# APDE: Seminars

## A brief overview of PDEs on graphs

Recent years have seen a lot of interest in partial differential equations discretised on arbitrary network structures. Especially driven by applications in data analysis and image processing, models such as the graph-based Ginzburg-Landau functional and variations on it have been used for processes such as data clustering, image segmentation, and maximum-cut computations. In this talk I will give a brief overview of some of the theory and applications in this area.

## Recent Results on Multiply Monotone Radial Functions

The purpose of this talk is to establish new results on interpolation to continuous functions of multiple variables. For this, radial basis function interpolation is most useful, as it provides always regular, indeed positive definite or conditionally positive definite collocation matrices, independent of the spatial dimension and the geometry of the data points we wish to work with.

These interpolants are used for example to solve partial differential equations by collocation.

Specifically, we aim to classify radial basis and other functions that are useful for scattered data interpolation from vector spaces spanned by translates of basis functions (kernels) in any (high) dimensional space, and to this end we study so-called multiply monotonic functions. We collect special properties of such multiply monotonic functions, generalise them and find larger classes than the well known completely monotone functions for multivariate interpolation. Furthermore, we generalise other results recently established for completely monotone functions to this class of functions. (Joint work with Janin Jäger.)

## Numerical solution of the radiative transfer equation with a posteriori error bounds

We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key ingredient is the formulation of an iteration in a suitable (infinite dimensional) function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is based on approximately realizing this outer iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. On the one hand, since in the course of this iteration the global scattering operator is only applied, this avoids solving linear systems with densely populated system matrices while only linear transport equations need to be solved. This, in turn, rests on a Discontinous Petrov—Galerkin scheme which comes with rigorous a posteriori error bounds. These bounds are crucial for guaranteeing the convergence of the outer iteration. Moreover, the application of the global (scattering) operator is accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with a non-trivial scattering kernel.

## TBA

Abstract not available

## Graduate Seminars (WE 15:00, MR5)

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