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Inverse problems in spaces of measures

Published online by Cambridge University Press:  27 March 2012

Kristian Bredies and
Hanna Katriina Pikkarainen
Kristian Bredies
Affiliation:
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria. kristian.bredies@uni-graz.at
Hanna Katriina Pikkarainen
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria; hanna.pikkarainen@ricam.oeaw.ac.at

Abstract

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The ill-posed problem of solving linear equations in the space of vector-valued finiteRadon measures with Hilbert space data is considered. Approximate solutions are obtainedby minimizing the Tikhonov functional with a total variation penalty. The well-posednessof this regularization method and further regularization properties are mentioned.Furthermore, a flexible numerical minimization algorithm is proposed which convergessubsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparsedeconvolution demonstrate the applicability for a finite-dimensional discrete data spaceand infinite-dimensional solution space.

Keywords

Inverse problemsvector-valued finite Radon measuresTikhonov regularizationdelta-peak solutionsgeneralized conditional gradient methoditerative soft-thresholdingsparse deconvolution

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013, pp. 190 - 218
Copyright
© EDP Sciences, SMAI, 2012

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