Schedule & Talks

Based on a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models, we first show the well-posedness of the considered class of scalar conservation laws. One reasonable extension of the nonlocal traffic model includes the investigation of a stochastic velocity function. A numerical analysis offers insights how the stochasticity affects the evolution of densities. Finally, we present recent results how learning techniques may affect the nonlocal flow dynamics.

This joint talk with Michael Multerer, Olaf Schenk, and Christoph Schwab is dedicated to recent developments in the field of wavelet analysis for scattered data. We introduce the concept of samplets, which are signed measures of wavelet type and may be defined on sets of arbitrarily distributed data sites in possibly high dimension. By employing samplets, we transfer well-known concepts known from wavelet analysis, namely the fast basis transform, data compression, operator compression and operator arithmetics to scattered data problems. Especially, samplet matrix compression facilitates the rapid solution of scattered data interpolation problems, even for kernel functions with nonlocal support. Finally, we demonstrate that sparsity constraints for scattered data approximation problems become meaningful and can efficiently be solved in samplet coordinates.

In this talk, we present new results on the sample size required to learn surrogates of nonlinear mappings between   infinite-dimensional Hilbert spaces. Such surrogate models have a wide range of applications and can be used in uncertainty quantification and parameter estimation problems in fields such as classical mechanics, fluid mechanics, electrodynamics, earth sciences etc. Here, the operator input determines the problem configuration, such as initial conditions, material properties, or forcing terms of a partial differential equation (PDE) governing the underlying physics. The operator output corresponds to the PDE solution. Our analysis shows that, for certain neural network architectures, empirical risk minimization can overcome the curse of dimensionality. Specifically, we show that both the number of network parameters and the quantity of input-output data pairs required for training remain manageable, with the error converging at an algebraic rate.

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