Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics

The aim of this study is an extension and employment of the concept of topological derivative as it pertains to the nucleation of infinitesimal inclusions in a reference (i.e. background) acoustic medium. The developments are motivated by the need to develop a preliminary indicator functional that would aid the solution of inverse scattering problems in terms of a rational initial 'guess' about the geometry and material characteristics of a hidden (finite) obstacle; an information that is often required by iterative minimization algorithms. To this end the customary definition of topological derivative, which quantifies the sensitivity of a given cost functional with respect to the creation of an infinitesimal hole, is adapted to permit the nucleation of a dissimilar acoustic medium. On employing the Green's function for the background domain, computation of topological sensitivity for the three-dimensional Helmholtz equation is reduced to the solution of a reference, Laplace transmission problem. Explicit formulae are given for the nucleating inclusions of spherical and ellipsoidal shapes. For generality the developments are also presented in an alternative, adjoint-field setting that permits nucleation of inclusions in an infinite, semi-infinite or finite background medium. Through numerical examples, it is shown that the featured topological sensitivity could be used, in the context of inverse scattering, as an effective obstacle indicator through an assembly of sampling points where it attains pronounced negative values. On varying a material characteristic (density) of the nucleating obstacle, it is also shown that the proposed methodology can be used as a preparatory tool for both geometric and material identification.