Shape and parameter reconstruction for the Robin transmission inverse problem

Antoine Laurain 1  and Houcine Meftahi 2
  • 1 Technical University of Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany; and Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, 05508-090 – São Paulo, Brazil
  • 2 Technical University of Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Antoine Laurain and Houcine Meftahi

Abstract

In this paper we consider the inverse problem of simultaneously reconstructing the interface where the jump of the conductivity occurs and the Robin parameter for a transmission problem with piecewise constant conductivity and Robin-type transmission conditions on the interface. We propose a reconstruction method based on a shape optimization approach and compare the results obtained using two different types of shape functionals. The reformulation of the shape optimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup combined with a function space parameterization technique. The reconstruction is then performed by means of an iterative algorithm based on a conjugate shape gradient method combined with a level set approach. To conclude we give and discuss several numerical examples.

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