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

Fu-Rong Lin and Shi-Wei Yang
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Fu-Rong Lin: Department of Mathematics, Shantou University, Shantou 515063, China
Shi-Wei Yang: Department of Mathematics, Shantou University, Shantou 515063, China

Mathematics, 2017, vol. 5, issue 2, 1-16

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.

Keywords: Fredholm integral equation of the first kind; regularization; piecewise-constant solution; total variation (TV) functional; Modica-Mortola (MM) functional (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2017
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