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

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    Type
    Article
    Authors
    Fornasier, Massimo
    Schönlieb, Carola-Bibiane
    KAUST Grant Number
    KUK-I1-007-43
    Date
    2009-01
    Permanent link to this record
    http://hdl.handle.net/10754/599796
    
    Metadata
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    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.
    Sponsors
    This work was based on work supported by Award KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
    Publisher
    Society for Industrial & Applied Mathematics (SIAM)
    Journal
    SIAM Journal on Numerical Analysis
    DOI
    10.1137/070710779
    ae974a485f413a2113503eed53cd6c53
    10.1137/070710779
    Scopus Count
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