Summer semester 23

​

• 03.05.23 Arnold Reusken (RWTH Aachen)
  Title: Partial differential equations on evolving surfaces: analysis and numerical methods 
  Abstract: In recent years there has been a strongly growing interest in the development and analysis of numerical simulation methods for PDEs on evolving surfaces. In the presentation we briefly discuss application fields in which such PDEs are relevant. We explain model problems for which well-posedness in a weak variational setting can be shown for the case of smoothly evolving surfaces. A main topic will be a class of numerical methods that can handle problems with topological singularities (e.g., droplet collision) in a robust way.

​

• 07.06.23 Chunxi Jiao (RWTH Aachen) in SFo 14
  Title: Solutions of stochastic Landau-Lifshitz-Slonczewski equations
  Abstract: The Landau-Lifshitz-Gilbert (LLG) equation and its stochastic counterpart have been intensively studied over the last two decades. On the other hand, the study of its variant: stochastic Landau-Lifshitz-Slonczewski (sLLS) equation, is still at an early stage. The sLLS equation models magnetisation under a spin-polarised fluctuating current which potentially triggers singularities and subsequent topological changes as observed in physics experiments. In this talk, we discuss the existence, (pathwise) uniqueness and regularity of solution of sLLS equation in the case of a ferromagnetic nanowire and a frustrated magnet.

​

• 21.06.23 Gaspard Kemlin (Uni Stuttgart)
  Title: Some recent work on the numerical analysis of two classes of nonlinear Schrödinger equations
  Abstract: This talk will be divided in two parts and I will discuss some recent results on the numerical analysis of some classes of nonlinear Schrödinger equations. 
  
  The first part of the talk will focus on Density functional theory (DFT): molecular simulation and electronic structure calculation are fundamental tools used in chemistry, condensed matter physics, molecular biology, materials science, nanoscience, etc. DFT is one of the most widely used methods today, offering a good compromise between efficiency and precision. It is a formidable problem, both mathematically and numerically, and requires a whole hierarchy of choices leading to a number of approximations and associated errors: choice of model, choice of discretisation basis, choice of solvers, truncation error, numerical error, etc. The aim of this first part is to present recent results dealing with these approximations. Firstly, I will introduce one of the most widely used models for electronic structure calculation. In concrete terms, this involves minimising a discrete energy on the Grassmann manifold, and I will then analyse the convergence of two classes of algorithms (direct minimisation methods and fixed point iterations) on this manifold. Then, we will see how the results established make it possible to obtain error bounds on quantities of practical interest (such as interatomic forces) or to improve the robustness of the calculation of response properties of materials. 
  
  The second part of the talk will focus on the numerical simulation of the Gross-Pitaevskii (GP) equation via vortex tracking, as part of the SFB 1481 CRC. The GP equation plays a central role in various models of superfluids and condensed matter physics. A dominating feature is the occurrence of quantized vortices that effectively evolve according to a Hamiltonian system in the limit of point-like vortices. In this part, I will present an abstract mathematical model for which the analytic theory of the limiting Hamiltonian system is well known. I will show recent updates on the numerical simulation of this Hamiltonian system as well as first ideas to efficiently localize numerically the vortices in order to bring the analytic theory to the computational level.

​

• 28.06.23 Michael Herty (RWTH Aachen)​
  Title: Trends in Nonlinear Optimization
  Abstract:We are interested in gradient-free optimization methods applicable also to non-differentiable cost functionals appearing in sparse recovery. The relation towards general inverse problems will also be discussed. The presented methods are based on the evolution of particles and convergence results are established using the corresponding mean field limit equation described by a possibly high-dimensional transport equations. Several extensions and relations to recent results will be discussed and numerical results will be presented. 

​

​

Winter semester 22/23

​

• 07.12.22 Arie Koster (RWTH Aachen)
  Title: Discrete Optimization - The high art of decision making
  Abstract: In this seminar, we will stride through the mathematical discipline of discrete optimization with sevenleague boots. We highlight not only the theoretical principles like complexity theory and approximation algorithms, but also the available toolbox to solve discrete optimization problems exactly. Optimization under uncertainty completes the talk.
   

• 14.12.22 Markus Bachmayr (RWTH Aachen)
  Title: Multilevel representations of random fields
  Abstract: We consider multilevel representations of stationary random fields on domains and of isotropic random fields on the sphere, implications for sampling such random fields, and how such representations can be utilized in the context of sparse polynomial expansions of solutions of random PDEs. Based on joint works with Albert Cohen, Ron DeVore, Ana Djurdjevac, Dinh Dung, Giovanni Migliorati, Christoph Schwab, and Igor Voulis.
   

• 25.01.23 Benjamin Berkels (RWTH Aachen)
  Title: From variational exit wave reconstruction to deep unrolling
  Abstract: We first revisit the so-called exit wave reconstruction problem in the variational setting. Here, exit wave reconstruction means to reconstruct the complex-valued electron wave in a transmission electron microscope (TEM) right before it passes the objective lens, i.e., the exit wave, from a series of real-valued TEM images acquired with varying focus. This is a non-linear inverse problem that is a variant of the well known phase retrieval problem. We will show existence of minimizers, discuss practical gradient based optimization algorithms and show results on real data. Moreover, we will show how a regularizer for this non-linear inverse problem can be learned from data in the form of the Total Deep Variation. 
  In the second part of the talk, we will turn to deep algorithm unrolling, a general strategy that allows to re-interpret iterative algorithms as deep neural networks with a very special structure inherited from the algorithm that is unrolled. In particular, unrolling allows to introduce data-driven learning to the unrolled algorithm. To this end, we will discuss the proximal gradient algorithm, which is suitable for unrolling and also applicable to exit wave reconstruction. 
   

• 01.02.23 Umberto Hryniewicz (RWTH Aachen)
  Title: Classical Morse theory and deep learning
  Abstract: The goal of this talk is to bring ideas from classical Morse theory closer to some problems arising in deep learning. We would like to discuss possible implications that certain statements about gradient flows of proper real-analytic functions might have in the analysis of “linear” deep learning schemes. An example of such a statement would be that a “typical” anti-gradient trajectory of a proper real-analytic function which is bounded from below converges necessary to a local minimum. The analogous version of this statement holds for a generic smooth function.

​