# Department Colloquium Winter 2018/19

## Talks in the winter semester 2018/19

At the Colloquium of the Department of Mathematics international researchers report on their work. The colloquium will be held on Wednesdays in "Hörsaal 3" (MI 00.06.011). We invite all interested parties.

**These are the lecture dates in the winter semester 2018/19:**

- 17 October 2018: Claudia Redenbach (16:00)
- 14 November 2018: Michael Dumbser (14:30), Holger Dullin (16:00)
- 9 January 2019: George Karniadakis (14:30), Vlad Vicol (16:00)
- 6 February 2019: Clotilde Fermanian Kammerer (14:30), Thomas Strohmer (16:00)

### Topics at the Department Colloquium

#### Taming non-convex optimization landscapes in data analysis

6 February, 16:00 - 17:00

**Prof. Thomas Strohmer, University of California Davis**

Non-convex optimization problems are the bottleneck in many appplications in science and technology. In my talk I will report on two recent breakthroughs in solving some important nonconvex optimization problems. The first example concerns blind deconvolution, a topic that pervades many areas of science and technology, including geophysics, medical imaging, and communications. Here, blind deconvolution refers to the problem of recovering a function f from the convolution of two unknown functions g and h. Blind deconvolution is obviously ill-posed and its optimization landscape is full of undesirable local minima.

I will first describe how I once failed to catch a murderer (dubbed the "graveyard murderer" by the media), because I failed in solving a blind deconvolution problem. I will then present a host of new algorithms to solve such nonconvex optimization problems.

The proposed methods come with theoretical guarantees, are numerically efficient, robust, and require little or no parameter tuning, thus making them useful for massive datasets.

The second example concerns the classical topics of data clustering and graph cuts.

I will discuss a convex relaxation approach, which gives rise to a rigorous theoretical analysis of graph cuts. I derive deterministic bounds of finding optimal graph cuts via a natural and intuitive spectral proximity condition. Moreover, our theory provides theoretical guarantees for spectral clustering and for community detection.

#### Microlocal interlude

6 February, 14:30 - 15:30

**Prof. Clotilde Fermanian Kammerer, Université Paris Est - Créteil Val de Marne, John von Neumann Professorin**

Some questions are at the threshold between harmonic analysis and partial differential equations and can be studied in the Euclidean space or in more complicated non-commutative settings like compact Lie groups such as the Heisenberg group. Our goal will be to use one of these questions in order to get familiar with the microlocal approach and to understand how it can be implemented in various frameworks.

#### Physics-Informed Learning Machines for Physical Systems

9 January, 14:30 - 15:30

**Prof. George Karniadakis, Brown University & MIT**

In this talk, we will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical systems and for discovering hidden physics from noisy data.

A key concept is the seamless fusion and integration of data of variable fidelity into the predictive models. First, we will present a Bayesian approach using Gaussian Process Regression (GPR), and subsequently a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between GPR and NNs and discuss the new powerful concept of meta-learning.

#### Wild solutions to the Navier-Stokes equation

9 January, 16:00 - 17:00

**Prof. Vlad Vicol, Courant Institute NYU**

We consider the uniqueness question for weak solutions of the 3D Navier-Stokes equation. We prove that solutions are not unique within the class of weak solutions with finite kinetic energy. Moreover, we prove that any continuous weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit.

#### New mathematical models and numerical algorithms for Newtonian and general relativistic continuum physics

14 November, 14:30 - 15:30

**Prof. Michael Dumbser, University of Trento**

In the first part of the talk we present a family of arbitrary high order accurate (ADER) finite volume and discontinous Galerkin finite element schemes for the numerical solution of a new unified first oder symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of Newtonian continuum physics, including a general description of fluid and solid mechanics as well as electro-magnetic fields in one single system of governing partial differential equations (PDE).

The model is based on previous work of Godunov, Peshkov and Romenski (so-called GPR model) on symmetric hyperbolic and thermodynamically compatible systems.

In the second part of the talk, we show a successful extension of the GPR model to general relativity, leading to a novel and unified first order hyperbolic formulation of general relativistic continuum mechanics. The model is able to describe nonlinear elasto-plastic solids, as well as ideal and non-ideal (viscous) fluids in full general relativity. Formal asymptotic expansion of the governing PDE reveals the structure of the viscous stress tensor in the asymptotic relaxation limit. The key features of the new model are its symmetric hyperbolicity and thermodynamical compatibility. The proposed PDE system is causal, covariant and has bounded signal speeds for all involved processes, including disspative ones. Since the new model also contains elastic solids as a special case, it should be understood as an *alternative *to existing models for vicous relativistic fluids that are usually derived from kinetic theory and extended irreversible thermodynamics. We present numerical results obtained with high oder ADER schemes for inviscid and viscous relativistic flows obtained in the stiff relaxation limit of the system, as well as results for solid mechanics.

In the last part of the talk we introduce a new, provably strongly hyperbolic first order reduction of the CCZ4 formalism of the Einstein field equations of general relativity and its solution with high order ADER discontinuous Galerkin finite element schemes.

#### The three body problem in four dimensions

14 November, 16:00 - 17:00

**Prof. Holger Dullin, University of Sydney**

The Newtonian three body problem has undergone a Renaissance in recent years. I will present an overview of old and new results on periodic solutions, symbolic dynamics, and chaos in this problem. Then I will describe new results about the symplectic symmetry reduction and dynamics of relative equilibria when the spatial dimension is at least four. In particular we will show that there are families of relative equilibria that are minima of the reduced Hamiltonian, and hence are Lyapunov stable. This establishes the first proof of Lyapunov stable periodic orbits in the three body problem, albeit in dimension four.

#### Anisotropy analysis of spatial point patterns

17 October, 16:00 - 17:00 Uhr

**Prof. Claudia Redenbach, Universität Kaiserslautern**

This talk will give an overview of techniques for detecting anisotropy in spatial point patterns. As an example of application, we will analyse the pore system in polar ice. In a depth below approx. 100 m, the ice contains isolated air bubbles which can be studied by using tomographic images of ice core samples. Interpreting the system of bubble centres as a realisation of a regular point process subject to geometric anisotropy, preferred directions and strength of compression can be estimated.

Previous topics at the Colloquium of the Department of Mathematics can be found on our Website.