, Volume 104, Issue 2, pp 177-203
Date: 22 Jul 2006

A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints

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Abstract

In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous (typically weighted ℓ p ) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem.