CNRS Senior Scientist with the Applied Mathematics of ENSTA
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 analysis for 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) A summarized presentation of the derivation of topological derivatives for acoustic and elastodynamic scattering, including a concise presentation of 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 (collaboration with Rémi Cornaggia).