Irregular Solutions of an Ill-Posed Problem

Abstract: Tikhonov regularization is a popular and effective method for the approximate solution of illposed problems, including Fredholm equations of the first kind. The Tikhonov method works well when the solution of the equation is well-behaved, but fails for solutions with irregularities, such as jump discontinuities. In this paper we develop a method that overcomes the limitations of the standard Tikhonov regularization. We present a criterion by which approximate solutions can be evaluated and use it in a search method that is effective in locating points of irregular behavior. Once the points of irregularity hav