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

TL;DR

This paper introduces domain decomposition preconditioners for integral equations in acoustic scattering, improving the efficiency of solving forward and inverse problems with iterative methods.

Contribution

It extends domain decomposition preconditioning techniques from PDEs to integral equations and develops a new preconditioner for the inverse problem's Gauss-Newton Hessian.

Findings

Preconditioners significantly accelerate iterative solvers.

The combined preconditioner with low-rank correction improves convergence.

Numerical results demonstrate enhanced performance in inverse scattering problems.

Abstract

We propose domain decomposition preconditioners for the solution of an integral equation formulation of forward and inverse acoustic scattering problems with point scatterers. We study both forward and inverse problems and propose preconditioning techniques to accelerate the iterative solvers. For the forward scattering problem, we extend 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}, to integral equations. We combine this domain decomposition preconditioner with a low-rank correction, which is easy to construct, forming a new preconditioner. For the inverse scattering problem, we use the forward problem preconditioner as a building block for constructing a preconditioner for the Gauss-Newton Hessian. We present numerical results that demonstrate the performance of both preconditioning strategies.