A wavelet-based method for solving discrete first-kind Fredholm equations with piecewise constant solutions

• Published in: International journal for numerical methods in engineering Vol. 57; no. 4; pp. 577 - 598
• Main Authors: Sánchez-Ávila, C., Sánchez-Reíllo, R.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 28.05.2003
• Subjects: Algorithms; Approximation; deconvolution problem; discrete ill-posed problems; edge detection; Hilbert space; Holder exponents; Integral equations; Iterative methods; Mathematical analysis; Mathematical models; Regularization; wavelets modulus maxima
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.697

The inverse problem of finding piecewise constant solutions to discrete Fredholm integral equations of the first kind arises in many applied fields, e.g. in geophysics. This equation is usually an ill‐posed problem when it is considered in a Hilbert space framework, requiring regularization techniques to control arbitrary error amplifications and to get adequate solutions. In this work, we describe an iterative regularizing method for computing piecewise constant solutions to first‐kind discrete Fredholm integral equations. The algorithm involves two main steps at each iteration: (1) approximating the solution using a new signal reconstruction algorithm from its wavelet maxima which involves a previous step of detecting discontinuities by estimation of its local Hölder exponents; and (2) obtaining a regularized solution of the original equation using the a priori knowledge and the above approximation. In order to check the behaviour of the proposed technique, we have carried out a statistical study from a high number of simulations obtaining excellent results. Their comparisons with the results coming from using classical Tikhonov regularization by the multiresolution support, total variation (TV) regularization and piecewise polynomial truncated singular value decomposition (PP‐TSVD) algorithm, serve to illustrate the advantages of the new method. Copyright © 2003 John Wiley & Sons, Ltd.