A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind

Abstract

A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f * TV . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f * TV as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach.