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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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An iterative thresholding-like algorithm for inverse problems with sparsity constraints in Banach space

K. Bredies1

1Center for Industrial Mathematics / Fachbereich 3, University of Bremen, Postfach 33 04 40, D-28334 Bremen, Germany. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 17, Issue 1, Pages 19–26, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/JIIP.2009.003, February 2009

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This paper addresses the problem of computing the minimizers for Tikhonov functionals associated with inverse problems with sparsity constraints in general Banach spaces. We present, based on splitting the Tikhonov functional into a smooth and a non-smooth part, a general iterative procedure for the Banach-space setting. In case of sparsity constraints, this algorithm yields a successive application of thresholding-like functions which generalizes the well-known iterative soft-thresholding procedure. The convergence properties of the proposed method are studied. Depending on the smoothness and convexity of the underlying spaces, convergence of asymptotic rate is obtained with the help of Bregman and Bregman–Taylor distance estimates. In particular, strong convergence can be achieved for a large class of linear inverse problems with sparsity constraints in Banach space.

Key words.: Iterative thresholding; sparsity constraints; Banach space; generalized gradient projection method; convergence analysis

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