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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Multilevel preconditioning and adaptive sparse solution of inverse problems
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by Stephan Dahlke, Massimo Fornasier and Thorsten Raasch PDF
Math. Comp. 81 (2012), 419-446 Request permission


We are concerned with the efficient numerical solution of minimization problems in Hilbert spaces involving sparsity constraints. These optimizations arise, e.g., in the context of inverse problems. In this work we analyze an efficient variant of the well-known iterative soft-shrinkage algorithm for large or even infinite dimensional problems. This algorithm is modified in the following way. Instead of prescribing a fixed thresholding parameter, we use a decreasing thresholding strategy. Moreover, we use suitable variants of the adaptive schemes derived by Cohen, Dahmen and DeVore for the approximation of the infinite matrix-vector products. We derive a block multiscale preconditioning technique which allows for local well-conditioning of the underlying matrices and for extending the concept of restricted isometry property to infinitely labelled matrices. The combination of these ingredients gives rise to a numerical scheme that is guaranteed to converge with exponential rate, and which allows for a controlled inflation of the support size of the iterations. We also present numerical experiments that confirm the applicability of our approach which extends concepts from compressed sensing to large scale simulation.
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Additional Information
  • Stephan Dahlke
  • Affiliation: Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Hans Meerwein Strasse, Lahnberge, 35032 Marburg, Germany
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  • Massimo Fornasier
  • Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A-4040, Linz, Austria
  • Address at time of publication: Faculty of Mathematics, Technical University of Munich, Boltzmannstrasse 3, D-85748 Garching, Germany
  • Email:
  • Thorsten Raasch
  • Affiliation: Johannes Gutenberg-Universität, Institut für Mathematik, Staudingerweg 9, 55099 Mainz, Germany
  • Email:
  • Received by editor(s): August 14, 2009
  • Received by editor(s) in revised form: November 18, 2010
  • Published electronically: May 25, 2011
  • Additional Notes: The work of Stephan Dahlke was supported by Deutsche Forschungsgemeinschaft (DFG), Grant DA 360/12–1.
    The work of Massimo Fornasier was supported by the FWF project Y 432-N15 START-Preis “Sparse Approximation and Optimization in High Dimensions” and Deutsche Forschungsgemeinschaft (DFG), Grant DA 360/12–1.
    The work of Thorsten Raasch was supported by Deutsche Forschungsgemeinschaft (DFG), Grants DA 360/7–1 and DA 360/11–1.
  • © Copyright 2011 American Mathematical Society
  • Journal: Math. Comp. 81 (2012), 419-446
  • MSC (2010): Primary 41A25, 65F35, 65F50, 65N12, 65T60
  • DOI:
  • MathSciNet review: 2833502