Identification of nonlinearities in transport-diffusion models of crowded motion

Martin Burger; Jan-Frederik Pietschmann; Marie-Therese Wolfram

doi:10.3934/ipi.2013.7.1157

Article Contents

2013, Volume 7, Issue 4: 1157-1182. Doi: 10.3934/ipi.2013.7.1157

This issue Previous Article Analysis of the Hessian for inverse scattering problems. Part III: Inverse medium scattering of electromagnetic waves in three dimensions Next Article Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization

Identification of nonlinearities in transport-diffusion models of crowded motion

1.
Department for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster
2.
DAMTP, Center for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA

Received: February 29, 2012

Revised: April 30, 2013

Published: October 2013

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Abstract

The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied. In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.

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Mathematics Subject Classification: 35R30, 65N21, 35K60.

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