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

Massimo Fornasier, Carola-Bibiane Schönlieb

Research output: Contribution to journalArticlepeer-review

34 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)3397-3428
Number of pages32
JournalSIAM Journal on Numerical Analysis
Volume47
Issue number5
DOIs
StatePublished - Jan 2009
Externally publishedYes

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