Termine Angewandte Mathematik Münster

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DUNE, the Distributed and Unified Numerics Environment, offers a rich body of functionality to implement all kinds of numerical methods to solve partial differential equations. A particular feature of DUNE are the well designed interfaces, which allow a clear separation between algorithm and data-structures. We introduce into the concepts of DUNE, discuss some of the interfaces and present examples.

One-dimensional Calogero-Moser-Sutherland particle models with N particles can be regarded as diffusions on suitable subsets of $\mathbb R^N$ like Weyl chambers and alcoves with second order differential operators as generators which are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman-Opdam processes which are related to special functions associated with root systems. These models include Dyson's Brownian motions and multivariate Jacobi processes and, for fixed times, $\beta$-Hermite, Laguerre, and Jacobi ensembles. The processes depend on parameters which have the interpretation of an inverse temperature. We review several freezing limits for fixed N when one or several parameters tend to $\infty$. Usually, the limits are normal distributions and, in the process case, Gaussian processes where the parameters of the limit distributions are described in terms of solutions of ordinary differential equations which appear as frozen versions of the particle diffusions. We also discuss connections of these ODEs with the zeros of the classical orthogonal polynomials and polynomial solutions of some associated one-dimensional inverse heat equations.

While attention for and usage of Deep Neural Network (DNN) based applications skyrocket, the mathematical understanding of their behavior and capabilities is still in its infancy. Contrary to traditional approaches, that depend on training by loss minimization algorithms, a method will be presented to explicitly construct DNNs that emulate multivariate Chebyshev polynomials and can be used to approximate a large class of functions. The theory of this method, it's accuracy and it's bounds on depth and size will be introduced as well as an implementation and comparison to training-based DNNs.

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I will discuss (compound) Poisson process approximation for stabilizing statistics of a stationary strongly mixing point process. The main results are formulated in a Wasserstein distance and are based on a general bound on the total variation distance of a stationary point process and its Palm measure. The new findings are applied to minimal angles in the stationary Poisson-Delaunay triangulation. In this example, the asymptotic cluster size distribution is explicit and compound Poisson process approximation is established with an explicit convergence rate. The talk is based on joint work with Nicolas Chenavier.

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