Unless otherwise stated, the colloquium talks will take place at 2:30pm in the lecture hall "SeMath" at Pontdriesch 14
28.10.22 Daniel Robertz (RWTH Aachen)
Title: Differential algebra for the study of singular behavior and of difference approximations of PDE systems
Abstract: Given a system of linear or nonlinear partial differential equations, various tasks like determining all power series solutions, finding all compatibility conditions, or deciding whether another given equation is a consequence of the system, require formal manipulation of the system. The Thomas decomposition method lends itself to answering such questions. It splits a differential system into finitely many so-called simple differential systems, which are formally integrable, and whose sets of analytic solutions form a partition of the original solution set. This talk gives an introduction to this method and presents a few applications: detection of singularities of differential systems, algebraic study of structural properties of nonlinear control systems, consistency check for finite difference approximations to PDE systems.
25.11.22 Sebastian Reich (Uni Potsdam)* in room R5
Title: Statistical inverse problems and affine-invariant gradient flow structures in the space of probability measures
Abstract: Statistical inverse problems lead to complex optimisation and/or Monte Carlo sampling problems. Gradient descent and Langevin samplers provide examples of widely used algorithms. In my talk, I will discuss recent results on sampling algorithms, which can be viewed as interacting particle systems, and their mean-field limits. I will highlight the geometric structure of these mean-field equations within the, so called, Otto calculus, that is, a gradient flow structure in the space of probability measures. Affine invariance is an important outcome of recent work on the subject, a property shared by Newton’s method but not by gradient descent or ordinary Langevin samplers. The emerging affine invariant gradient flow structures allow us to discuss coupling-based Bayesian inference methods, such as the ensemble Kalman filter, as well as invariance-of-measure-based inference methods, such as preconditioned Langevin dynamics, within a common mathematical framework. The talk is based joint research work with Nik Nüsken, Simon Weißmann, Andrew Stuart, Daniel Zhengyu Huang, Jiaoyang Huang, Zifan Chen, and Edoardo Calvello. Most results have been summarised in a recent survey paper available from arXiv2209.11371.
9.12.22 Virginie Ehrlacher (Ecole des Ponts Paris Tech)* in room epH
Title: Moment constrained optimal transport problem: application to quantum chemistry
Abstract: This work is motivated by applications in quantum chemistry, for the computation of the electronic structure of molecules. The so-called Density Functional Theory (DFT) is a very powerful framework which enables to carry out such computations. Within this theory, a key role is played by the so-called Levy-Lieb functional, the computation of which remains unaffordable for systems with a large number of electrons. This is why a full zoology of approximate DFT models, relying on the use of various approximations of this Levy-Lieb functional, have been proposed in the chemistry literature. In this talk, a specific focus will be made on one particular DFT model which makes use of the so-called semi-classical limit of the Lévy-Lieb functional, which happens to read as a symmetric multimarginal optimal transport problem with Coulomb cost, the number of marginals being equal to the number of electrons in the system. In this talk, I will present recent results about a new approach for the resolution of multi-marginal optimal transport problems which consists in relaxing the marginal constraints into a finite number of moment constraints. Using Tchakhaloff's theorem, it is possible to prove the existence of minimizers of this relaxed problem and characterize them as discrete measures charging a number of points which scales independently of the number of electrons. This opens the way towards the design of new numerical schemes which can hopefully circumvent the curse of dimensionality for this problem. Preliminary numerical results will be presented.
16.12.22 Gitta Kutyniok (LMU München)
Title: Reliable AI: Successes, Challenges, and Limitations
Abstract: Artificial intelligence is currently leading to one breakthrough after the other, both in public life with, for instance, autonomous driving and speech recognition, and in the sciences in areas such as medical diagnostics or molecular dynamics. However, one current major drawback is the lack of reliability of such methodologies. In this lecture we will first provide an introduction into this vibrant research area, focussing specifically on deep neural networks. We will then survey recent advances, in particular, concerning generalization guarantees and explainability. Finally, we will discuss fundamental limitations of deep neural networks and related approaches in terms of computability, which seriously affects their reliability.
27.1.23 Bernd Sturmfels (MPI Leipzig and UC Berkeley)
*: The colloquium talk will take place in a different lecture hall.