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

Keywords: inverse problems, vector-valued finite Radon measures, Tikhonov regularization, delta-peak solutions, generalized conditional gradient method, iterative soft-thresholding, sparse deconvolution

@article{COCV_2013__19_1_190_0, author = {Bredies, Kristian and Pikkarainen, Hanna Katriina}, title = {Inverse problems in spaces of measures}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {190--218}, publisher = {EDP-Sciences}, volume = {19}, number = {1}, year = {2013}, doi = {10.1051/cocv/2011205}, mrnumber = {3023066}, zbl = {1266.65083}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2011205/} }

TY - JOUR AU - Bredies, Kristian AU - Pikkarainen, Hanna Katriina TI - Inverse problems in spaces of measures JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 190 EP - 218 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011205/ DO - 10.1051/cocv/2011205 LA - en ID - COCV_2013__19_1_190_0 ER -

%0 Journal Article %A Bredies, Kristian %A Pikkarainen, Hanna Katriina %T Inverse problems in spaces of measures %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 190-218 %V 19 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2011205/ %R 10.1051/cocv/2011205 %G en %F COCV_2013__19_1_190_0

Bredies, Kristian; Pikkarainen, Hanna Katriina. Inverse problems in spaces of measures. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 190-218. doi : 10.1051/cocv/2011205. http://www.numdam.org/articles/10.1051/cocv/2011205/

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