Vorträge

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.

Zoomlink:
https://tuhh.zoom.us/j/2180596134

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.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

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.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

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.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09