On the penalization by the perimeter in shape optimization applied to Dirichlet inverse obstacle problem

2024/04/04 by Caubet, Fabien, Dambrine, Marc, Dardé, Jérémi
#35R30 35B35 49Q10 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2404.03536

Abstract

This paper is devoted to the understanding of regularisation process in the shape optimization approach to the so-called Dirichlet inverse obstacle problem for elliptic operators. More precisely, we study two different regularisations of the very classical shape optimization approach consisting in minimizing a mismatched functional. The first one is an implicit regularisation when working in the class of inclusion having a uniform ε-cone property, a natural class in shape optimization. As this regularity is not trivial to guarantee numerically, we discuss the regularisation by perimeter penalization. We show that this second regularisation provides a stability gain in the minimization process.

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