Kolloquium für Angewandte Mathematik

Peter Düben

European Centre for Medium-Range Weather Forecasts,

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.

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Philip Freese

Institut für Mathematik, Universität Augsburg,

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.

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Martin Weiser

ZIB,

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.

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Winnifried Wollner

Universität Hamburg,

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)

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Giovanni Samaey

KU Leuven,

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

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Wisdom Agboh

University of Leeds,

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

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Ivan Veselic

TU Dortmund, Fakultät für Mathematik, Lehrstuhl LSIX,

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.

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Malin Lachmann

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.

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Dr. Helmut Linde

Merck KGaA, Darmstadt, Germany

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

Helmut Linde is the Global Head of Data Science at Merck KGaA (Darmstadt, Germany). He is responsible to drive innovation in the field of artificial intelligence and to ensure that the company leverages the potential of these modern technologies. Before joining Merck, Helmut has held various leadership roles at the software company SAP in the area of data science, overseeing the development and implementation of customer-specific forecasting and optimization solutions. Helmut has an academic research background in mathematical physics.

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Thibaut Lunet

Université de Genève,

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.

https://tuhh.zoom.us/j/81267321365?pwd=LzQ1cWtESXlRYXFpSFU3VkxPL08zdz09

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Dr. Adeleke Bankole

Institute of Mathematics, Hamburg University,

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).

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Prof. Dr. Marek Behr

Chair for Computational Analysis of Technical Systems (CATS), RWTH Aachen University,

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.

- Univ.-Prof. (C4) at the Chair for Computational Analysis of Technical Systems in the Faculty of Mechanical Engineering of the RWTH Aachen University since 2004
- Founding director of the Center for Simulation and Data Science since 2017
- Speaker of International Research Training Group 2379 Modern Inverse Problems since 2018
- Scientific director of Aachen Institute for Advanced Study in Computational Engineering Science since 2006
- Adjunct Professor of Chemical and Biomolecular Engineering at Rice University since 2005

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Dr. Sebastian Götschel

Zuse Institut Berlin (ZIB),

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).

https://www.zib.de/members/goetschel

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Sabine Le Borne

Technische Universität Hamburg, Institut für Mathematik, Lehrstuhl Numerische Mathematik, Am Schwarzenberg-Campus 3, Gebäude E, 21073 Hamburg

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Christian Budde

Bergische Universität Wuppertal, Arbeitsgruppe Funktionalanalysis,

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.

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Marco Peruzzetto

Christian-Albrechts-Universität zu Kiel, Arbeitsbereich Analysis,

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).

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Ines Dorschky

Fachbereich Mathematik, Universität Hamburg,

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.

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Christian Seifert

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.

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Dennis Gallaun

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.

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Karsten Kruse

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.

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Jan Meichsner

Institut fuer Mathematik, Lehrstuhl angewandte Analysis, TUHH,

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.

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Sebastian Bechtel

Arbeitsgruppe Analysis, TU Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt

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.

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Benedikt Diederichs

Fachbereich Mathematik, Universität Hamburg,

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.

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Albrecht Seelmann

Fakultät für Mathematik - Technische Universität Dortmund,

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.

http://www.mathematik.tu-dortmund.de/lsix/people/seelmann/index.php

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Jan Meichsner

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.

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Stefan Kunis

Institut für Mathematik, Uni Osnabrück,

Academic positions

since 10/2010 Professor for Applied and numerical analysis, University Osnabrück

10/2009 - 05/2016 Head of the young investigators group: Fast algorithms for biomedical imaging, jointly at Helmholtz Center Munich
02/2010 - 09/2010 Juniorprofessor for Fourier analysis, Chemnitz University of Technology
04/2006 - 02/2010 Research assistant, Chemnitz University of Technology

Education

08/2006 Doctorate degree, University Lübeck
05/2005 - 03/2006 University Vienna
09/2003 - 04/2005 University Lübeck

08/2003 Diplom in Computer science
01/2002 - 06/2002 University Linköping
10/1998 - 08/2003 University Lübeck

07/1997 Abitur

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Jan Meichsner

Institut fuer Mathematik, Lehrstuhl angewandte Analysis, TUHH,

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.

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Jörn Behrens

UHH,

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.

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Karsten Kruse

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.

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Jan Meichsner

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.

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Albrecht Böttcher

TU Chemnitz,

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.

214 Paper
9 Bücher
- weiß *alles* über Toeplitzmatrizen und -operatoren
- hält großartige Vorträge

https://www-user.tu-chemnitz.de/~aboettch/

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Dr. Katharina Kormann

Technische Universität München, Zentrum Mathematik - M16, Boltzmannstraße 3, 85747 Garching, Germany

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.

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Ralf Hiptmair

ETH Zürich,

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.

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Prof. Lothar Reichel

Department of Mathematical Sciences, Kent State University, Ohio, USA

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.

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Hagen Eichel

Eröffnung des Promotionsverfahrens,

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.

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Miro Rozložník

Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic

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).

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Steffen Börm

Christian-Albrechts-Universität Kiel,

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.

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Wolfgang Hackbusch

Max-Planck-Institut für Mathematik in den Naturwissenschaften,

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.

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Lars Grasedyck

RWTH Aachen,

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.

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Leo Rebholz

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.

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Martin Stoll

MPI Magdeburg,

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Alexander Linke

WIAS Berlin,

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.

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Wolfgang Hackbusch

Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

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

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Dominik Enseleit

UHH, UHH

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.

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Jussi Behrndt

TU Graz, Österreich

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.

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Cedric Effenberger

École polytechnique fédérale de Lausanne EPFL, Lausanne

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.

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Prof. Dr. Daniel Kressner

École polytechnique fédérale de Lausanne EPFL, Lausanne

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.

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Peter Stollmann

TU Chemnitz, TU Chemnitz, Fakultät für Mathematik, 09107 Chemnitz

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.

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Martin Gutknecht

ETH Zurich; Seminar for Applied Mathematics, LEO D3 (Leonhardstrasse 27), 8092 Zurich, Switzerland

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).

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Martin van Gijzen

Delft University of Technology; Faculty of Electrical Engineering, Mathematics and Computer Science, Mekelweg 4; 2628 CD Delft; The Netherlands

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.

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Ivo Panayotov

Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, England

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.

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Prof. Sarka Necasova

Institute of Mathematics of the Academy of Sciences, Praha, Czech Republic

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.

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Prof. Jan Filo

Comenius University, Bratislava, Slovak Republic

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.

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Prof. Dr. Karl Meerbergen

Katholieke Universiteit, Leuven

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.

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Prof. Jiequan Li

School of Mathematics, Capital Normal University, Beijing, China

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).

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Dr. Bippine Appadu

University of Mauritius, Reduit, Mauritius

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.

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Prof. D. C. Sorensen

Rice University, Houston, Texas

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.

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Prof. Martin Gutknecht

Seminar for Applied Mathematics, ETH Zurich

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.

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