Fast methods for three-dimensional inverse obstacle scattering problems

Abstract

We study the inverse problem to reconstruct the shape of a three dimensional sound-soft obstacle from measurements of scattered acoustic waves. To solve the forward problem we use a wavelet based boundary element method and prove fourth order accuracy both for the evaluation of the forward solution operator and its Fréchet derivative. Moreover, we discuss the characterization and implementation of the adjoint of the Fréchet derivative. For the solution of the inverse problem we use a regularized Newton method. The boundaries are represented by a class of parametrizations, which include non star-shaped domains and which are not uniquely determined by the obstacle. To prevent degeneration of the parametrizations during the Newton iteration, we introduce an additional penalty term. Numerical examples illustrate the performance of our method. © 2007 Rocky Mountain Mathematics Consortium.