Direct Implementation of Tikhonov Regularization for the First Kind Integral Equation

Abstract

A common way to handle the Tikhonov regularization method for the first kind Fredholm integral equations, is first to discretize and then to work with the final linear system. This unavoidably inflicts discretization errors which may lead to disastrous results, especially when a quadrature rule is used. We propose to regularize directly the integral equation resulting in a continuous Tikhonov problem. The Tikhonov problem is reduced to a simple least squares problem by applying the Golub-Kahan bidiagonalization (GKB) directly to the integral operator. The regularization parameter and the iteration index are determined by the discrepancy principle approach. Moreover, we study the discrete version of the proposed method resulted from numerical evaluating the needed integrals. Focusing on the nodal values of the solution results in a weighted version of GKB-Tikhonov method for linear systems arisen from the Nyström discretization. Finally, we use numerical experiments on a few test problems to illustrate the performance of our algorithms.