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Journal of Electrical and Computer Engineering
Volume 2010, Article ID 930218, 16 pages
http://dx.doi.org/10.1155/2010/930218
Research Article

Smoothing and Regularization with Modified Sparse Approximate Inverses

Technische Universität München, Boltzmannstraße 3, 80748 Garching, Germany

Received 20 September 2010; Accepted 22 September 2010

Academic Editor: Owe Axelsson

Copyright © 2010 T. Huckle and M. Sedlacek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Sparse approximate inverses 𝑀 which satisfy m i n 𝑀 β€– 𝐴 𝑀 βˆ’ 𝐼 β€– 𝐹 have shown to be an attractive alternative to classical smoothers like Jacobi or Gauss-Seidel (Tang and Wan; 2000). The static and dynamic computation of a SAI and a SPAI (Grote and Huckle; 1997), respectively, comes along with advantages like inherent parallelism and robustness with equal smoothing properties (Bröker et al.; 2001). Here, we are interested in developing preconditioners that can incorporate probing conditions for improving the approximation relative to high- or low-frequency subspaces. We present analytically derived optimal smoothers for the discretization of the constant-coefficient Laplace operator. On this basis, we introduce probing conditions in the generalized Modified SPAI (MSPAI) approach (Huckle and Kallischko; 2007) which yields efficient smoothers for multigrid. In the second part, we transfer our approach to the domain of ill-posed problems to recover original information from blurred signals. Using the probing facility of MSPAI, we impose the preconditioner to act as approximately zero on the noise subspace. In combination with an iterative regularization method, it thus becomes possible to reconstruct the original information more accurately in many cases. A variety of numerical results demonstrate the usefulness of this approach.