Iterative Regularization Methods for Inverse Problems in Acoustics

Abstract

We consider the use of conjugate-gradient-like iterative methods for the solution of integral equations arising from an inverse problem in acoustics in a bounded three dimensional region. The inverse problem is the computation of the normal velocities on the boundary of a region from pressure measurements on an interior surface. The pressure satisfies the Helmholtz equation in the region. Two formulations are considered: one based on the representation of pressures by a single layer potential and the other based on the Helmholtz-Kirchhoff integral equation. Both formulations can be used to approximate the Neumann Green’s function as an alternative. The integral equations are all ill-posed and are discretized by a boundary element method. The resulting liner systems are ill-conditoned and a (smooth) regularized solutions must be sought. Two regularization rules, including a new one, for conjugate-gradient-like methods are applied and found to have advantages over a standard method based on the truncated singular value decomposition using generalized cross validation. Due to the occurence of multiple singular values for our integral operators, conjugate gradient methods compute the optimal solution in the first few iterations and prove to be particularly fast for these large scale acoustics problems.