3RGp5NTBQYjhVdz09

Meeting ID: 875 3553 8628
Passcode: 750232

Production processes are usually investigated using models and methods from queueing theory (queue = line where people wait for goods or services). Control of warehouses and their optimization rely on models and methods from inventory theory. Both theories are fields of Operations Research, but they comprise quite different methodologies and techniques. In classical Operations Research these theories are considered as disjoint research areas. Today's emergence of complex supply chains (=production-inventory networks) calls for integrated production-inventory models, which are focus of my research. We have developed Markov process models for several production-inventory systems and derived the steady state distribution of the global system. For most of the production-inventory systems the obtained steady state is of product form. This enables us to analyse the long term average costs with the aim to find the optimal inventory size.
In my talk, I focus on a basic production-inventory model and present the essentials of the other models. Furthermore, I show the connection to the industrial project “Robotic Mobile fulfillment system”.

*title by Karsten Kruse

An overview about the master thesis will be given, treating numerical methods for solving a PDE-constrained optimization problem. Afterwards, an outlook on advanced numerical methods for PDEs and modelling of tsunamis will be presented.

Stein's method is a powerful technique to establish convergence in distribution of a sequence of random variables to a standard Gaussian random variable. After an introduction to this approach, its application to several problems from stochastic geometry is discussed.

This will be an introductory talk on how the fundamental assumptions of quantum mechanics are modeled and how this relies on the spectral theory of unbounded self-adjoint operators on separable Hilbert spaces.

I will introduce myself and present the topic of my master thesis.

In my thesis, I derived a method for model order reduction of parametric linear time-invariant systems. With this method we can compute parametric reduced-order models that are optimal with respect to the $\mathcal{H}_2 \otimes \mathcal{L}_\infty$-error. The method combines interpolatory methods with numerical optimization. We furthermore discuss the computation of the $\mathcal{H}_2 \otimes \mathcal{L}_\infty$-error and have a look at some numerical results.

Deformable image registration is a key technology in medical imaging; there the goal is to compute a meaningful spatial correspondence between two or more images of the same scene. One approach is to use an optimal control formulation to compute a stationary velocity field that parameterize the deformation map. The same methods can be used to estimate the motion of contrast agents from 3d ultrasound images.

This is work-in-progress; in the talk I’ll introduce the application problem and discuss computational techniques for its solution, with a focus on using parallelization in time to reduce the time-to-solution. It should be accessible for a broad audience.

An overview talk about interesting topics in mathematical physics I used over the last years. It should be accessible for a broader audience.

This will be an overview talk on Evolution Equations (and a bit on Evolutionary Equations).

We consider the Caffarelli-Silvestre problem in a Banach space $X$ which is finding a solution $u$ to the problem
\[
u''(t) + \frac{1-2\alpha}{t} u'(t) = Au(t), \quad u(0)=x
\]
where $\alpha \in (0,1)$ is a given parameter and $A \in \mathcal{S}_{\omega}$ is a sectorial operator. Goal of the presentation will be to sketch of a proof that a solution got to be unique (we will not deal with existence but this is a much easier anyway).
The proof is simpler and independent of what can be found in

J. Meichsner and C. Seifert. On the Harmonic Extension Aproach to Fractional Powers in Banach Spaces. arxiv preprint https://arxiv.org/abs/1905.06779

All existing QR decompositions for hierarchical matrices suffer from numerical drawbacks that limit their use in many applications. In this talk, I will present a new method based on the recursive WY-based QR decomposition by Elmroth and Gustavson. It is an extension of an already existing method for a subclass of hierarchical methods developed by Kressner and Susnjara.

I will try to keep things as simple as possible and give a short introduction to hierarchical matrices as well. Previous knowledge of hierarchical matrices is not necessary to understand the basic ideas and main obstacles of the new algorithm. Although this talk is similar to my last one in November I will try to focus on some aspects we have only touched upon and present some new results as well.

Partial differential equations can be solved numerically by the radial basis function-generated finite difference (RBF-FD) method, which can be viewed as a generalization of the finite difference method to unstructured point sets.
A so-called stencil is computed for each interior node and radial basis functions are used for the computation of the stencil weights. The discretization error depends on the type of the point set (i.e. on the number of interior and boundary nodes and their distribution), the stencil size, the RBF type and the shape parameter of the RBF.
In this talk, I present an introduction of the RBF-FD method and a numerical analysis of the influence of the various parameters on the discretization error. I focus on Poisson's equation and on the convection-diffusion equation in three-dimensions.

I will introduce myself and talk about my master thesis.

The topic of my thesis was "On periodic finite sections", which are an approximation method based on the regular finite section method. The methods are used to approximate (the inverses of) infite matrices by finite matrices.

Since most of you couldn't attend my master thesis defense due to the university closure, I would like to use this talk to present you some results of my master thesis "Many-body localization: A spectral theoretic investigation of spin chains". Spin chains are a class of quantum-mechanical models well-suited to study many-body localization (MBL) phenomena due to their one-dimensional structure. After a brief introduction to one-particle (Anderson) localization and spectral properties of infinite-dimensional operators, we will see possible definitions and manifestations of MBL. The proofs of MBL for two specific spin chains will be sketched, illustrating how one-particle and many-body techniques can be combined. Furthermore, I would like to use this talk to properly present myself in case you wondered who that person in the guest office was.

Since I am new to the institute, I want to use this talk to introduce myself to you, and talk a little bit about my master thesis.

For the thesis I studied the approximability of the inverse of finite element matrices, i.e. matrices which are gained from the discretization of elliptic PDE's with the finite element method, by hierarchical matrices. So, after introducing myself, i will give an overview about the construction of the approximation and the error analysis.

supervision by Prof. Ernst Brinkmeyer (retired 2013, hence not hosted by him)

no maths topic

Talk (PDF, 219KB)

For a regular coupled cell network, we define an index of integer tuples for its associated lattice of synchrony subspaces, and use this index for identifying equivalent synchrony subspaces to be merged to each other. Based on this equivalence, the initial lattice of synchrony subspaces can be reduced to a lattice of synchrony subspaces which corresponds to a simple eigenvalue case discussed in previous work. The result is a reduced lattice of synchrony subspaces, which affords a well-defined non-negative integer index that can be used for bifurcation analysis in regular coupled cell networks.

After only two weeks of being able to get to know you in person, I will use this talk to introduce myself and talk about present and past work. My presentation will be divided into three parts. After shortly introducing myself (1), I'll cover the following topics related to my masters thesis (2) and my PhD research (3).

2. Amplitude dynamics of resonant Rossby wave triads in Nambu form:
Non-linear interactions play a fundamental role in the redistribution of energy amongst Rossby waves. For weakly interacting waves, geometric constraints govern the dynamics of forced and unforced resonant Rossby wave triads. These constraints allow to cast the dynamical equations in Nambu formulation.

3. How Semi-Implicit Spectral Deferred Correction (SISDC) can improve Numerical Weather Prediction (NWP) models and climate projections:
The multitude of time scales associated with atmospheric waves (e.g. Rossby-, gravity-, sound-waves) poses difficulties in modeling the full governing equations. The CFL condition usually requires the adaption of the time step to the fastest waves to prevent instabilities. However, the fastest waves (i.e. sound waves) in general transport a negligible amount of energy and therefore have minimal impact on the dynamics. By treating the fastest waves implicitly, instabilities can be prevented despite a CFL-number > 1. Using SISDC to integrate the linearized Boussinesq equations has proven to be valuable alternative to implicit-explicit Runge-Kutta and diagonally implicit Runge-Kutta methods. Applying SISDC to the full compressible governing equations may improve cost and accuracy of state of the art NWP models.

The presentations aims to give a rather short non-technical introduction in the general concept of a functional calculus including several examples and applications.

The presentation will (mostlikely) make use of the tool 'Zoom'. The audience will not have to do much but simply follow a link the speaker provides everybody with who asks in advance (jan.meichsner@tuhh.de). Members of the institute will get the link via the common email list.

Quite often, we wonder about the people around us but we are too shy to actually ask. On Thursday 27.02.2020, the Institute für Mathematik has organised a presentation about the one that is – up to the current date and not for very long – its latest “outstanding” acquisition.

During the presentation, you will finally be able to respond to the following questions:

- Why is he so fascinating?

- What was his last piece of work?

- What has he done during his first month?

These and any other question will be discussed during the meeting and who knows, maybe one day, in your closest cinema.

Moving-boundary flow simulations are an important design and analysis tool in many areas, including civil and biomedical engineering, as well as production engineering. Interface-capturing offers flexibility for complex free-surface motion, while interface-tracking is very attractive due to its mass conservation properties at low resolution. We focus on these alternatives in the context of flow simulations based on stabilized finite element discretizations of Navier-Stokes equations, including space-time formulations that allow extra flexibility concerning grid design at the interface.

Space-time approaches offer some not-yet-fully-exploited advantages; among them, the potential to allow some degree of unstructured space-time meshing. A method for generating simplex space-time meshes has been developed, allowing arbitrary temporal refinement in selected portions of space-time slabs. The method increases the flexibility of space-time discretizations, even in the absence of dedicated space-time mesh generation tools. The resulting tetrahedral and pentatope meshes are being used in the context of cavity filling flow simulations, such as those necessary to design injection molding processes.

We show the existence of solutions of nonlinear non-autonomous Cauchy problems
\[
\partial_t u(t,x) - \nabla_x \cdot (a(t,x)\nabla_xu(t,x))= f(t,x,u(t,x),\nabla u(t,x)),
\qquad u(0,\cdot)=u_0
\]
for a bounded open set $\Omega\subseteq \mathbb R^n$.
The coefficient matrix $a$ is supposed to be symmetric, uniformly elliptic,
Lipschitz continuous w.r.t.\ $t\in(0,\tau)$ and measurable w.r.t.\ $x\in\Omega$;
the nonlinearity $f$ is required to satisfy a linear growth condition.
We show that, given $u_0\in H_0^1(\Omega)$, there exists $u\in L_2(0,\tau;H_0^1(\Omega))
\cap H^1(0,\tau;L_2(\Omega))$ solving the problem mentioned above.

The proof relies on Schaefer's fixed point theorem. In the
course of the proof one uses maximal regularity properties of solutions of
inhomogeneous linear problems and compact embeddings of vector-valued Sobolev spaces.

The result partly generalises [ArCh10].

The talk is based on joint work with Wolfgang Arendt and Jürgen Voigt.


[ArCh10] W. Arendt, R. Chill: Global existence for quasilinear
diffusion equations in isotropic nondivergence form. Ann. Sc. Norm. Super. Pisa
Cl. Sci. (5) IX, 523-539 (2010).

The question of possible generalisations of the operation of differentiation towards fractional powers can be traced back to a letter from L'Hospital to Leibniz in 1695 ([1]).
Since this time, mathematicians developed plenty of different approaches to fractional differentiation and integration generalising different aspects of the known theory.
The possibly most prominent examples are the fractional derivatives (and integrals) of Riemann-Liouville and Weyl ([4]).
Both can also be understood as instances of the sectorial functional calculus of sectorial operators as it was introduced in [2] and further promoted in [3].
Nonetheless, a direct use of the abstract techniques from operator theory seems to be rare in applications.
Therefore, the talk aims for introducing the audience in the basic principles of functional calculus and how to use it to recover the above mentioned instances of fractional derivatives.


$\mathbf{References}$

[1] B. Ross. The Development of Fractional Calculus 1695--1900. $\mathit{\text{Historia Math., 4(1):}}$ 75--89, 1977.

[2] A. McIntosh. Operators which have an $H_{\infty}$ functional calculus. $\mathit{\text{Miniconference on operator theory and partial differential equations:}}$ 210--231, 1986.

[3] M. Haase. $\mathit{\text{The Functional Calculus for Sectorial Operators,}}$ volume 169 of $\mathit{\text{Operator Theory: Advances and Applications.}}$ Birkhäuser Basel, 2006.

[4] K. S. Miller and B. Ross. $\mathit{\text{An Introduction to the Fractional Calculus and Fractional Differential Equations.}}$ John Wiley & Sons, 1993.

Im Maschinen- und Automobilbau werden für mechanisch extrem beanspruchte, temperatur- und chemikalienbeständige Bauteile Elastomerwerkstoffe benötigt. Typische Beispiele sind Reifen, Zahn- und Keilriemen, Motor- und Aggregatelager, Luftfedern und Fahrwerkbuchsen sowie statisch und dynamisch beanspruchte Dichtungen. Die Anforderungen bezüglich Zuverlässigkeit und Leistungsdichte sind speziell in Verbindung mit den hochsensiblen, sicherheitsrelevanten Funktionen der Bauteile besonders hoch.
Insbesondere im Rahmen der Digitalisierung in der Produktion gewinnen Simulationsmodelle verstärkt an Bedeutung. Viele Verarbeitungsschritte in der Herstellung von Elastomerbauteilen beginnend mit dem Mischen, dem Walzen und der Extrusion oder des Spritzgießens, über die Vulkanisation beeinflussen die endgültigen mechanischen Eigenschaften. Im Laufe ihres Einsatzlebens verändern sich diese Eigenschaften auf Grund von thermo-oxidativer Alterung, so dass auch Lebensdauervorhersagen zur einer Herausforderung werden. Die zuverlässige Erstellung von „Digitalen Zwillingen“ für Elastomerbauteile bedarf so einer Beschreibung vieler auch untereinander gekoppelter Effekte.
Dieser Vortrag bietet Einblicke in verschiedene Modellierungsansätze einzelner Abschnitte des Leben von Elastomeren. Hauptfokus ist hierbei die Beschreibung der mechanischen Eigenschaften unter Berücksichtigung der Vernetzung und Alterung.

Language of the talk is going to be either German or English depending on the audience preferences.

Positive Kernels provide methods to solve multivariate interpolation problems, but standard kernel methods usually lead to ill-conditioned linear systems. Therefore, a suitable choice of interpolation centers and basis functions reqiures particular care.

In this talk, i will give an introduction to kernel based approximation and discuss greedy point selection strategies, which will improve the stability of the interpolation method.

Compressed Sensing deals with reconstructing some unknown vector from few linear measurements in high dimension by additionally assuming sparsity, i.e. many entries are zero. Recent results guaranteed recovery even when just signs of the measurements are available (one-bit CS). A natural generalization of classical CS replaces sparse vectors by vectors lying on manifolds having low intrinsic dimension. In this talk I introduce the one-bit problem and proposes a tractable strategy to solve one-bit CS problems for data lying on manifolds. This is based on joint work with Johannes Maly and Mark Iwen.

Large-scale optimization problems governed by partial differential equations (PDEs) occur in a multitude of applications, for example in inverse problems for non-destructive testing of materials and structures, or in individualized medicine. Algorithms for the numerical solution of such PDE-constrained optimization problems are computationally extremely demanding, as they require multiple PDE solves during the iterative optimization process. This is especially challenging for transient problems, where methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the associated optimality system. The evaluation of the reduced gradient then requires one solve of the state equation forward in time, and one backward-in-time solve of the adjoint equation. In order to tackle real-life applications, it is not only essential to devise efficient discretization schemes, but also to use advanced techniques to exploit computer architectures and decrease the time-to-solution, which otherwise is prohibitively long.

One approach is to utilize the increasing number of CPU cores available in current computers. In addition to more common spatial parallelization, time-parallel methods are receiving increasing interest in the last years. There, iterative multilevel schemes such as PFASST (Parallel Full Approximation Scheme in Space and Time) are currently state of the art and achieve significant parallel efficiency. In this talk, we investigate approaches to use PFASST for the solution of parabolic optimal control problems. Besides enabling time parallelism, the iterative nature of the temporal integrators within PFASST provides additional flexibility for reducing the cost of solving nonlinear equations, re-using previous solutions in the optimization loop, and adapting the accuracy of state and adjoint solves to the optimization progress. We discuss benefits and difficulties, and present numerical examples.

This is joint work with Michael Minion (Lawrence Berkeley National Lab).

Molecular-continuum methods, as referred to in my talk, employ a domain decomposition and compute fluid flow either by means of molecular dynamics (MD) or computational fluid dynamics (CFD) in the sub-domains. This enables multiscale investigations of nano- and microflows beyond the limits of validity of classical CFD.

In my talk, I will focus on latest developments in the macro-micro-coupling tool (MaMiCo). MaMiCo enables the coupling of arbitrary CFD and MD solvers, hiding the entire coupling algorithmics from the actual single-scale solvers. After a brief discussion of the limits of the MD method, I will focus on various aspects of the molecular-continuum coupling and its realization in MaMiCo, including parallelization, multi-instance sampling for MD (that is ensemble averaging) and filtering methods that extract smooth responses from the fluctuating MD description to enhance consistency on the side of the continuum solver. I will further present preliminary results from a study which aims to generate open boundary force models for MD using machine learning.

All existing QR decompositions for hierarchical matrices suffer from numerical drawbacks that limit their use in many applications. In this talk, I will present a new method based on the recursive WY-based QR decomposition by Elmroth and Gustavson. It is an extension of an already existing method for a subclass of hierarchical methods developed by Kressner and Susnjara.

I will try to keep things as simple as possible and give a short introduction to hierarchical matrices as well. Previous knowledge of hierarchical matrices is not necessary to understand the basic ideas and main obstacles of the new algorithm.

In the talk, we shall develop a general framework for the treatment of both deterministic and stochastic homogenisation problems for evolutionary equations. The versatility of the methods allow the unified treatment of static, dynamic as well as mixed type problems.

The efficient use of modern supercomputers has become one of the key challenges in computational science. New mathematical concepts are needed to fully exploit massively parallel architectures. For the numerical solution of time-dependent processes, time-parallel methods have opened new ways to overcome scaling limits. With the "parallel full approximation scheme in space and time" (PFASST), multiple time-steps can be integrated simultaneously. Based on spectral deferred corrections (SDC) methods and nonlinear multigrid ideas, PFASST uses a space-time hierarchy with various coarsening strategies to maximize parallel efficiency. In numerous studies, this approach has been used on up to 448K cores and coupled to space-parallel solvers with finite differences, spectral methods or even articles for discretization in space. Yet, since the integration of SDC or PFASST into an existing application code is not straightforward and the potential gain is typically uncertain, we will present in this talk our Python prototyping framework pySDC. It allows to rapidly test new ideas and to implement first toy problems more easily. We will also discuss the transition from pySDC to application-specific implementations and show recent use cases.

The plasma in a fusion reactor is heated by neutral beam injection: injecting high energy neutrons which quickly ionize and swirl around in the reactor's magnetic fiel. Modelling this process requires solving the Lorentz equations numerically over long times (up to a second) with very small time steps (order of nanoseconds), which means very many time steps and thus long simulation times (from days up to a week). The talk will introduce GMRES-Boris-SDC (GBSDC), a new time stepping algorithm that can reduce computational cost compared to the currently used Boris method. The method is a potpourri of various numerical techniques, including the GMRES linear solver, spectral deferred corrections, the velocity Verlet scheme and the Boris trick. I will describe the algorithm and show examples of its performance for benchmarks with varying degree of realism.

This is joint work with Dr Krasymyr Tretiak, School of Mathematics, University of Leeds.

During their work-term at TUHH the two Canadian students worked on projects relating to current research in the institute.
As their term comes to an end they will present their ongoing work in short talks.

In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an observability estimate with explicit dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on Lp spaces. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.
The talk is based on joint work with Christian Seifert and Martin Tautenhahn.

We give a unified approach to handle the problem of extending functions with values in a locally convex Hausdorff space $E$ over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, which have weak extensions in a space $\mathcal{F}(\Omega,\mathbb{K})$ of scalar-valued functions on a set $\Omega$, to functions in a vector-valued counterpart $\mathcal{F}(\Omega,E)$ of $\mathcal{F}(\Omega,\mathbb{K})$. The main tool is the representation of vector-valued functions as linear continuous operators.

Ich präsentiere einen allgemeinen Ansatz für das Jacobi-Davidson-Verfahren basierend auf einem beliebigen iterativen Verfahren zum Lösen eines linearen oder nichtlinearen Eigenwertproblems. Die Auswirkung eines inexakten Lösens der Korrekturgleichung wird betrachtet und hieraus kann lineare Konvergenz für drei Fälle von verschiedene Vorbedingungen bewiesen werden.

We give an overview or recent developments on Deep Learning, its relations to wavelet
theory and applications to image analysis with interactions with persistent homology.

We consider differential-algebraic equations on (possibly) infinite dimensional Hilbert spaces, that is, we consider equations of the form
\begin{align*}
(Eu)'+Au & =0\quad(\text{on }\mathbb{R}_{\geq0}),\\
u(0) & =u_{0},
\end{align*}
where $E,A$ are linear operators on a Hilbert space $H$ and $E$ is bounded and allowed to have a non-trivial kernel. These equations cannot have a unique solution for each $u_{0}\in H$ (just look at the case $E=0$). Thus, finding the ``right'' space of initial values arises as a natural question. Imposing growth conditions on the operator pencil
\[
z\mapsto(zE+A)^{-1}
\]
we determine the maximal space of admissible initial values. First, we treat the case of bounded $A$ and then generalise the results to the case of unbounded $A$. In particular, we discuss whether we can find a $C_{0}$-semigroup yielding the mild solutions of the above problem.

Es geht um eine Fortführung des Vortrages von Anfang Februar:

Kann man Pseudoeigenvektoren der unendlichen Matrix $A$ bzw. ihrer endlichen Ausschnitte $A_n$ aus den jeweils anderen gewinnen? Wir hatten u.a. gesehen, dass die sogenannte untere Norm von $A_n$ für große $n$ mit der von $A$ in Verbindung steht. (Entsprechende Aussagen übertragen sich auf die Resolvente.)

Diesmal soll das Ganze in Abhängigkeit von $n$ quantifiziert werden.

Talk (PDF, 159KB)

Radiale Basisfunktionen (RBF) dienen der Interpolation und Approximation mehrdimensionaler verteilter Daten. In diesem Vortrag werden RBF motiviert, die positive Definitheit und damit eindeutige Lösbarkeit der Interpolationsaufgabe einiger RBF hergeleitet, sowie Erweiterungen, wie bedingt definite RBF und flache RBF, vorgestellt. Der Fokus liegt hierbei auf den Beweistechniken und den Ideen hinter RBF.

Based on 6 + 2 assumptions, we will derive a model (a system of PDEs) with the purpose to describe the movement of a fluid. Ideally, at the end of the talk, we will have arrived at the incompressible Navier-Stokes equations.

Part II of the presentation on general relativity. In this part we will talk about the basic equations and how physical quantities are described in terms of mathematical objects.

I am not an expert on the field but during my studies I spent some time on understanding the mathematical basics of the general theory of relativity. I would present them in two parts. In the first part on the 26th of April I would concentrate on basics of differential geometry which are needed to describe the mathematics of the theory. In a second part on the 3rd of May I would explain how the before introduced structures are used to create a mathematical model of general relativity.

In the classical potential theory on the Euclidean space and in the potential theory of transient Markov chains a unique decomposition of superharmonic functions into a harmonic and a potential part is well-known. In this talk the basic concepts and ideas to gain such a decomposition for Schrödinger operators on graphs will be shown. The talk will show results of my master's thesis supervised by Matthias Keller.

Wie verändert sich das Spektrum des Laplace-Operators (oder allgemeiner von Schrödinger-Operatoren) auf metrischen Graphen unter Variation der Graphenparameter? Einige Antworten auf die Frage gibt es im Vortrag.

I will give a short summary of my master thesis as well as a quick introduction of myself.

Was wird aus (Pseudo-)Eigenwerten und -vektoren beim Abschneiden einer unendlichen Matrix? (Sie bleiben welche.)
Gibt es auch Aussagen in die umgekehrte Richtung?
Wie gut lassen sich diese Aussagen quantifizieren?

Talk (PDF, 1.0MB)

We construct extrapolation spaces for non-densely defined (weak) Hille--Yosida operators. In particular, we discuss extrapolation of bi-continuous semigroups. As an application we present a Desch--Schappacher type perturbation result for this kind of semigroups. This talk is based on joint work with B. Farkas.

For two Banach spaces $X,Y$ let $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$ be an operator valued function and $\mathtt{P}$ a regularity property. Assume that each orbit $t\mapsto u(t)x$ has the regularity property $\mathtt{P}$ on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we prove a Baire-type theorem, which allows to remove the dependency of $x$ in certain situations. Afterwards, we provide some applications which are of interest in semigroup theory. In particular, we generalize and explain the result obtained by Bárta in his article ``\emph{Two notes on eventually differentiable families of operators}'' (Comment. Math. Univ. Carolin. 51,1 (2010), 19-24).

In this presentation we will talk about the application of hierarchical matrices to solve the least squares problem arising in the RBF Approximation of scattered data.

We will shortly introduce hierarchical matrices as well as some central aspects of the RBF approach to scattered data. The main part will be about different ideas regarding the QR decomposition of hierarchical matrices.

I will present an introduction of the RBF-FD method and properties of the arising linear systems.

In this talk we present results of a comparative study, where we analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices, i.e., the van Roosbroeck system.

The relation between the quasi-Fermi levels and the densities of electrons and holes is given by the equation of state. Three common challenges, that can corrupt the precision of numerical solutions of the van Roosbroeck system, will be discussed: boundary layers of the quasi-Fermi potentials at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.

The Hot Spots Conjecture of J. Rauch asserts that the hottest and coldest points of an insulated body should move towards its boundary for large times, if the insulation is perfect. Via the semigroup associated with the Neumann Laplacian this reduces to proving that maximum and minimum of the eigenfunction(s) associated with the smallest positive eigenvalue are located on the boundary. This conjecture is not true in full generality but is currently open, for example, for convex domains.

In this talk we will examine the corresponding question on metric graphs: for the Laplacian on a finite metric graph with standard (continuity and Kirchhoff) vertex conditions we consider the possible distribution of maxima and minima of eigenfunctions associated with the smallest nonzero eigenvalue. Among other things, we give examples to show that the usual notion of “boundary” of a metric graph, namely the set of vertices of degree one, has limited relevance for determining the “hottest” and “coldest” parts of a graph.

This is joint work with James Kennedy (Lisbon).

In this talk we study approximation techniques for input-output systems, which appear in the modeling process of mechanical systems. So, the focus will be on linear dynamical systems with a second derivative term.
These system can become very large in practice and therefore, expensive to be used for simulations and controller design.
Since this frequently happens to all control systems coming from real-live application, model order reduction became a major field in control theory over the last decades.
Here however, beside approximating the input-output behavior of the original system, the special structure should be preserved in the reduced-order model.
So far, reduction techniques designed for the linearized model fail in this aspect. On the other hand, there is a wide variety of methods that directly treat the second order control system. However, up to this point none of those methods deliver reasonable error-bounds for the approximation.
In this talk an approximation method is presented for the special class of passive mechanical systems. Roughly speaking passivity for control systems means that the system itself cannot produce energy. For this class the special canonical structure, given by so called Jordan triples for matrix polynomials, can be exploited.
In the end an error bound in the gap metric will be derived. The gap metric is used as a measure for the distance of two linear systems. It is defined via the distance of the closed subspaces of stable trajectories corresponding to zero initial conditions of the systems. Hence, the gap metric error-bound ensures the quality of the approximation of the state/signal system.

Abstract
We study two standard biased (1 : b) Maker-Breaker positional games
| the Perfect Matching game and the Hamilton Cycle game, played on
the edge set of the complete graph on n vertices, Kn. Given Breaker's bias
b, possibly depending on n, our goal is to determine the minimal number
of moves in which Maker can win in each of these two graph games.

This is joint work with Miloš Stojakovic.

In this talk we outline classical connections between such mathematical objects as operator semigroups, evolution equations and Markov processes. Further, we present a method to approximate operator semigroups with the help of the Chernoff theorem. Many \emph{Chernoff approximations} lead to representations of solutions of (corresponding) evolution equations in the form of limits of $n$-fold iterated integrals of elementary functions when $n$ tends to infinity. Such representations are called \emph{Feynman formulae}. They can be used for direct computations, modelling of the related dynamics, simulation of underlying stochastic processes.
In some cases, Chernoff approximations can be understood as a version of the operator splitting method (known in the numerics of PDEs); some Feynman formulae provide Euler--Maruyama schemes for SDEs. Moreover, the limits in Feynman formulae sometimes coincide with path integrals with respect to probability measures (\emph{Feynman-Kac formulae}) or with respect to Feynman type pseudomeasures (\emph{Feynman path integrals}). It is planned to discuss different Chernoff approximations for semigroups corresponding to some Markov processes (e.g., subordinate Feller diffusions on star graphs and Riemannian manifolds) and for Schr\''{o}dinger groups.
Furthermore, the constructed Chernoff approximations for operator semigroups can be used to approximate solutions of some time-fractional evolution equations describing anomalous diffusion (solutions of such equations do not posess the semigroup property).

The original problem of counting domino towers was first studied by G. Viennot in 1985, see also D. Zeilberger (The Amazing 3^n Theorem). We analyse a generalisation of domino towers that was proposed by T. M. Brown (J. Integer Seq. 20.3 (2017), Art. 17.3.1), which we call S-omino towers. After establishing an equation that the generating function must satisfy and applying the Lagrange Inversion Formula, we find a closed formula for the number of towers.

The talk should hopefully also be accessible to people not used to this kind of mathematics.

This talk is divided into two parts. The first part will be given on Thursday 08.11.18 by Dennis Gallaun.
In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an obserbability estimate with explicite dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on Lp(Rd) and on Lp(Td) for 1 < p < ∞. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.

This talk is divided into two parts. The second part will be given on Thursday 15.11.18 by Christian Seifert.
In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an obserbability estimate with explicite dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on Lp(Rd) and on Lp(Td) for 1 < p < ∞. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.

My talk will consist of three short, independent parts, the first one being a quick introduction of myself. In the second and the third part, I will give an ''easy-to-digest'' survey of my graduate theses [1,2].

References:

[1] Implementation and Performance Analyses of a Highly Efficient Algorithm for Pressure-Velocity Coupling. Master Thesis Computational Engineering, Darmstadt, 2015

[2] On the L^p Theory of the Stokes Operator in Lipschitz Domains. Master Thesis Mathematics, Darmstadt, 2018

Talk (PDF, 2.1MB)

It is a classical result that every $\mathbb{C}$-valued holomorphic function has a local power series representation.
This even remains true for holomorphic functions with values in a locally complete locally convex Hausdorff space $E$ over
$\mathbb{C}$. Motivated by this example we try to answer the following question. Let $E$ be a locally convex Hausdorff space
over a field $\mathbb{K}$, $\mathcal{FV}(\Omega)$ be a locally convex Hausdorff space of $\mathbb{K}$-valued functions on a set $\Omega$ and $\mathcal{FV}(\Omega,E)$ be an $E$-valued counterpart of $\mathcal{FV}(\Omega)$
(where the term $E$-valued counterpart needs clarification itself).
For which spaces is it possible to lift series representations of elements of $\mathcal{FV}(\Omega)$ to elements of $\mathcal{FV}(\Omega,E)$?
We derive sufficient conditions for the answer to be affirmative which are applicable for many classical spaces of functions
$\mathcal{FV}(\Omega)$ having a Schauder basis.

BA-Vortrag

This talk will be an extended version of the talk I gave on the SOTA 2018 in Poland.
I will discuss existence and uniqueness of the so-called Harmonic extension approach to fractional powers of linear operators.

Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods are designed to increase the smoothness and improve the convergence rate of the DG solution form p+1 to 2p+1 through post-processing. However, introducing these filters can be challenging for multi-dimensional data since a tensor product filter grows in support size as the field dimension increases [(3p+2)*h]^d, where p + the polynomial order and d is the dimension. This becomes computationally prohibitive as the dimension increases. An alternative approach is to utilize a one-dimensional univariate filter. In this talk we introduce the Line SIAC filter and explore how the orientation, structure and filter size affect the order of accuracy and global errors. We show how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate, given an appropriate rotation. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs.

First, we give a short introduction to abstraction interpolation theory and
relate it to the well-known interpolation results from Riesz--Thorin and
Marcinkiewicz. Then we apply the abstract methods to concrete spaces
incorporating (mixed) boundary conditions and give an overview on arising
challenges and ways to resolve them.

This talk is divided into two. In the first part we discuss a so-called scale-free and quantitative unique continuation principle for spectral projectors of Schr\''odinger operators.
Let $\Omega = \Lambda_L = (-L,L)^d$ or $\Omega = \mathbb{R}^d$, and $H = -\Delta + V$ be a Schr\''odinger operator on $L^2 (\Omega)$ with a bounded potential $V$. If $\Omega = \Lambda_L$ we impose Dirichlet, Neumann, or periodic boundary conditions. The unique continuation principle states that for any $E \geq 0$, and any $\phi \in \operatorname{Ran} \chi_{(-\infty , E]} (H)$ we have
\begin{equation} \label{quc}
\lVert \phi \rVert_{L^2 (\Omega)}^2 \leq C_{\rm sfuc} \lVert \chi_{S_\delta \cap \Omega} \phi \rVert_{L^2 (\Omega)}^2,
\end{equation}
where $S_\delta \subset \mathbb{R}^d$ is a union of equidistributed $\delta$-balls, and $C_{\rm sfuc} = C_{\rm sfuc} (d , E ,\allowbreak \delta , \lVert V \rVert)$ some explicitly given constant.
\par
In the second part of the talk we will discuss an applications thereof to control theory. On the time interval $[0,T]$ we consider the controlled heat equation
\begin{equation} \label{eq:parabolic}
\partial_t u + H u = f\chi_{S_\delta \cap \Omega} ,
\end{equation}
where $u,f \in L^2([0,T] \times \Omega)$, and $u (0,\cdot) \in L^2 (\Omega)$.
Note that the control function $f$ acts on the set $S_\delta$ only. Our aim is to study null-controllability in time $T > 0$, i.e.\ there is a control function $f$ such that $u(T,\cdot) = 0$. We provide explicit estimates on the costs of the form $\lVert f \rVert_{L^2([0,T]\times \Omega )} \leq C \lVert u_0 \rVert_{L^2 (\Omega)}$.

Prony's problem - estimating the frequencies of an exponential sum - and its higher dimensional
analogs have attracted a lot of attention in recent years. A somewhat neglected question is whether
this problem is well-posed. In this talk, some results in this direction will be presented.
We start by giving a brief introduction to stability in compressed sensing. Compressed sensing is
concerned with solving nite dimensional linear systems under a priori sparsity assumptions. Stability
follows from the so-called restricted isometric property (RIP) of the system matrix.
We then discuss sparse frequency estimation. Due to the continuous nature, proving an analogue of
the RIP is more dicult. To this end, we briey introduce specic functions, which are well localized
in the spatial and frequency domain. Then we deduce stability results as well as a posteriori error
estimates.
This talk is based on joint work with Armin Iske.

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan sin(2\Theta) theorem.

This talked is based on joint work with I. Nakic, M. Täufer, M. Tautenhahn, and I. Veselic.

I'll start with a short recap of the lattice Anderson
model, with a focus on Minami's estimate and its applications. In
particular it implies Poissonian local eigenvalue statistics, which
is believed to be a characteristic feature of spectrally localized
quantum mechanical systems. In the second part of the talk I'll
present our main technical result, a level-spacing estimate for
continuum random Schrödinger operators, and argue why it implies
Poissonian local eigenvalue statistics. If time permits I'll comment
on the proof's methods.
The talk is based on joint work with Alexander Elgart.

We are going to discuss (again) the approach of describing fractional powers of linear operators on
Banach spaces as it was performed by Silvestre and Caffarelli when they were studying the fractional
Laplacian. Even though useful it is still an open problem whether this is possible for all sectorial
operators and, if so, whether it is unique.
The presented content is work in progress.

The study of electromagnetic fields in 2D circuits often leads to resonances. We use a polynomial chaos expansion (due to uncertain circuit parameters), which is analytically and numerically troublesome near the resonance frequencies. As a toy model for the convergence of the polynomial chaos expansion, we look at the parallel RLC circuit with uncertain capacitance and give $L^2$ error bounds depending on the degree of the expansion, the random distribution, the distance to resonance and the so-called quality factor of the circuit (which is a measure for the damping).

Im Rahmen des Lothar-Collatz-Seminars spreche ich am Geomatikum (Uni Hamburg) über strukturierte Pseudospektren in der Systemtheorie.

Abstract: https://www.c3s.uni-hamburg.de/en/news-events/seminar-c3s/gallaun.pdf

Interpolationsaufgaben mit radialen Basisfunktionen fuehren auf vollbesetzte Sattelpunktprobleme, deren iterative Loesung eine Praekonditionierung erfordert. Die Systemmatrizen koennen als H-Matrizen approximiert und fuer die Konstruktion algebraischer Praekonditionierer verwendet werden. Als Praekonditionierer verwenden wir die Nullraummethode sowie ein Gebietszerlegungsverfahren. Mittels der Nullraummethode kann die Loesung des indefiniten Systems im Wesentlichen auf die Loesung eines positiv definiten Systems geringfügig kleinerer Groesse zurueckgefuehrt werden. Die positiv definiten Systeme koennen mit einer approximativen Cholesky-Zerlegung in der Arithmetik hierarchischer Matrizen praekonditioniert werden. Kleinere Probleme werden auf diese Art zufriedenstellend geloest, aber fuer groessere Punktzahlen nimmt die Effektivität der Cholesky-Praekonditionierung ab.
Im Gebietszerlegungsverfahren wird die Kombination aus Nullraummethode und Cholesky-Praekondiitonierung nur in jedem Teilgebiet angewendet und das globale System mit einer aeusseren GMRes-Iteration geloest. Ein weiterer Vorteil der Gebietszerlegung liegt in der Parallelisierbarkeit der Konstruktion des Praekonditionierers.

Let \Omgea\subseteq\R^2 be a curvilinear polygon and Q_\Omega^\gamma be the Laplacian in L_2(\Omega) with the Robin boundary condition \partial_\nu \psi = \gamma \psi, where \partial_\nu is the outer normal derivative and \gamma>0. We are interested in the behavior of the eigenvalues of Q_\Omega^\gamma as \gamma becomes large. We prove that there exists N_\Omega \in\N such that the asymptotics of the N_\Omega first eigenvalues of Q_\Omega^\gamma is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with \partial\Omega. In the particular case of a polygon with straight edges the N_\Omega first eigenpairs are exponentially close to those of the model operators. Moreover, if the polygon admits only non-resonant or concave corners, we prove that, for any fixed j\in\N, the N_\Omega+j eigenvalue E_{N_\Omega+j}(Q_\Omega^\gamma) behaves as E_{N_\Omega+j}(Q_\Omega^\gamma) = -\gamma^2+\mu_j^D+o(1) as \gamma\to\infty, where \mu_j^D stands for the jth eigenvalue of the operator D_1\oplus\ldots\oplus D_M and Dn denotes the one-dimensional Laplacian on (0,l_n), where l_n is the length of the nth side of \Omega, with the Dirichlet boundary condition. Finally, we prove a Weyl asymptotics for the eigenvalue
counting function of Q_\Omega^\gamma for a threshold depending on \gamma, and show that the leading term is the same as for smooth domains.

Populations-Bilanzen und ihre Simulation spielen in vielen Prozessen der Chemie, Pharmazie und Biotechnolgie eine zunehmend wichtige Rolle. Partikel werden dabei anhand bestimmter Eigenschaften wie Masse oder Volumen gezählt. Ein wichtiger Teil dieser Simulationen ist die Aggregation.
In diesem Vortrag wird dieser Vorgang in multivariaten Problemen behandelt, eine Diskretisierung und ein effizienter Algorithmus vorgestellt.

Kooperation mit der Basler AG

Teil 2 des Vortrags über lineare Relationen und Randtripel.

Auf Homepage hochgeladen.

Talk (PDF, 4.3MB)

Ist S ein symmetrischer Operator in einem Hilbertraum, so stellt sich oft die Frage, welche selbstadjungierten Erweiterungen der Operator S hat und ob sich Aussagen über die Spektren (beispielsweise über die Eigenwerte) dieser Erweiterungen machen lassen. Ein mathematisches Konzept, welches hierbei hilfreich sein kann, ist das Konzept der Randtripel. Dabei stellt es sich als hilfreich heraus, nicht nur Operatoren sondern auch lineare Relationen (''mehrwertige Operatoren'') zu betrachten.

Der Vortrag soll einen einführenden Charakter haben. Es werden also die grundlegenden Definitionen und Sätze angegeben und anhand von einfachen Beispielen illustriert.

In diesem Vortrag stelle ich mich und meine Masterarbeit kurz vor.

Im Rahmen meiner Masterarbeit habe ich mich mit strukturierten Pseudospektren und deren Bezug zur Systemtheorie beschäftigt.
Strukturierte Pseudospektren sind ein wichtiges graphisches Werkzeug in der Stabilitätstheorie endlich-dimensionaler linearer Systeme mit ungenauen Parametern. In diesem Vortrag beschäftigen wir uns mit der Verallgemeinerung strukturierter Pseudospektren auf unendlich-dimensionale Systeme und gehen auf den Bezug zur Stabilität stark stetiger Halbgruppen ein.

We prove a locally uniform lower bound on the density of states of continuum random Schrödinger operators in the localised regime. The main technical ingredient is a new bound on the expectation of the spectral shift function for random Schrödinger operators in the localised regime, corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a finite volume. The bound scales with the surface area. (Joint with Adrian Dietlein, Abel Klein, Peter Hislop, Peter Müller)

Wir beiden stellen uns und unsere Masterarbeiten jeweils kurz vor.

Im ersten Vortrag (von Vincent Griem) geht es um gewichtete positiv definite Kernel und
ihre Anwendung in der Interpolation. Nach einer kleinen Einführung in
die Interpolation durch positiv definite Funktionen wird die
Erweiterung durch zusätzliche Gewichtsfunktionen vorgestellt und der Einfluss und mögliche
Nutzen dieses Vorgehens untersucht. Es wird insbesondere eine
Verbindung zum diagonalen Skalieren zur Verbesserung der Kondition
einer Matrix hergestellt. Abschließend folgen noch einige numerische
Beispiele.

Im zweiten Vortrag (von Willi Leinen) geht es um die Simulation des Kühlvorgangs eines
Fluid-gefüllten Behälters mit OpenFOAM. Dabei werden einerseits
die mathematischen Grundlagen der Simulation, wie z.B. die
Finite-Volumen-Methode sowie die PDEs zur Modellierung von Wärmeleitung
und Strömung, vorgestellt. Andererseits wird auf die Software OpenFOAM,
mit deren Hilfe die Simulation durchgeführt wurde, eingegangen. Als
Anwendungsbeispiel wird die Kühlung einer Weinflasche im Gefrierschrank
untersucht. Zum Abschluss werden die numerischen Ergebnisse
der Simulation vorgestellt.

Strukturierte Pseudospektren sind ein wichtiges graphisches Werkzeug in der Stabilitätstheorie endlich-dimensionaler linearer Systeme mit ungenauen Parametern. In diesem Vortrag beschäftigen wir uns mit der Verallgemeinerung strukturierter Pseudospektren auf unendlich-dimensionale Systeme und gehen auf den Bezug zur Stabilität stark stetiger Halbgruppen ein.

We all know that sometimes problems get easier by generalizing them. In this talk we want to present several recent results on radii functionals of convex bodies. This results were possible allowing non-symmetric gauge bodies, where in the past only
symmetric ones were studied (via general Minkowski spaces).

In dem Vortrag wird es um bi-stetige Halbgruppen gehen. Das Konzept geht auf die Dissertation
'Bi–Continuous Semigroups on Spaces with Two Topologies: Theory and Applications' von
F. Kühnemund aus dem Jahre 2001 zurueck. Betrachtet werden Halbgruppen auf einem
Banachraum, welche nicht stark-stetig sind. Das wird behoben, indem man sich eine groebere
Topologie betrachtet.
Da ich Anfaenger auf dem Feld bin, wird der Vortrag eine Einfuehrung enthalten und die Nuetzlichkeit
des Konzepts an einigen Beispielen illustriert.

Die Sturm’sche Oszillationstheorie stammt von Charles-François Sturm um 1830, und bezieht sich meistens auf sogenannte Sturm-Liouville-Probleme, d.h. Eigenwertprobleme für gewisse Differentialgleichungen. Im Vortrag wird das diskrete Analogon davon betrachtet und in Verbindung mit dem spektralen Fluss in von-Neumann-Algebren gebracht.

In dem Vortrag werde ich ein abstraktes Konzept vorstellen, um selbstadjungierte Operatoren mit singulären Störungen zu untersuchen und dieses anschließend auf Schrödingeroperatoren mit Delta-Interaktionen anwenden.

Partial differential equations model a wide range of physical phenomena.
Unfortunately, most of them cannot be solved directly, making it necessary
to develop efficient and robust numerical solution methods. In this talk,
we focus on two different ones: Radial basis functions (RBFs) and finite volume
methods (FVM). The former allow to solve differential equations without the
cumbersome generation of a grid. Moreover, RBFs can be used to improve flawed grids.
The latter are particularly useful in the context of semiconductor device simulation.
They yield robust numerical solutions even in the presence of boundary layers. The presented
finite volume scheme additionally satisfies a discrete maximum principle, just like the
continuous semiconductor equations (the van Roosbroeck system).

Tsunami modeling is - to first (and very accurate) approximation - performed with the help of shallow water theory and equations. This is still the method of choice for many applications, including forecasting, hazard assessment and inundation modeling. However, for long propagation distances as well as highly nonuniform topographies dispersive effects become important. While truly dispersive model equations are fully three-dimensional and therefore expensive with respect to computational requirements, a common approach to dispersive modeling comprises a non-hydrostatic correction of shallow water equations. In order to derive this correction term, a linear system of equations needs to be solved in each time step - even when the time-stepping scheme is explicit.

In the presentation we will introduce the basic modeling concepts for tsunami simulation, will show the derivation of non-hydrostatic correction terms and motivate further research on solvers for linear systems of equations.

We give new conditions for strong convergence of positive operator
semigroups as time tends to infinity. This is achieved by a new approach
that combines the splitting theorem by Jacobs, de Leeuw and Glicksberg
with a purely algebraic result about positive group representations.
Thus, we obtain convergence theorems not only for one-parameter
semigroups but also for a much larger class of semigroup representations
without any continuity or regularity assumption in time.
In particular, this generalizes results from the literature that, under
technical assumptions, a bounded positive strongly continuous semigroup
that contains or dominates a kernel operator converges strongly as time
tends to infinity. One can also derive a generalization of a famous
theorem by Doob for operator semigroups on the space of measures.

Ich werde in einem ca. 45 minütigen Vortrag meine Masterarbeit, die ich im Juni 2016 an der Uni Lund verteidigt habe, vorstellen.
Die Arbeit mit dem Titel ''Evaluation of a Gradient Free and a Gradient Based Optimization Algorithm for Industrial Beverage Pasteurisation Described by Different Modeling Variants'' entstand in Zusammenarbeit mit der Firma Krones AG in Kopenhagen.
Ziel ist die Optimierung thermaler Behandlung von Getränken und flüssigen Dosenkonserven. Dazu wurden Pasteurisierungsprozesse mathematisch formuliert, simuliert und die Optimierung mit Python durchgeführt.

Consider the usual function space L^2(D) on the unit disk D and
the (closed) subspace of holomorphic functions A^2(D). An important class of bounded
linear operators arises by restricting multiplication operators M_f on L^2(D) to A^2(D). More
precisely, if P denotes the orthogonal projection onto A^2(D), one considers operators of
the form PM_f in A^2(D), so-called Toeplitz operators. In this talk we are going to study the
essential spectrum of these Toeplitz operators. It is a classical result that if the defining
symbol f is continuous up to the boundary, the essential spectrum can be obtained by
evaluating f at the boundary. As it turns out, this statement can be generalized to more
general symbols by using techniques that were developed to solve a similar problem on
the sequence space \ell^2(\Z).

Im Vortrag wird es darum gehen, wie man eine vektorwertige Gleichung löst, wenn man die entsprechende Gleichung schon einmal skalarwertig gelöst hat. Typische Beispiele hierfür sind elliptische Differentialgleichungen. Hierbei geht es dann weniger darum, den Differentialoperator selbst zu untersuchen, sondern die Eigenschaften der Räume, auf denen er lebt.

Im wesentlichen ein 60 bis 90 minütiger Arbeitsstandbericht. Es werden grundlagen der Theorie fraktionaler Operatoren erläutert und danach auf die Problematik der Einführung durch harmonische Erweiterung eingegangen.

Vier konkrete Anwendungen von Toeplitzoperatoren

Es werden vier konkrete und sehr unterschiedliche Anwendungen von Toeplitzoperatoren vorgestellt. Diese sind (1) ein Problem aus der optimalen ell-eins-Kontrolle, (2) Spektralfaktorisierung von Polynomen vom Grad 20000, (3) Berechnung des Volumens der Fundamentalgebiete gewisser hochdimensionaler Gitter, und (4) Bestimmung der Grenzmenge der Nullstellen von Polynomen vom Fibonacci-Typ in der Hausdorffmetrik. Der Vortrag erlaubt es, viermal abzuschalten und ebenso oft wieder einzusteigen.

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Four concrete applications of Toeplitz operators

I present four concrete and very different applications of Toeplitz operators. These applications are (1) a problem in optimal ell-one control, (2) spectral factorization of polynomials of degree 20000, (3) computation of the volume of the fundamental domains of some high-dimensional lattices, and (4) the determination of the Hausdorff limit of the zero set of polynomials of the Fibonacci type. The talk allows you to switch off four times and to re-enter the same number of times.

The behaviour of particulate flow is mathematically modelled by population balance equations. The various terms of the equation model phenomena including particle transport, nucleation, growth, and aggregation. Their efficient numerical simulation requires sophisticated techniques, and various approaches proposed in the literature vary not only in computational complexity but also in the accuracy of the computed solutions. We will focus on the numerical treatment of aggregation integrals, the terms that model the aggregation process and which oftentimes dominate the overall simulation time. Within such a process, particles are characterized by a property coordinate x, e.g. the particle mass, the particle area, or the chemical composition, to mention only a few, and their distribution is quantified by a density distribution function f(x,t), which describes the property distribution of the particles at a given time t.

First, we discuss the evaluation of univariate aggregation integrals, where only one of particle characteristics is considered, and we discretise the property coordinate x through equidistant grids and approximate the density distribution f(x,t) through piecewise constant functions. Then, we extend the approach to grids with nested structures and approximation the density distribution through higher order polynomials (of degree p), which allow a better approximation. This novel approach reduces the quadratic complexity of its direct computation to an almost optimal complexity of order pNlogN with the problem size N. Furthermore, we also discuss examples of bivariate problems, where also a second property of particles is considered.

The key components of the developed algorithms are a separable approximation of the aggregation kernel, a nested grid consisting of piecewise uniform portions, application of FFT to compute the aggregation (convolution) on such uniform portions and orthogonality of basis functions which in combination lead to efficient recursion formulas. We provide extensive numerical tests for different initial setups to illustrate the performance of the developed algorithms with respect to their accuracy and efficiency, leading to (heuristic) strategies for the choice of discretization parameters.

Talk (PDF, 228KB)

Scattered data interpolation refers to an interpolation problem where the data sites are distributed irregularly within some domain. An interpolant may be constructed as a linear combination of radial basis functions centered at the data sites. Finding the coefficients in this representation leads to linear equations where the system matrices are large, dense, indefinite and ill-conditioned. These matrices can be approximated using the framework of hierarchical matrices. We will compare different approximation methods and discuss how to construct algebraic preconditioners.

Es werden große dünnbesetzte Sattelpunktprobleme betrachtet, wie sie z.B. in der Strömungsmechanik auftreten. Diese i.A. unsymmetrischen und indefiniten Systeme können mittels iterativer Krylovraum-Verfahren, inkl. geeigneter Präkonditionierer, gelöst werden. Insbesondere werden sogenannte induzierte Dimensions-Reduktions-Methoden (IDR), im Speziellen QMRIDR(s), verwendet, welche zusätzlich mit einem Deflationsansatz gepaart werden. Dabei werden Informationen aus früheren Durchläufen derart recycelt, sodass es möglich ist, Sequenzen von linearen Systemen effektiv zu lösen. Als Beispiel werden die diskretisierten Oseen-Gleichungen betrachtet; weitere Anwendung kann dies darüber hinaus z.B. bei inneren Punkte-Verfahren in der linearen Optimierung finden.

The goal of an a posteriori error analysis for an approximate PDE
solution is to establish the equivalence of error and a posteriori
estimator. Unfortunately, this equivalence is often only up to so-
called oscillation terms.

In this talk we shall clarify the reasons for the presence of
oscillation. Moreover, we propose a new approach to a posteriori error
estimation, where oscillation can be bounded by the error and so does
not longer spoil the aforementioned equivalence.

This is joint work with Christian Kreuzer (Bochum).

We present a space-time discontinuous Galerkin (DG) method for linear
wave propagation problems.
The special feature of the scheme is that it is a Trefftz method,
namely that trial and test functions are solution of the partial
differential equation to be discretised in each element of the
(space-time) mesh.
The DG scheme is defined for unstructured meshes whose internal faces
need not be aligned to the space-time axes.
The Trefftz approach can be used to improve and ease the
implementation of explicit schemes based on ``tent-pitched'' meshes.
We show that the scheme is well-posed, quasi-optimal and dissipative,
and prove a priori error bounds for general Trefftz discrete spaces.
A concrete discretisation can be obtained using piecewise polynomials
that satisfy the wave equation elementwise, for which we show high
orders of convergence.
If time allows, we will describe a similar Trefftz-DG method for the
Helmholtz equation, i.e. wave equation in time-harmonic regime, for
which non-polynomial basis functions are used and quite a complete
theory has been established.

Projektarbeit

The evolution of a plasma in external and self-consistent fields is modelled by the Vlasov equation for the distribution function in six dimensional phase space. Due to the high dimensionality and the development of small structures the numerical solution is very challenging. Grid-based methods
for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two or four dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the hierarchical Tucker format provides efficient basic linear algebra routines for low-rank representations of tensors.

In this talk, I will present a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Interpolation formulas for the low-parametric tensor format as well as efficient implementations will be discussed. Numerical simulations for the Vlasov-Poisson equation are shown for the Landau damping test case in two, four, and six dimensional phase space as well as simulations with a constant magnetic field. Depending on the test case, the memory
requirements reduce by a factor $10^2$-$10^3$ in four and a factor $10^5$-$10^6$ in six dimensions compared to the full-grid method.

Auxiliary space preconditioning targets elliptic boundary value problems discetized by means of finite elements. The idea is to use a related discrete boundary value problem, for which efficient solvers are available, as a preconditioner. The connection between both problems is established by means of a suitable prolongation operator.

We apply this strategy to variational problems for the bilinear form $(\alpha(x)\cdot,\cdot)_0+(\beta(x)curl\cdot,curl\cdot)_0$ ($\alpha,\beta$ uniformly positive coefficient functions) posed on the function space $H(curl)$ (or $H_0(curl)$).
These are commonly encountered in magneto-quasistatic models for electromagnetic phenomena (eddy current models). Finite element Galerkin discretization usually relies on Nedelec's $H(curl)$-conforming edge elements, but discontinuous Galerkin (DG) methods are a viable option, too. In any case, one faces large sparse linear systems of equations, for which efficient preconditioners are badly needed. Three settings will be discussed:

I) When edge elements are used on a single unstructured mesh, coarser meshes needed for the application of geometric multigrid solvers may not be available. They may be easy to construct, however, for a semi-structured mesh, suggesting the use of an auxiliary edge element space on that mesh.
II) In the same setting as (I), algebraic multigrid methods (AMG) could look promising. Alas, AMG schemes for edge finite element discretizations that match the performance of those for $H^{1}$-conforming finite elements are not available. To harness standard nodal AMG schemes one may use an auxiliary space of continuous piecewise polynomial vectorfields.
III) Using a DG discretization on a standard triangulation, which may be required in the context of magneto-hydrodynamics, an edge element space may serve as auxiliary space.

For all these cases we present theoretical results about the performance of the preconditioner with focus on $h$-independence and robustness with respect to jumps of the coefficients. The main ideas needed to verify the abstract assumptions of the theory of auxiliary space preconditioning will be outlined.

Die Untersuchung des Elektronen- und Quantentransports von Quasikristallen führt auf das Spektrum des Fibonacci Hamilton Operators. Auch mathematisch ist das Spektrum interessant: Es ist eine Cantor-Menge mit Lebesgue-Maß Null.
Mit Hilfe eines Algorithmus zur Bestimmung der Faktoren des Fibonacci-Wortes lässt sich das Spektrum, mit einer in dieser Arbeit vorgestellten Methode, approximieren.

Einige iterative Lösungsverfahren für lineare Gleichungssysteme sind auf die Anwendung auf symmetrisch (positiv definite) Systeme beschränkt. Wir werden theoretische Ansätze aus der Literatur diskutieren, wie nicht-symmetrische Gleichungssysteme symmetrisiert werden können, Möglichkeiten der Realisierung ausarbeiten und diese auf ihre Rechenzeit testen. Motiviert durch diese Ansätze und deren Resultate werden wir im Rahmen dieser Arbeit eine Modifizierung bzw. Kombination der Ansätze vornehmen und vergleichende Tests durchführen.

The standard Arnoldi method is one of the most popular schemes for reducing a large matrix A to a small one. The method requires the evaluation of matrix-vector products with A. Rational Arnoldi methods reduce the matrix A by both evaluating matrix-vector products and solving linear systems of equations with A. Rational Arnoldi methods are attractive to use when A has a structure that allows efficient solution linear systems of equations with A. They are commonly applied to the computation of an invariant subspace of A and to the approximation of matrix functions. We discuss implementations of rational Arnoldi methods and compares their properties.

Bei der numerischen Simulation physikalischer Prozesse treten häufig große parameterabhängige nichtlineare Gleichungssysteme auf. Zur Verringerung des Rechenaufwands werden oft Reduzierte-Basis-Methoden verwendet, die sich in lokale und globale Methoden unterscheiden lassen, wobei letztere Umkehrpunkte bezüglich des Parameters gewöhnlich nicht zulassen. In dieser Arbeit wird ein globaler, interpolationsbasierter Ansatz für Probleme mit Umkehrpunkten entwickelt und es werden die Vorteile und Grenzen dieser Methode aufgezeigt.

Viele Variationsmethoden in der mathematischen Bildverarbeitung nutzen die 1-Norm des Gradienten, die sogenannte Totalvariation, als Regularisierungsterm. Diese Totalvariation hat die Eigenschaft, Kanten im Bild zuzulassen und zu erhalten. Sie kann aber auch zur Entstehung von unerwünschten Kanten beitragen, dem sogenannten Staircasing-Effekt. Diese Arbeit soll die Huber-Funktion, eine Kombination zweier Normen, als mögliche Alternative vorstellen.

Bachelor-Vortrag

Bachelorvortrag

A linear operator $T$ on a Banach space is called Tadmor-Ritt if its spectrum is contained in the closed unit disc and the resolvent satisfies $C(T)=\sup_{|z|>1} \|(z-1)R(z,T)\|<\infty$. Such operators can be seen as discrete analog for sectorial operators.
We prove corresponding $H^{\infty}$-functional calculus estimates, which generalize and improve results by Vitse. Moreover, they are in conformity with the best so-far known power-bound for Tadmor-Ritt operators in terms of the constant $C(T)$.
We furthermore show the effect of having discrete square function estimates on the derived estimates.

We say a graph $G$ is universal for a set of graphs $\mathcal{H}$ if for each $H\in\mathcal{H}$ we have $H\subset G$. There are several results stating that the random graph $G(n,p)$ is universal for various classes of graphs $\mathcal{H}$, for appropriate functions $p=p(n)$. In order for $p$ not to be very close to one, we need the graphs in $\mathcal{H}$ to be quite sparse. There are then (at least) three natural graph classes one could consider: trees, graphs with bounded degree, and graphs with bounded degeneracy. I will outline the current state of knowledge (mainly due to other people) and sketch one or two proofs

Bachelor-Vortrag

After the introduction of random matrices to nuclear physics by Eugene Wigner in 1955, random quantum systems have grown in popularity. Wigner's idea was to consider families of Hamiltonians that underlie a certain probability distribution to describe overly complicated systems. Of particular interest are, of course, the spectra of these Hamiltonians. In this talk we consider random, in general non-self-adjoint, tridiagonal operators on the Hilbert space of square-summable sequences. To model randomness, we use an approach by Davies that eliminates all probabilistic arguments.

Despite the rising interest, not much is known about the spectra of non-self-adjoint random operators. The Feinberg-Zee random hopping matrix reveals this in a beautiful manner. The boundary of its spectrum appears to be fractal, but a proof has not been found yet. While we can not give a proof either, we present a reason why this is very plausible. Certain tridiagonal operators share remarkable symmetries that allow us to enlarge known subsets of the spectrum by sizeable amounts. In some cases like the Feinberg-Zee random hopping matrix, this implies that the spectrum contains an infinite sequence of Julia sets.

One of the most important and frequently used preconditioning techniques for solving symmetric positive definite systems is based on computing the approximate inverse factorizations. It is also a well-known fact that such factors can be computed column-wise by the orthogonalization process applied to the unit basis vectors provided that we use a non-standard inner product induced by the positive definite system matrix A. In this contribution we consider the classical Gram-Schmidt algorithm (CGS), the modified Gram-Schmidt algorithm (MGS) and also yet another variant of sequential orthogonalization, which is motivated originally by the AINV preconditioner and which uses oblique projections.

The orthogonality between computed vectors is crucial for the quality of the preconditioner constructed in the approximate inverse factorization. While for the case of the standard inner product there exists a complete rounding error analysis for all main orthogonalization schemes, the numerical properties of the schemes with a non-standard inner product are much less understood. We will formulate results on the loss of orthogonality and on the factorization error for all previously mentioned orthogonalization schemes.

This contribution is joint work with Jiří Kopal (Technical University Liberec), Miroslav Tůma and Alicja Smoktunowicz (Warsaw University of Technology).

Die von Peter Sonneveld erdachten Methoden, allen voran die neueste, IDR(s), können zur approximativen Eigenwertberechnung linearer Operatoren herangezogen werden. Im Gegensatz zu klassischen Krylovraumverfahren, welche Tridiagonal- oder Hessenbergmatrizen berechnen, berechnen Sonneveld-Methoden Büschel aus einer Band-Hessenbergmatrix und einer oberen Band-Dreiecksmatrix, von denen einige Eigenwerte bekannt sind. Basierend auf einer trivialen Beobachtung präsentieren wir Wege, die anderen Eigenwerte stabil zu berechnen.

Abstract:
We study the Decomposition Conjecture posed by Barát and Thomassen (2006), which states that for every tree T there exists a natural number k_T such that, if G is a k_T-edge-connected graph and |E(T)| divides |E(G)|, then G admits a decomposition into copies of T. This conjecture was verified for stars, some bistars, paths whose length is a power of 2, and paths of length 3. We verify the Decomposition Conjecture for paths of length 5. In this talk I will discuss the ideas behind the proof of this result.

Die von Peter Sonneveld erdachten Methoden, allen voran die neueste, IDR(s), können zur approximativen Eigenwertberechnung linearer Operatoren herangezogen werden. Im Gegensatz zu klassischen Krylovraumverfahren, welche Tridiagonal- oder Hessenbergmatrizen berechnen, berechnen Sonneveld-Methoden Büschel aus einer Band-Hessenbergmatrix und einer oberen Band-Dreiecksmatrix, von denen einige Eigenwerte bekannt sind. Basierend auf einer trivialen Beobachtung präsentieren wir Wege, die anderen Eigenwerte stabil zu berechnen.

Abstract:
The following is NP-hard in general:
given an edge-weighted finite graph and a set of special vertices,
compute a minimum-weight set of edges whose removal disconnects
any special vertex from any other special vertex.
Very efficient algorithms via LP-duality are known for natural subsets of graphs, though,
such as finite trees. Basic theoretical duality-type questions remain open for infinite trees.
To prepare for future talks on the problems about infinite trees,
I will explain an efficient algorithm solving the problem for finite trees.

Die von Peter Sonneveld erdachten Methoden, allen voran die neueste, IDR(s), können zur approximativen Eigenwertberechnung linearer Operatoren herangezogen werden. Im Gegensatz zu klassischen Krylovraumverfahren, welche Tridiagonal- oder Hessenbergmatrizen berechnen, berechnen Sonneveld-Methoden Büschel aus einer Band-Hessenbergmatrix und einer oberen Band-Dreiecksmatrix, von denen einige Eigenwerte bekannt sind. Basierend auf einer trivialen Beobachtung präsentieren wir Wege, die anderen Eigenwerte stabil zu berechnen.

Boundary integral equations are an important tool for analyzing elliptic partial differential equations arising, e.g., in structural mechanics or the simulation of acoustic or electromagnetic fields. Standard discretization techniques lead to large and densely populated matrices that require special algorithms.

The H²-matrix method offers efficient compression schemes for large matrices and can also perform algebraic operations like multiplication, inversion or factorization directly on the compressed matrices.

This talk gives an introduction to the basic concepts of H²-matrices and routlines two recent results: the Green hybrid compression scheme can be used to construct compressed approximations of discretized boundary element systems. Preconditioners for these systems can be constructed by applying a sequence of local low-rank updates to H²-matrices.

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead one may use a block decomposition. Approximating the smaller
block matrices by low-rank matrices and agglomerating them into a new, coarser
block decomposition, one obtains a recursive method. The required computation work is O(rnm) where r is the desired rank and n x m is the size of the matrix. New estimates are presented for the errors A-B and M-A,
where A is the result of the recursive truncation applied to M, while B is the best approximation.

Homogenization theory comprises the study of heterogeneous
materials. In mathematical terms this goes along with the discussion of
differential equations with oscillatory coefficients and the behavior of the
respective solutions, when the oscillations become infinitely fast. The aim in homogenization theory is to show convergence of the solutions for infinitely fast oscillations and to find an effective equation satisfied by the limit. In a Hilbert space setting, we discuss homogenization of ordinary differential equations and give an operator-theoretic reason, when it is likely that the limit equation is of integro-differential type -- in contrast to the equation one started out with. We also discuss possible generalizations to non-autonomous and/or partial differential equations.

Bachelor-Vortrag

Bachelor-Vortrag

In this talk, we discuss our recent progress on the famous directed cycle double cover conjecture of Jaeger. We define the class of trigraphs and prove that a graph connections conjecture formulated on trigraphs implies general Jaeger's conjecture. In addition, we give supporting evidence for our conjecture. This is joint work with Martin Loebl.

Given an $n$-element set which contains a known number $d$ of unknown special elements, we are allowed to use an oracle which accepts queries of size $k$ and responds positively if at least one of the elements of the queried set is in our set of unknowns. The case of a single unknown element has been studied and solved in the past by Rényi (1961), Katona (1966) and more recently by Hosszu, Tapolcai and Wiener (2013). We generalise their results in both the adaptive (on-line) and non-adaptive (parallelised) case for general d. Our approach provides new links between separability and (hyper-)graph girth, as well as new bounds for the problem.
This is joint work with F. Benevides, D. Gerbner and C. Palmer.

In Zusammenhang mit der Nullstellenlage von Polynomen welche ausschließlich nichtnegative Koeffizienten aufweisen wurden von Holtz und Tyaglov (SIAM Review, 2012) speziell strukturierte, unendliche Matrizen betrachtet, deren Minoren sämtlich nicht-negativ sind genau dann, wenn das Polynom nur negative Nullstellen besitzt.

Wir werden zum einen diese aufwendige Charakterisierung der
Nullstellenlage von Polynomen deutlich vereinfachen, desweiteren den Satz von Holtz und Tyaglov auf eine Klasse ganzer Funktionen ausweiten sowie den Bezug zu bekannten Klassen total nichtnegativer Matrizen herstellen.

Als mathematische Anwendungen erhalten wir einen einfachen, unabhängigen Beweis der Charakterisierung von Holtz-Tyaglov, eine neue Verknüpungseigenschaft der betrachteten Matrizen sowie eine Charakterisierung der Nullstellenlage spezieller ganzer Funktionen.

We consider the solution of a rather simple class of eigenvalue problems $Ax=\lambda{Mx}$ for symmetric positive definite matrices $A$,$M$ that stem, e.g., from the discretisation of a PDE eigenvalue problem. Thus, the problem is in principle simple, but the matrices $A$ and $M$ are large-scale and we would like to compute all relevant eigenvalues, where relevant is to be understood in the sense that all eigenvalues should be computed that can be captured by the discretisation of the continuous PDE eigenvalue problem.

We propose a new method for the solution of such eigenvalue problems.
The new method combines ideas of domain decomposition, as in the automated multi-level substructuring (short AMLS) or component mode synthesis, with the concept of hierarchical matrices (short $\cal{H}$-matrices) in order to obtain a solver that scales almost linearly (linear up to logarithmic factors) in the size of the discrete space, i.e. the size $N$ of the linear system times the number of sought eigenvectors. Whereas the classical AMLS method is very effective for PDEs posed in two dimensions, it is getting very expensive in the three-dimensional case, due to the fact that the interface coupling in the domain decomposition requires dense matrix operations. We resolve this problem by use of data-sparse hierarchical matrices. In addition to the discretisation error our new approach involves a projection error due to AMLS and an arithmetic error due to $\cal{H}$-matrix approximation. We will shortly analyse the complexity in theory and practice, and consider several numerical examples that underline the performance of the solver.

We prove that in finite element settings where the divergence-free subspace of the velocity space has optimal approximation properties, the solution of Chorin/Temam projection methods for Navier-Stokes equations equipped with grad-div stabilization with parameter $\gamma$, converge to the associated coupled method solution with rate $\gamma^{-1}$ as $\gamma\rightarrow \infty$. We prove this first for backward Euler schemes, and then extend the results to BDF2 schemes, and finally to schemes with outflow boundary conditions. Several numerical experiments are given which verify the convergence rate, and show how using projection methods in this setting with large grad-div stabilization parameters can dramatically improve accuracy.

Sometimes it is convenient to have a bi-infinite enumeration of the basis elements in the domain and image spaces of an operator A - leading to a representation of A by a bi-infinite matrix.
Shifting one of these enumerations shifts the matrix and hence changes the main diagonal. So which diagonal is ''the'' main diagonal? Isreal Gohberg once diplomatically said that in a bi-infinite matrix, it is every diagonal's right to claim to be the main diagonal. However, there are concrete problems in numerics and in matrix algebra that require a concrete choice - and, as it turns out, the choices coincide: From a certain point of view, there is one distinguished diagonal that deserves being called the main diagonal (a bit more than the others). We show how to find it and we discuss examples.

This is joint work with Gilbert Strang.

We adapt the grammar introduced by Chapuy, Fusy, Kang and Shoilekova to study bipartite graph families which are defined by their 3-connected components. More precisely, in this talk I will explain how to get the counting formulas for bipartite series-parallel graphs (and more generally of the Ising model over this family of graphs), as well as asymptotic estimates for the number of such graphs with a fixed size. This talk is based in a work in progress joint with Kerstin Weller.

... ist vielleicht nur für die Analytiker interessant.

We describe a general approach to the strong hypergraph regularity lemma, which we call 'regular slices', which avoids many of the usual technical complications and retains the features one would like to use in extremal hypergraph theory. This talk will avoid painful technical details in so far as that is possible and focus on an application, proving a hypergraph extension of the Erdos-Gallai theorem.

This is joint work with Julia Böttcher, Oliver Cooley and Richard Mycroft.

The blow-up lemma of Komlós, Sárközy and Szemerédi is an important tool for embedding large graphs H into dense graphs G. We recently obtained versions of this lemma for subgraphs G of sparse random and pseudo-random graphs. This has important applications in extremal graph theory on random graphs, but can also be used for the analysis of certain maker-breaker games.

In the talk I will explain our blow-up lemmas and describe their connection to maker-breaker games, after giving some necessary background.

Joint work with P. Allen, H. Hàn, Y. Kohayakawa, Y. Person.

High dimensionalities are a major roadblock for the numerical solution of problems in computational sciences. Straightforward discretizations are severely limited by the curse of dimensionality, the exponential dependency of the overall computational effort on the number of dimensions. It is therefore typically not feasible to treat more than four dimensions. In this talk, I will give a short introduction to Sparse Grids, which provide a versatile way to overcome the curse of dimensionality to a large extent, and show some of their applications. A special focus will be on spatially adaptive refinement, which adapts to the peculiarities of the problem at hand, and on adapted basis functions. Both are crucial whenever only few grid points can be spent, or where real-world problems do not meet the underlying smoothness requirements. The hierarchical basis formulation of the direct Sparse Grid approach conveniently provides a reasonable criterion for spatially adaptive refinement practically for free. This can serve as a starting point to develop suitable and problem-adapted modifications.

In this talk we survey the research activities and interests of the discrete maths group at TUHH.
We will discuss various topics such as colourings of embeddable graphs and hypergraphs, computational convexity, minimum bisection problems, and random graphs.

This talk gives an overview of some classical and recent uniqueness results in Discrete Tomography. In the first part we will review the concept of J-additivity and apply it to settle a problem of Kuba on 3-dimensional lattice sets and a conjecture of Brunetti and Daurat on planar lattice convex sets. The second part of the talk deals with the computational complexity of finding a smallest set of lattice positions of a given lattice set whose disclosure yields uniqueness w.r.t. some given X-rays. It turns out that this problem is already NP-hard in the plane and for the two standard directions.

This is joint work with Peter Gritzmann and Markus Wiegelmann.

According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields. Therefore, a new variational crime for the nonconforming Crouzeix-Raviart element is proposed, where divergence-free, lowest-order Raviart-Thomas velocity reconstructions reestablish L2-orthogonality. This approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

The numerical treatment of tensors and the use of tensors for various numerical problem has rapidly increased in the last time. It is now applied to many fields in analysis (treatment of pdes, representation of multivariate functions, etc.). The key for an efficient numerical treatment is a suitable format. We discuss the various formats, their properties, and operations with tensors.

Literature: W. H.: Tensor spaces and numerical tensor calculus. Springer 2012

Die Technik der Hierarchischen Matrizen H-Matrizen) ermöglicht die Berechnung einer approximativen H-Inversen oder H-LU-Zerlegung in fast linearer Komplexität und kann auf diese Weise zur effizienten Lösung linearer Gleichungssysteme eingesetzt werden. Vor der Verwendung der H-Matrix-Technik ist zu untersuchen, ob eine H-Matrix Approximation der Inversen bzw. der Faktoren der LU-Zerlegung existiert.
Resultate dieser Form konnten bereits für diverse Matrizen (z.B. Finite-Element-Matrizen) gezeigt werden, im Finite-Differenzen-Kontext sind jedoch keine Veröffentlichungen zum Einsatz der H-Matrix-Technik bekannt. Mit der Zielsetzung die Anwendbarkeit der H-Matrix-Technik für eine Finite-Differenzen-Matrix aus dem meteorologischen Transport- und Strömungsmodell METRAS zu untersuchen, wird in diesem Vortrag ein Resultat für Finite-Differenzen-Matrizen vorgestellt. Aufbauend auf dem methodischen Ansatz für Finite-Element-Matrizen wird die Existenz einer H-Matrix Approximation der Inversen von Finite-Differenzen-Matrizen gezeigt.
Die Ergebnisse können mittels numerischer Tests bestätigt werden. Bei Testproblemen, die in Anlehnung an das Gleichungssystem aus dem Modell METRAS aufgestellt werden, lässt sich im Einklang mit den theoretischen Ergebnissen jedoch eine Verschlechterung des Fehlerverlaufs in Abhängigkeit von einem Parameter feststellen. Für diese Fälle wird eine modifizierte Partitionierungsstrategie vorgestellt, deren Verwendung zu deutlich besseren Ergebnissen führt.

Die effiziente Darstellung und Approximation von Tensoren gewinnt in vielen Anwendungsbereichen der Mathematik wie Quantenchemie und -physik und auch generell innerhalb der Numerik immer mehr an Bedeutung. In diesem Vortrag werden wir ein neues Format zur Darstellung von hochdimensionalen
Tensoren vorstellen - das sogenannte Hierarchische Format oder auch Hierarchische Tucker-Format - und die grundlegende Arbeitsweise einer darauf basierenden inexakten Arithmetik erläutern. Der Schwerpunkt liegt auf der Approximation von Tensoren, sowie den Vorteilen des neuen Formates im Vergleich zu Standardformaten wie dem kanonischen Format oder der
Unterraum-/Tucker-Darstellung.

In this talk we describe the spectral properties of selfadjoint Schrödinger operators on unbounded domains with
an associated Dirichlet-to-Neumann map. In particular, a
characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogues of classical facts from singular
Sturm–Liouville theory.

We consider eigenvalue problems which are nonlinear in the eigenvalue
parameter. Newton-based methods are well-established techniques for determining individual eigenpairs of such nonlinear eigenvalue problems. If a larger number of eigenpairs is sought, however, the tendency of these methods to re-converge to previously discovered eigenpairs is a hindrance. In this talk, a deflation strategy for nonlinear eigenvalue problems will be presented, which overcomes this limitation in a natural way. Furthermore, we will comment on how this deflation approach can be implemented in a Jacobi-Davidson framework with only minimal overhead.

The concept of invariant subspaces is fundamental to linear eigenvalue problems and provides an important theoretical foundation in the design of numerical eigenvalue solvers. It turns out that there is no straightforward extension of this concept to eigenvalue problems that are nonlinear in the eigenvalue parameter. One obstacle is that eigenvectors belonging to different eigenvalues may become linearly dependent in the nonlinear case. Invariant pairs offer an elegant way to avoid this obstacle and appear to be the most natural extension of invariant subspaces. In this talk, we give an overview of the properties of invariant pairs and explain how they can be used in the design of numerical algorithms for nonlinear eigenvalue problems, as they arise for example in band diagram calculations for photonic crystals and fluid-structure interaction problems.

Ein klassischer Satz von Friedrichs besagt, dass Schrödingeroperatoren kompakte Resolvente besitzen, wenn das zugrundeliegende Potential bei Unendlich gegen Unendlich geht. In diesem Vortrag werden wir einen einfachen Beweis einer Verallgemeinerung präsentieren, basierend auf einer gemeinsamen Arbeit mit D. Lenz (Jena) und D. Wingert.

In their seminal 1952 paper on the conjugate gradient (CG) method Hestenes and Stiefel pointed out that their method, which is applicable to linear systems of equations with symmetric positive definite matrix only, is closely related to certain orthogonal polynomials, the corresponding Gauss quadrature formulas, certain continued fractions, and their convergents (or `partial sums'). The latter can be seen to be Padé approximants of a function that involves the resolvent of the matrix.

Around the same time, in 1950 and 1952, Cornelius Lanczos published two related articles, of which the second one introduced a precursor of the biconjugate gradient (BCG or BiCG) method, which generalizes CG to the case of a nonsymmetric system. Here, the residual polynomials are formal orthogonal polynomials only, but the connections to continued fractions and Padé approximants persist. Moreover, there is a relation to the qd algorithm of Rutishauser (1954). The understanding of all these connections became probably the key to Rutishauser's discovery of the LR algorithm (1955, 1958), which was later enhanced by John G.F. Francis to the ubiquitous QR algorithm (1961/62).

The IDR(s) method is a family of fast algorithms for iteratively solving large nonsymmetric linear systems. In the talk we will discuss an IDR(s) variant that is specifically tuned for parallel and grid computing. In particular in grid computing the inner product is a bottleneck operation. We will discuss three techniques that we have used to alleviate this bottleneck in IDR(s). Firstly, the efficient and stable IDR(s)-biortho method is reformulated in such a way that it has a single global synchronisation point per iteration step. Secondly, the so-called test matrix is chosen so that the work, communication, and storage involving this matrix is minimised in multi-cluster environments. Finally, a methodology is presented for a-priori estimation of the optimal value of s using only problem and machine--based parameters. We will also discuss a preconditioned version of IDR(s) that is particularly suited for grid computing. We will illustrate our results with numerical experiments on the DAS--3 Grid computer, which consists of five cluster computers located at geographically separated places in the Netherlands.

This is joint work with Tijmen Collignon.

The Lanczos algorithm was introduced in 1950 as means of solving eigenvalue problems. Despite its apparent elegance, the algorithm was initially neglected by the scientific community because it was observed to depart from its theoretical properties due to the effects of finite-precision computer arithmetic. The algorithm regained popularity several decades later when it was shown that despite its departure from theory, it nevertheless produces highly accurate eigenvalue estimates.

In my talk, I will briefly introduce the Lanczos algorithm and will present bounds characterizing the quality of eigenvalue estimates generated by the algorithm in exact arithmetic. Then, I will describe the difficulties of producing similar bounds in finite-precision arithmetic, and will present rounding error results, including recent ones, which overcome these difficulties.

We devised an IDR(s)-based SOR method and presented its effectiveness in view of efficiency and robustness by comparison with other iterative methods one year ago. In this talk, we consider the preconditioned IDR(s)-based Residual Reduction (R2) method as an extension of the IDR(s)-based SOR method in view of robust preconditioning. Moreover, we present numerical experiments that clearly show that our proposed IDR(s)-R2 method outperforms other approaches.

Lur'e equations are a generalization of algebraic Riccati equations and they arise in linear-quadratic optimal control with cost functional being singular in the input.
For Riccati equations, it is well-known that there is a one-to-one correspondence between set of solutions and certain Lagrangian eigenspaces of a Hamiltonian matrix.
The aim of this talk is to generalize this concept to Lur'e equations. We are led to the consideration of deflating subspaces of even matrix pencils.

1. A brief introduction about our group in Peking University
2. Seismic design of buildings with accidental eccentricity
3. Structural design of wind turbine blades

The Induced Dimension Reduction method was proposed in 1980 by Peter Sonneveld as an iterative method for solving large non-symmetric linear systems of equations. IDR can be considered as the predecessor of methods like CGS (Conjugate Gradient Squared [Sonneveld '89]) and Bi-CGSTAB (Bi-Conjugate Gradients STABilized [van der Vorst '92]). All three methods are based on efficient short recurrences. An important similarity between the methods is that they use orthogonalization with respect to a fixed `shadow residual'. Of the three methods, Bi-CGSTAB has gained the most popularity, and is probably still the most widely used short recurrence method for solving non-symmetric systems.

Recently, Sonneveld and van Gijzen revived the interest for IDR. In 2008, they demonstrate that a higher dimensional shadow space, defined by an n by s matrix tR_0, can easily be incorporated into IDR, yielding a highly effective method. Convergence (in terms of steps, or, equivalently, in terms of matrix-vector multiplications) is often comparable to GRMES, but in contrast to GMRES, this ''s version'' of IDR relies on short recurrences and all steps are equally fast.

The original IDR method is closely related to Bi-CGSTAB. It is therefore natural to ask whether Bi-CGSTAB can be extended to an ''s-version'' in a way similar to IDR. To answer this question we explore the relation between IDR and Bi-CGSTAB. Our findings lead to an abstract description of the IDR method. It shows that there is a lot of freedom in implementing , leading to variants that are mathematically equivalent. The implementational variants, however, may have different stability and efficiency properties.

Bi-CGSTAB relies on degree 1 stabilization polynomials. Higher degree stabilization polynomials can also be exploited as is shown by Sleijpen and Fokkema in 1993. The resulting method BiCGstab(L) is often more stable than Bi-CGSTAB leading the much faster convergence. As shown by Sleijpen, van Gijzen 2009 and Tanio, Sugihara 2009, higher degree stabilization polynomials can also be incorporated in IDR and it can greatly improve stability of IDR with degree 1 stabilization polynomials. We argue that this is another implementational variant of IDR.

This is joint work with Martin van Gijzen, Delft University of Technology, Delft, The Netherlands

Bekanntlich liefert ein Schriitt $(u,\theta) \mapsto u_+^{InvIt}$ der Inversen Iteration für das nichtlineare Eigenwertproblem $T(\lambda)x=0$ dieselbe Richtung wie ein Schritt $(u,\theta) \mapsto (u_+^{Newt},\theta_+^{Newt})$ des Newtonverfahrens für das erweiterte System $T(\lambda)x=0,\;w^Hx=1$ mit einem geeigneten Skalierungsvektor $w$, d.h., es gilt $\mbox{span}\,\{u_+^{InvIt}\}=\mbox{span}\,\{u_+^{Newt}\}$. Es liegt daher nahe, zur Abschätzung der Verbesserung der Eigenvektorapproximation $u$ durch die Inverse Iteration Newton-Techniken zu verwenden. Es wird gezeigt, dass dies zu genauen Abschätzungen führt, wenn explizit mit dem Restglied zweiter Ordnung gearbeitet und dessen spezielle Produktstruktur berücksichtigt wird wie das von \textsc{Heinz Unger} [50] erstmalig (und ohne publizierten Beweis) für das lineare Problem $T(\lambda)=A-\lambda I$ getan worden ist.

Durch Kombination mit neuen Abschätzunegn für das nichtlineare klassische bzw. verallgemeinerte Rayleigh-Funktional läßt sich dann einfach die quadratische Konvergenz
der nichtlinearen Rayleigh-Funktional-Iteration wie auch die kubische Konvergenz der nichtlinearen Verallgemeinerung der zweiseitigen Ostrowskischen Rayleigh-Quotienten-Iteration herleiten.

The motion of one or several rigid bodies in a viscous fluid occupying a bounded domain ­$\Omega in R^3$ represents an interesting theoretical problem featuring, among others, possible contacts of two or more solid objects. We consider the motion of several rigid bodies in a non-Newtonian fluid of a power-law type. Our main result establishes the existence of global-in-time solutions of the associated evolutionary system, when collisions of two or more rigid objects do not appear in a finite time unless they were present initially.

We solve a particular system of nonlinear ODEs defined on the two different components of the real line connected by the nonlinear contact condition
\[
w^\prime =h^\prime \;,\qquad h=\psi(w)\qquad\text{at the point $\,x=0\,$}.
\]
We show that, for a prescribed power-law nonlinearity $\psi$ and using the solution $(w,h)$, a self-similar solution to the porous medium equation in the two-component domain can be constructed.

The detection of a Hopf bifurcation in a large scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer et. al. (SINUM, 34, (1997) pp. 1-21) proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilises a certain sum of Kronecker products and involves the solution of matrices of squared dimension, which is impractical for large scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearisation process). The problem is formulated as an inexact inverse iteration method that requires the solution of a sequence of Lyapunov equations with low rank right hand sides. It is this last fact that makes the method feasible for large systems. The power of our method is tested on numerical examples.

The conventional SOR (Successive Over-Relaxation) method originated from the dissertation by D. Young in 1950. After that, the SOR method has been often used for the solution of problems which stem from various applications. The SOR method, however, has many issues on possibility of the solution because of no robustness of convergence of the SOR method.

Recently Sonneveld and van Gijzen brought epoch-making and renewed interest in the Induced Dimension Reduction (IDR) method in 2008. In addition, the Bi_IDR(s) method which was proposed by them is more elegant and stable than IDR(s) method. Furthermore, in 2009, IDR(s)Stab(L) and GBiCGStab(s,L) methods were independently proposed as one of the generalized version of IDR(s) method with polynomial of high degree L by Sleijpen and Tanio et al.

In my talk, we extend IDR Theorem to designing of the residual of the Jacobi, Gauss-Seidel and SOR methods, and accelerate their convergence rate and robustness. Through numerical experiments, we make clear improvement of performance of IDR-based Jacobi, Gauss-Seidel and SOR methods with parameters.

In this talk I will briefly review the generalized Riemann problem (GRP) method for compressible fluid flows. There were originally two versions of this method:
Lagrangian and Eulerian. The latter is always derived via a passage from the former. In our recent efforts, we developed a direct Eulerian GRP method using the ingredient of Riemann invariants. The main advantage is (1) to avoid the passage from the Lagrangian to Eulerian and thus easily to be extended into multidimensional cases; (2) treat sonic cases easily; and (3) conveniently combine with other techniques such as adaptive meshes.
We will also report some stability, convergence properties, and applications to shallow water equations on the sphere (earth).

In CFD, Atmospheric Sciences and Computational Aeroacoustics, many problems involve regions of discontinuity. When used to solve problems involving regions of shocks, dispersive schemes give rise to oscillations while dissipative schemes cause smearing, close to these regions of sharp gradients.

Based on the results of the 1-D shallow water problem, when solved using MCLF2, we observe that different cfl numbers yield results with different amount of dispersion and dissipation. This led us to devise a technique in order to locate the cfl number at which we can obtain results with efficient shock-capturing properties. This new technique involves the control of numerical effects of dispersion and dissipation in numerical schemes. We baptise this technique as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES. The cfl number at which dissipation curbs dispersion optimally is then located. It is termed as the optimal cfl.

We extend the concept of CDDES to that of Minimised Integrated Square Difference Error,(MISDE). The latter is an improved technique over the CDDES technique since it can be used to obtain two optimal parameters which are generally the cfl number and another variable, for efficient-shock capturing. Another technique of optimisation is devised which enables better control over the grade and balance of oscillation and dissipation to optimise parameters which regulate dispersion and dissipation effects. This technique is baptised as Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation, (MIEELDLD) and has advantages over the previous technique, MISDE.

A dimension reduction method called Discrete Empirical Interpolation (DEIM) will be presented and shown to dramatically reduce the computational complexity of the popular Proper Orthogonal Decomposition (POD) method for constructing reduced-order models for unsteady and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem.

I will describe DEIM as a modification of POD that reduces the complexity as well as the dimension of general nonlinear systems of ordinary differential equations (ODEs). It is, in particular, applicable to ODEs arising from finite difference discretization of unsteady time dependent PDE and/or parametrically dependent steady state problems. Our contribution is a greatly simplified description of Empirical Interpolation in a finite dimensional setting. The method possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the 1-D FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures non-linear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

The lecture will be concerned with the simulation of inviscid and viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow. The system of the governing equations is discretized in space by the discontinous Galerkin method. The time discretization is based on a semi-implicit linearized time stepping scheme, which leads to the solution of a linear algebraic system on each time level. As a result we get an efficient and robust numerical process. The applicability of the developed method will be demonstrated by some computational results obtained for flow in a channel with a moving wall and past an oscillating airfoil.

These results were obtained in cooperation with Vaclav Kucera and Jaroslava Prokopova from Charles University in Prague, Faculty of Mathematics and Physics, and Jaromir Horacek from Institute of Thermomechanics of Academy of Sciences of the Czech Republic.

The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was first developed by Sonneveld around 1979 and documented on three and a half pages of a 1980 proceedings paper by Wesseling and Sonneveld. Soon after IDR, Sonneveld introduced his widely applied Conjugate Gradient Squared (CGS) algorithm. Then, in 1990, van der Vorst suggested Bi-CGSTAB that he claimed to improve both those methods.

Bi-CGSTAB has become a method of choice for nonsymmetric linear systems, and it has been generalized in various ways in the hope of further improving its reliability and speed. Among these generalizations there is the ML(k)BiCGSTAB method of Yeung and Chan, which in the framework of block Lanczos methods can be understood as a variation of Bi-CGSTAB with right-hand side block size 1 and left-hand side block size k.

In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR is equally fast but preferable to Bi-CGSTAB, and that IDR(s) may be much faster than IDR = IDR(1). It turned out that IDR(s) is closely related to BiCGSTAB if s = 1 and to ML(s)BiCGSTAB if s > 1. In 2008, a new, particularly ingenious and elegant variant of IDR(s) has been proposed by the same authors.

In this talk we first try to explain the basic, seemingly quite general IDR approach, which differs completely from traditional approaches to Krylov space methods. Then we compare the basic properties of the above mentioned methods and discuss some of their connections.

I will first give a brief description of finite-volume, Godunov-type methods for hyperbolic systems of conservation laws. These methods consist of two types of schemes: upwind and central. My lecture will focus on the second type -- non-oscillatory central schemes.

Godunov-type schemes are projection-evolution methods. In these methods, the solution, at each time step, is interpolated by a (discontinuous) piecewise polynomial interpolant, which is then evolved to the next time level using the integral form of conservation laws. Therefore, in order to design an upwind scheme, (generalized) Riemann problems have to be (approximately) solved at each cell interface. This however may be hard or even impossible.

The main idea in the derivation of central schemes is to avoid solving Riemann problems by averaging over the wave fans generated at cell interfaces. This strategy leads to a family of universal numerical methods that can be applied as a black-box-solver to a wide variety of hyperbolic PDEs and related problems. At the same time, central schemes suffer from (relatively) high numerical viscosity, which can be reduced by incorporating of some upwinding information into the scheme derivation -- this leads to central-upwind schemes, which will be presented in the lecture.

During the talk, I will show a number of recent applications of the central schemes.

Uniqueness for weak solutions of conservation laws is based on the sign of the entropy production across discontinuos solutions. Although the entropy plays a fundamental role in the theory of hyperbolic systems, it is generally not used as a computational tool.
In this talk I describe how the numerical production of entropy induced by the discretization of the equations is a reliable indicator of the quality of the numerical solution. Thus the entropy production can be used as a regularity indicator, identifying the cells in which non linear limiters must be used to prevent the onset of spurious oscillations.
More quantitatively, when the solution is smooth, the entropy production has the same size of the local truncation error and can therefore be used as an a-posteriori error indicator to drive the construction of adaptive grids.

We start with a brief introduction to Conservation Laws and to their numerical solutions. Then we discuss the construction and manipulation of non-uniform meshes, using geometric properties of the numerical solution under consideration. Next, we examine properties (such as consistency, stability and order of accuracy) of numerical schemes over both uniform and non-uniform meshes. Finally, we combine a proper mesh selection mechanism with Entropy Conservative or oscillatory numerical schemes for the evolution step.

We will show the new rigid rod-like model in a polymeric fluid. The constitutive relations considered are motivated by the kinetic theory. The micro equation has five spatial freedom variables, two of them are in the configuration domain and the others are in the macro flow domain. It is obtained the local existence of the solution with large initial data and global existence of the solution with small Deborah and Reynolds constants in periodic domains. For the case of no flow we will give the structure of stationary solutions to the micro equation with Maier-Saupe potential on the sphere. The stationary solutions are shown to be necessarily a set of axially symmetric functions, and a complete classification of parameters for phase transitions to these stationary solutions is obtained. It is shown that the number of stationary solutions hinges on whether the potential intensity crosses two critical values 6.731393 and 7.5. Furthermore, we present explicit formulas for all stationary solutions. It is first theoretically proven that there is a hysteresis phenomenon when the non-dimensional potential intensity among particles changes. In the weak shear flow, we show that there exist many stable dynamic states: flow-aligning, tumbling, log-rolling and kayaking, which depend on the initial concentrated orientation of liquid crystal particles. Theoretical analysis is reported the first time that the Kayaking state does not circulate around a fixed direction but the asymmetric axis will periodically change.

In this article we compare the set of integer points in the homothetic copy ${n\Pi}$ of a lattice polytope ${\Pi\subseteq{{\mathbb R}}^d}$ with the set of all sums${ x_1+\ldots +xn}$ with ${x_1,\ldots,x_n\in \Pi\cap{{\mathbb Z}}^d}$ and ${n\in{{\mathbb N}}}$. We give conditions on the polytope ${\Pi}$ under which these two sets coincide and we discuss two notions of boundary for subsets of${{{\mathbb Z}}^d}$ or, more generally, subsets of a finitely generated discrete group.