A domain decomposition preconditioning for the integral equation formulation of the inverse scattering problem

Carlos Borges, George Biros

Published 2019-01-26Version 1

We propose domain decomposition preconditioners for the solution of the integral equation formulation of forward and inverse acoustic scattering problems with points scatterers. We study independently both problems and propose preconditioning techniques to accelerate the iterative solvers. In the forward scattering case, first, we extend to integral equations the domain decomposition based preconditioning techniques presented for partial differential equations in {\em "A restricted additive Schwarz preconditioner for general sparse linear systems", SIAM Journal on Scientific Computing, 21 (1999), pp. 792--797}. Next, we propose a new preconditioner that is a low-rank correction of the domain decomposition based preconditioner. In the inverse scattering case, we use the low-rank corrected preconditioner proposed for the forward problem as the building block for constructing a preconditioner for the Gauss-Newton Hessian. Our numerical results show that both preconditioning strategies work well. In particular, for the inverse scattering problem, our preconditioner outperform low-rank approximations, which are the state of the art.