# Subspace Correction Methods for Total Variation and $\ell_1$-Minimization

Handle URI:
http://hdl.handle.net/10754/599796
Title:
Subspace Correction Methods for Total Variation and $\ell_1$-Minimization
Authors:
Fornasier, Massimo; Schönlieb, Carola-Bibiane
Abstract:
This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a seminorm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via oblique thresholding, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for degenerate elliptic PDEs arising in total variation minimization and in accelerated sparse recovery algorithms based on 1-minimization. We include numerical examples which show e.cient solutions to classical problems in signal and image processing. © 2009 Society for Industrial and Applied Physics.
Citation:
Fornasier M, Schönlieb C-B (2009) Subspace Correction Methods for Total Variation and $\ell_1$-Minimization. SIAM J Numer Anal 47: 3397–3428. Available: http://dx.doi.org/10.1137/070710779.
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
SIAM Journal on Numerical Analysis
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Jan-2009
DOI:
10.1137/070710779
Type:
Article
ISSN:
0036-1429; 1095-7170
This work was based on work supported by Award KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Publications Acknowledging KAUST Support

DC FieldValue Language
dc.contributor.authorFornasier, Massimoen
dc.contributor.authorSchönlieb, Carola-Bibianeen
dc.date.accessioned2016-02-28T06:10:01Zen
dc.date.available2016-02-28T06:10:01Zen
dc.date.issued2009-01en
dc.identifier.citationFornasier M, Schönlieb C-B (2009) Subspace Correction Methods for Total Variation and $\ell_1$-Minimization. SIAM J Numer Anal 47: 3397–3428. Available: http://dx.doi.org/10.1137/070710779.en
dc.identifier.issn0036-1429en
dc.identifier.issn1095-7170en
dc.identifier.doi10.1137/070710779en
dc.identifier.urihttp://hdl.handle.net/10754/599796en
dc.description.abstractThis paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a seminorm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via oblique thresholding, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for degenerate elliptic PDEs arising in total variation minimization and in accelerated sparse recovery algorithms based on 1-minimization. We include numerical examples which show e.cient solutions to classical problems in signal and image processing. © 2009 Society for Industrial and Applied Physics.en
dc.description.sponsorshipThis work was based on work supported by Award KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.subject11-Minimizationen
dc.subjectConvex optimizationen
dc.subjectDegenerate elliptic PDEsen
dc.subjectDiscontinuous solutionsen
dc.subjectDomain decomposition methoden
dc.subjectImage and signal processingen
dc.subjectParallel computationen
dc.subjectSubspace correctionsen
dc.subjectTotal variation minimizationen
dc.titleSubspace Correction Methods for Total Variation and $\ell_1$-Minimizationen
dc.typeArticleen
dc.identifier.journalSIAM Journal on Numerical Analysisen
dc.contributor.institutionJohann Radon Institute for Computational and Applied Mathematics, Linz, Austriaen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en