Fast algorithms for systems of equations in wavelet-based solution of integral equations with Toeplitz kernels

Abstract

The wavelet transform is applied to integral equations with Toeplitz kernels. Such integral equations arise in inverse scattering and linear least-squares estimation. The result is a system of equations with block-slanted-Toeplitz structure. In previous approaches, this linear system was sparsified by neglecting all entries below some threshold. However, in inverse scattering, the Toeplitz kernel may not be a rapidly decreasing function due to reflections from great depths. In this case, neglecting entries below a threshold will not work since the system matrix is ill-conditioned. We use the different approach of exploiting the block-slanted-Toeplitz structure to obtain fast algorithms similar to the multichannel Levinson and Schur algorithms. Since it is exact to within the wavelet-basis approximation, this different approach should prove to be a valuable alternative to the approximate approach of sparsification in cases when the latter does not work.