# Inverse problems in spaces of measures

Kristian Bredies; Hanna Katriina Pikkarainen

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 1, page 190-218
- ISSN: 1292-8119

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topBredies, Kristian, and Pikkarainen, Hanna Katriina. "Inverse problems in spaces of measures." ESAIM: Control, Optimisation and Calculus of Variations 19.1 (2013): 190-218. <http://eudml.org/doc/272801>.

@article{Bredies2013,

abstract = {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.},

author = {Bredies, Kristian, Pikkarainen, Hanna Katriina},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {inverse problems; vector-valued finite Radon measures; Tikhonov regularization; delta-peak solutions; generalized conditional gradient method; iterative soft-thresholding; sparse deconvolution; ill-posed problem; linear equation; Hilbert space; numerical results},

language = {eng},

number = {1},

pages = {190-218},

publisher = {EDP-Sciences},

title = {Inverse problems in spaces of measures},

url = {http://eudml.org/doc/272801},

volume = {19},

year = {2013},

}

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

PB - EDP-Sciences

VL - 19

IS - 1

SP - 190

EP - 218

AB - 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.

LA - eng

KW - inverse problems; vector-valued finite Radon measures; Tikhonov regularization; delta-peak solutions; generalized conditional gradient method; iterative soft-thresholding; sparse deconvolution; ill-posed problem; linear equation; Hilbert space; numerical results

UR - http://eudml.org/doc/272801

ER -

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