Symmetric Cauchy-like Preconditioners for the Regularized Solution of 1-D Ill-Posed Problems

The discretization of integral equations can lead to systems involving symmetric Toeplitz matrices. We describe a preconditioning technique for the regularized solution of the related discrete ill-posed problem. We use discrete sine transforms to transform the system to one involving a Cauchy-like matrix. Based on the approach of Kilmer and O'Leary, the preconditioner is a symmetric, rank $m^{*}$ approximation to the Cauchy-like matrix augmented by the identity. We shall show that if the kernel of the integral equation is smooth then the preconditioned matrix has two desirable properties; namely, the largest $m^{*}$ magnitude eigenvalues are clustered around and bounded below by one, and that small magnitude eigenvalues remain small. We also show that the initialization cost is less than the initialization cost for the preconditioner introduced by Kilmer and O'Leary. Further, we describe a method for applying the preconditioner in $O((n+1) \lg (n+1))$ operations when $n+1$ is a power of 2, and describe a variant of the MINRES algorithm to solve the symmetrically preconditioned problem. The preconditioned method is tested on two examples