Application of the hybrid stochastic-deterministic minimization method to a surface data inverse scattering problem

Abstract

Abstract. A method for the identification of small inhomogeneitiesfrom a surface data is presented in the framework of an inverse scat-tering problem for the Helmholtz equation. Using the assumptions ofsmallness of the scatterers one reduces this inverse problem to an iden-tification of the positions of the small scatterers. These positions arefound by a global minimization search. Such a search is implementedby a novel Hybrid Stochastic-Deterministic Minimization method. Themethod combines random tries and a deterministic minimization. Theeffectiveness of this approach is illustrated by numerical experiments.In the modeling part our method is valid when the Born approximationfails. In the numerical part, an algorithm for the estimate of the numberof the small scatterers is proposed. 1 IntroductionIn many applications it is essential to find small inhomogeneities from surfacedata. For example, such a problem arises in ultrasound mammography, where smallinhomogeneities are cancer cells. Current X-ray mammography will be replaced bythe ultrasound one because X-ray mammography has a high probability of creatingnew cancer cells in a woman’s breast in the course of taking the mammography test.Other examples include the problem of finding small holes and cracks in metals andother materials, or the mine detection. The scattering theory for small scatterersoriginated in the classical works of Lord Rayleigh. It was developed in [15] and [16],where analyticalformulas forthe scattering matrix werederived for the acoustic and