High Resolution Inverse Scattering in Two Dimensions Using Recursive Linearization

We describe a fast, stable algorithm for the solution of the inverse acoustic scattering problem in two dimensions. Given full aperture far field measurements of the scattered field for multiple angles of incidence, we use Chen's method of recursive linearization to reconstruct an unknown sound speed at resolutions of thousands of square wavelengths in a fully nonlinear regime. Despite the fact that the underlying optimization problem is formally ill-posed and nonconvex, recursive linearization requires only the solution of a sequence of linear least squares problems at successively higher frequencies. By seeking a suitably band-limited approximation of the sound speed profile, we ensure that each least squares calculation is well-conditioned so that an iterative solver can be effectively applied. Each matrix-vector product involves the solution of a large number of forward scattering problems, for which we have created a new, spectrally accurate, fast direct solver. For the largest problems considered, involving 19,600 unknowns, approximately 1 million partial differential equations were solved, requiring approximately 2 days to compute using a parallel MATLAB implementation on a multicore workstation.