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Total variation and phase field regularisations of an inverse problem with quasilinear magentostatic equations

Abstract

We tackle the inverse problem of reconstructing a discontinuous coefficient in magnetostatic equations from measurements in a subdomain. This problem is motivated from non-invasive/non-destructive testing based on electromagnetic phenomena identifying the location of magnetic materials (e.g. iron) in a bounded domain containing also non-magnetic materials (e.g. copper). As the inverse problem is likely to be ill-posed, we reformulate it into a constraint minimisation problem with perimeter penalisation and also with phase field regularisation. We show existence of minimisers, stability with respect to data perturbation, and connect these problems via asymptotic limits as the penalisation parameters tend to zero.  Our results also include the convergence of optimality conditions, which demonstrates the usefulness of the phase field approach as a method to solve this geometric inverse problem.  This is a joint work with Irwin Yousept from Duisburg-Essen, Germany.