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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.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, Linz, 4040, Austria

    Ronny Ramlau

  2. Konrad–Zuse–Zentrum für Informationstechnik Berlin (ZIB), Takustr. 7, Berlin-Dahlem, 14195, Germany

    Gerd Teschke

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  1. Ronny Ramlau
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  2. Gerd Teschke
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Correspondence to Ronny Ramlau.

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Ramlau, R., Teschke, G. A Tikhonov-based projection iteration for nonlinear Ill-posed problems with sparsity constraints. Numer. Math. 104, 177–203 (2006). https://doi.org/10.1007/s00211-006-0016-3

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  • DOI: https://doi.org/10.1007/s00211-006-0016-3

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