Smoothing and regularization with modified sparse approximate inverses
type: Item
instance of: scholarly article
title: Smoothing and regularization with modified sparse approximate inverses
zbMATH Open document ID: 1229.65058
DOI: 10.1155/2010/930218
published in: Journal of Electrical and Computer Engineering
publication date: 2011-06-20T00:00:00Z
review text: Summary: Sparse approximate inverses \(M\) which satisfy \(\min_{M}\| AM-I\|_{F}\) have shown to be an attractive alternative to classical smoothers like Jacobi or Gauss-Seidel [\textit{W.-P. Tang} and \textit{W. L. Wan}, SIAM J. Matrix Anal. Appl. 21, No. 4, 1236-1252 (2000; Zbl 1049.65146)]. The static and dynamic computation of a SAI and a SPAI [\textit{M. J. Grote} and \textit{T. Huckle}, SIAM J. Sci. Comput. 18, No. 3, 838--853 (1997; Zbl 0872.65031)], respectively, comes along with advantages like inherent parallelism and robustness with equal smoothing properties. 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 [\textit{T. Huckle} and \textit{A. Kallischko}, Int. J. Comput. Math. 84, No. 8, 1225--1248 (2007; Zbl 1126.65035)] 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.
Mathematics Subject Classification ID: 65F08
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65Y05
zbMATH DE Number: 5910099
author: Thomas K. Huckle
author: Matous Sedlacek
describes a project that uses: RestoreTools
describes a project that uses: MSPAI
describes a project that uses: Regularization tools
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2010/930218
OpenAlex ID: W2054744919
Wikidata QID: Q58652014
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DBLP publication ID: journals/jece/HuckleS10
description: scientific article; zbMATH DE number 5910099