Februar 2017

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### 07.02.2017 14:15 Axel Voigt (TU Dresden):Orientational order on surfaces - the coupling of topology, geometry and dynamicsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

We consider the numerical investigation of surface bound orientational order using unit tangential vector fields by means of a gradient-flow equation of a weak surface Frank-Oseen energy. The energy is composed of intrinsic and extrinsic contributions, as well as a penalization term to enforce the unity of the vector field. Four different numerical discretizations, namely a discrete exterior calculus approach, a method based on vector spherical harmonics, a surface finite-element method, and an approach utilizing an implicit surface description, the diffuse interface method, are described and compared with each other for surfaces with Euler characteristic 2. We demonstrate the influence of geometric properties on realizations of the Poincare-Hopf theorem and show examples where the energy is decreased by introducing additional orientational defects. .

### 07.02.2017 15:30 Fred Chazal (INRIA, France):Persistent homology for data: stability and statistical propertiesMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Computational topology has recently seen an important development toward data analysis, giving birth to Topological Data Analysis. Persistent homology appears as a fundamental tool in this field. It is usually computed from filtrations built on top of data sets sampled from some unknown (metric) space, providing "topological signatures" revealing the structure of the underlying space. When the size of the sample is large, direct computation of persistent homology often suffers two issues. First, it becomes prohibitive due to the combinatorial size of the considered filtrations and, second, it appears to be very sensitive to noise and outliers. The goal of the talk is twofold. First, we will briefly introduce the notion of persistent homology and show how it can be used to infer relevant topological information from metric data through stability properties. Second, we will present a method to overcome the above mentioned computational and noise issues by computing persistent diagrams from several subsamples and combining them in order to efficiently infer robust and relevant topological information.

### 07.02.2017 16:00 Kiryung Lee (Georgia Tech):Nonconvex optimization methods for multichannel blind deconvolution with FIR sparse/subspace channel models02.12.20 (Boltzmannstr. 3, 85748 Garching)

Multichannel blind deconvolution resolves unknown input signal from multiple channel outputs with unknown impulse responses. It is often easier to estimate only the unknown channel impulse responses when they are modeled with only few parameters. This is the case with channel estimation problems in wireless communications and underwater acoustics. In particular with FIR models, various reconstruction methods based on statistics of the input signal and/or the commutativity of the convolution operator have been proposed with algebraic performance guarantees in 1990s. However, these guarantees are restricted to the case where the observations are noise-free or the input has infinite length. In fact, with finitely many noisy observations, the empirical performance of these classical methods deteriorates dramatically as the length of observation decreases. This weakness restricts their utility in estimating time-varying channels. Motivated from the observation that parametric models with few parameters can be embedded into low-dimensional subspaces, we propose modifications on the classical methods leveraging these subspace models. Our proposed method provides significant improvement over the original method in terms of providing stable estimates from finitely many noisy measurements. Furthermore, using recently developed tools in applied probability, we derive rigorous performance guarantees under certain random subspace models. We verify that the numerical results are aligned with our theory through Monte Carlo simulations. Furthermore, we also verify the empirical success in a realistic setup of channel delay estimation. Finally, we extend the methods and theory to the sparsity case and present corresponding theoretic and numerical results. This is the joint work with Felix Krahmer and Justin Romberg.

### 09.02.2017 15:45 Julian Fischer (IST Wien):Global existence and weak-strong uniqueness of renormalized solutions to entropy-dissipating reaction-diffusion systemsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

Consider a single reversible chemical reaction with mass-action kinetics. The corresponding reaction-diffusion equation then formally satisfies an entropy inequality. Nevertheless, in general the global existence of any kind of solution remained an open problem - even for smooth data - due to the possibly steep growth of the reaction terms. In the present talk, we establish the global existence of renormalized solutions for such entropy-dissipating reaction-diffusion equations under quite general assumptions on the data. The main challenge in our construction of solutions is the only weak convergence of the gradient of approximate solutions. Finally, we establish a weak-strong uniqueness result for renormalized solutions, based on a careful adaption of relative entropies.

### 09.02.2017 16:30 Bálint Tóth:Central Limit Theorem for Random Walks in Doubly Stochastic Random EnvironmentA 027 (Theresienstr. 39, 80333 München)

We prove annealed and quenched CLT under diffusive scaling for the displacement of a random walk on $Z^d$ in stationary and ergodic doubly stochastic random environment, under the $H_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. Based on joint work with Gady Kozma (Weizmann Institute).

### 09.02.2017 17:15 Dominik Jüstel (TU München):Radiation design: retrieval of structural symmetry from X-ray diffraction patternsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

X-ray crystallography is the method of choice for the analysis of molecular structures. Most known protein structures have been inferred from the characteristic discrete diffraction patterns of their crystalline forms. Based on a mathematical model for the diffraction of electromagnetic radiation, concepts of X-ray crystallography can be generalized to certain non-crystalline molecular structures by designing suitable radiation (joint work with Gero Friesecke, TUM, and Richard James, University of Minnesota, [1,2]). Potentially, these ideas might help to analyze molecular structures that are not accessible by current methods. In this talk I will discuss the possibility to retrieve 3D symmetry information from the 2D X-ray diffraction patterns, using methods from abstract harmonic analysis. More precisely, a non-abelian generalization of the Zak transform [3] allows to implement the abstract operator-valued Fourier transform on symmetry groups in three-dimensional space via time-harmonic solutions of Maxwell's equations. An application of a non-abelian generalization of the Poisson summation formula [3] reveals the symmetry information contained in the resulting diffraction patterns.

### 10.02.2017 13:00 Alexey Chernov, Universität Oldenburg:High order approximation with the Virtual Element MethodMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

The Virtual Element Method (VEM) is a very recent generalization of the Finite Element Method (FEM). VEM utilizes polygonal/polyhedral meshes in lieu of the classical triangular/tetrahedral and quadrilateral/hexaedral meshes. This automatically includes nonconvex elements, hanging nodes (enabling natural handling of interface problems with nonmatching grids), easy construction of adaptive meshes and efficient approximations of geometric data features.

In this talk we review the basic construction of the method and discuss an extension of VEM to * approximations of high order on quasiuniform polygonal grids (p-VEM) and * variable order approximations on geometrically refined polygonal grids (hp-VEM).

(Joint work with L. Mascotto, L. Beirao da Veiga, A. Russo)

### 10.02.2017 16:00 Bernhard Heim (GUTech Oman):Hurwitzsche Klassenzahlen, Verallgemeinerungen und neueste EntwicklungenMI HS 3 (Boltzmannstr. 3, 85748 Garching)

Adolf Hurwitz (1859-1919) trug wesentlich zur Entwicklung der Mathematik bei. Der Schüler von Felix Klein und Lehrer von David Hilbert beschaeftigte sich unter anderem mit Fragestellungen aus der Funktionentheorie (Riemannsche Flaechen) und algebraischer Zahlentheorie (Klassenzahlen). Er hatte die Einsicht, dass durch eine kleine aber wesentliche Neudefinition der schon von Gauss betrachteten Klassenzahlen, deren Eigenschaften transparenter werden und mit neuen Methoden untersucht werden können.

Meinen Vortrag moechte ich den Hurwitzschen Klassenzahlen widmen. Es soll deren Zusammenhang zu den klassischen Klassenzahlen hergestellt und erste grundlegende Eigenschaften vorgestellt werden.

H. Cohen hat in seiner Doktorabeit vor 40 Jahren einen Zusammenhang zu Modulformen halbganzen Gewichts hergestellt und als weitere Verallgemeinerung die heute nach ihm benannten Cohen Zahlen untersucht. Kürzlich ist es mir gelungen zu diesen Resultaten einen neuen Zugang zu finden. Diesen werde ich in dem Vortrag skizzieren.

### 10.02.2017 17:30 Christian Kühn (TUM, Antrittsvorlesung):Three Facets of Multiscale DynamicsMI HS 3 (Boltzmannstr. 3, 85748 Garching)

Multiscale dynamical systems occur in a wide variety of contexts in the natural sciences and engineering. Processes at different spatial and/or temporal scales provide a doubled-edged challenge for applied mathematics. On the one hand, having small parameters available in differential equations is one of the few highly efficient handles to treat a wide variety of nonlinear problems analytically. On the other hand, multiscale models are frequently more challenging numerically in comparison to single-scale models. In this talk I shall outline three deceptively simple cases, where the role of small parameters is crucial for theoretical as well as practical results: (I) a model describing autocatalytic chemical reactions with multiple time scales, (II) an evolution equation with small spatial nonlocality from mathematical biology, and (III) a model from climate science with small noise.

### 13.02.2017 16:15 Manon Baudel, Université d'Orléans, Frankreich:Spectral theory for random Poincaré mapsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting N asymptotically stable periodic orbits. To quantify the rare transitions between periodic orbits, we construct a discrete-time, continuous-space Markov chain, called a random Poincaré map. We show that this process admits exactly N eigenvalues which are exponentially close to 1, and provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighbourhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasistationary distributions of processes killed upon hitting some of these neighbourhoods. The proofs rely on Feynman-Kac -type representation formulas for eigenfunctions, Doob’s h-transform, spectral theory of compact operators, and a detailed balance property satisfied by committor functions. Joint work with Nils Berglund (Orléans).

### 13.02.2017 16:30 Dr. Benedikt Jahnel (WIAS Berlin):The Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocalityB 252 (Theresienstr. 39, 80333 München)

We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices.

We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-almost sure quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time $t_G>0$ the model is almost sure quasilocal. For the color-symmetric model there is no reentrance. On the constructive side, for all $t>t_G$, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.

### 20.02.2017 14:00 Tina Schmidt (TUHH):Über die Struktur von Bäumen ohne kleine BisektionMI 02.04.011 (Boltzmannstr. 3, 85748 Garching)

Eine Bisektion eines Graphen ist eine Partition seiner Knotenmenge in zwei gleich große Klassen, wobei die Weite angibt, wie viele Kanten des Graphens zwischen diesen Klassen verlaufen. Minimum Bisection ist das NP-schwere Problem, in einem gegebenen Graphen eine Bisektion mit kleinst möglicher Weite zu finden. Ziel dieses Vortrags ist es, die Struktur von Bäumen mit beschränktem Grad, die keine Bisektion mit konstanter Weite erlauben, zu untersuchen. Zuerst wird gezeigt, dass Bäume auf n Knoten mit einem Pfad der Länge n/4 eine Bisektion mit konstanter Weite erlauben. Danach wird erläutert, wie diese Ideen für beliebige Bäume verallgemeinert werden können um Bisektionen zu konstruieren, deren Weite vom Durchmesser des Baumes abhängt. Dies wiederum kann mit Hilfe von Baumzerlegungen für beliebige Graphen verallgemeinert werden.

### 20.02.2017 16:15 Philipp Würl (LMU):Talagrand’s Inequality and ApplicationsB 133 (Theresienstr. 39, 80333 München)

We consider probabilities of deviations for functions, which depend on multiple independent random variables, from a certain value, usually the expected value. In order to find upper bounds for these probabilities one initially used approaches depending on martingales. In 1994, however, M. Talagrand showcased a new way to prove such concentration inequalities in his paper "Concentration of measure and isoperimetric inequalities in product spaces". This meant a significant progress in this subject and in many cases also provided better results than previous methods. The paper at hand presents this approach and thereby Talagrand's convex distance inequality as well as two proofs of it. Moreover, the variety of application possibilities of Talagrand's convex distance inequality will be demonstrated with the help of several examples such as Bin Packing and the traveling salesman problem.

### 21.02.2017 14:00 Tina Schmidt (TUHH):Approximation des Minimum k-Section Problems in Bäumen mit linearem DurchmesserMI 02.04.011 (Boltzmannstr. 3, 85748 Garching)

Minimum k-Section bezeichnet das Problem, die Knotenmenge eines gegebenen Graphen in k gleich große Klassen aufzuteilen, sodass die Anzahl der Kanten zwischen diesen Klassen kleinst möglich ist. Selbst für Bäume auf n Knoten ist es NP-schwer, eine optimale Lösung des Minimum k-Section Problems bis auf einen Faktor von n^c zu approximieren, für jedes beliebige c<1. Ziel dieses Vortrags ist es zu zeigen, dass jeder Baum T auf n Knoten eine k-Sektion der Weite (k-1) (2 + 16n / diam(T)) * Delta(T) erlaubt. Diese Schranke impliziert einen polynomialen Algorithmus, der für Bäume mit beschränktem Grad und linearem Durchmesser eine optimale Lösung des Minimum k-Section Problem bis auf einen konstanten Faktor approximiert. Außerdem wird erläutert, wie die Methoden mit Hilfe von Baumzerlegungen verallgemeinert werden können.

### 22.02.2017 17:00 Piotr Swierczynski (TUM):Applications of energy-corrected finite element to optimal control and parabolic problemsGebäude 33, Raum 1401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

It is a well-known fact that the presence of re-entrant corners, i.e. corner with angle $\Theta > \pi$, in polygonal domains leads to the loss of regularity of solutions of elliptic problems[Kondratjev 1967]. This, in turn, means that only a suboptimal order of convergence of their standard piecewise linear finite element approximation can be obtained. Recently, an effective method of recovering the full second-order convergence for elliptic equations on domains with re-entrant corners, when measured in locally modified $L_2$ and $H^1$ norms, known as energy-correction, has been proposed[Egger, R\"ude, Wohlmuth 2014]. This method is based on a modification of a fixed number of entries in the system's stiffness matrix. In this talk, we present two applications of the energy-correction method.\\ Firstly, we show how the energy-correction method can be applied to finding an approximation of optimal Dirichlet boundary control problem on non-convex domains. We present the saddle-point structure of the problem and investigate the convergence properties of the method building on the work conducted in[Of, Phan, Steinbach 2015].\\ Secondly, we show how the energy-correction method can be applied to regain optimal convergence in weighted norms for parabolic problems and introduce a post-processing strategy yielding optimal convergence order in standard Sobolev norms. Standard discretization approach involving graded meshes results in a very restrictive form of a CFL condition, making the use of explicit time stepping practically impossible. On the other hand, the energy-correction can be used on uniform meshes, allowing for application of explicit time stepping scheme with relatively large time steps. This, combined with mass-lumping strategy, leads to a very efficient discretization of parabolic problems, where at each time step only one vector multiplication with a scaled stiffness matrix needs to be performed. Finally, we extend this idea to higher-order finite element methods.\\ All theoretical results are confirmed by the numerical tests.