Optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements

Abstract

We propose an original optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenvectors associated with the transmission problem for the Laplacian. Let Ω be a smooth domain and D be an inclusion contained in Ω whose boundary is also assumed to be smooth. Shape deformation of D causes a perturbation of modal parameters. The aim of this work is to show how this information can be used to reconstruct the unknown deformation. For doing so, we rigorously derive an asymptotic formula for the perturbations in the eigenvalues of the transmission problem for the Laplacian that are due to small deformations of the interface of an inclusion. Based on this formula, we design an efficient reconstruction algorithm from modal measurements. Our algorithm consists on minimizing a functional whose minimizer yields certain geometric properties of the unknown inclusion. It naturally follows from a key identity that is in some sense dual to the asymptotic formula. Numerical experiments showing the viability of our algorithm are presented. Our results in this paper extend those established in the context of small volume inclusions as well as those for the conductivity interface problem. In fact, on one hand, in a series of recent papers [5,3,1,2] we have derived high-order asymptotic expansions of the eigenvalue perturbations due to the presence of small inclusions and used them for locating the inclusions and identifying some of their geometric features. On the other hand, in [4], we have derived highorder terms in the asymptotic expansions of the boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion. Based on these derivations, we have designed an effective algorithm to determine some geometric features of the shape perturbation of the inclusion based on boundary measurements.