Recent developments on topological derivative methods for qualitative inverse scattering

The topological derivative quantifies the sensitivity of objective functionals whose evaluation involves the solution of a PDE with respect to the nucleation of a small feature (e.g. cavity, inclusion, crack) at a prescribed location in the (e.g. acoustic, elastic, electromagnetic) medium of interest. Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for wave-based identification of either small or finite-sized objects. In that approach, hidden objects are deemed to lie at locations where the topological derivative is most negative. Topological derivatives need asymptotic analysisfor their derivation, but are then very simple to implement and entail computational costs that are much lower than straightforward optimization-based inversion methods. Focusing on acoustic and elastodynamic scattering, and stressing main concepts and results rather than technical detail, the following topics will be addressed: 1) An overview of inverse scattering approaches relying on asymptotic expansions for small scatterers. 2) Asummarized presentation of the derivation of topological derivatives for acoustic and elastodynamic scattering, including small-inclusion asymptotics based on the expansion of Lippmann-Schwinger volume integral equations. 3) The implementation of topological derivative and numerical experiments. 4) Going beyond the above heuristic-based use of the topological derivative, a (partial in scope) justification is presented for the acoustic
case involving far-field measurements (collaboration with Cedric Bellis and Fioralba Cakoni). 5) Higher-order expansions, providing approximations of objective functionals that are polynomial in the defect diameter (with coefficients depending on trial defect location and assumed physical properties) and permitting quantitative identification within moderate computational costs, will finally be addressed
Bio :  Marc Bonnet, a CNRS senior scientist, is with the POEMS group of the Applied Mathematics department of ENSTA since 2011, and was before that with the Solid Mechanics laboratory (LMS) of Ecole Polytechnique. He completed master studies in solid mechanics in 1983 (université Paris 6, concurrently with the engineering degree of ENPC, Paris), obtained his doctor degree from ENPC in 1986 and his habilitation degree from Université Paris 6 in 1995. His research interests include inverse problems, integral equation methods and asymptotic models for elastic and acoustic wave propagation problems. He is associate editor or editorial board member of several international research journals (Inverse Problems, Engineering Analysis for Boundary Elements, Journal
of Integral Equations and Applications, Inverse Problems in Engineering and Sciences, Computational Mechanics, Journal of Optimization Theory and Applications).