An adaptive LSQR algorithm for computing discontinuous solutions in deconvolution problems

Abstract

Ill-posed problems described by first-kind Fredholm equations appear in many interesting practical cases in engineering or mathematical physics, such as the inverse problem of signal deconvolution, and require regularization techniques to get adequate solutions ([C. Sánchez-Ávila, A.R. Figueiras-Vidal, J. Comp. Appl. Math. 72 (1996) 21–39] and [A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers, Dordrecht, 1995]). In this work we consider the noisy discrete version of this equation which describes a typical problem in signal processing (e.g., in geophysics): recovering the discontinuities of a solution showing sharp edges. Such solutions are suited to describe models, e.g., geological layers, where the coarse structure is more important than the fine structure. We present a new adaptive algorithm which is capable of computing solutions that are piecewise constant, without having to specify a priori the positions of the break points between the constant pieces. This iterative algorithm is built on the base of regularizing LSQR method [P.C. Hansen, Numerical Aspects of Linear Inversion, SIAM, USA, 1997] and consists of two steps at each iteration: 1. detecting the discontinuities by an adaptive procedure, and 2. solving the original equation by regularizing LSQR iteration. We have carried out a high number of simulations to check the performance of proposed technique considering different signal-to-noise ratios in order to study its capacity of recovering edges in very noisy environments. Here we show some representative examples.