Recompression techniques for adaptive cross approximation
type: Item
instance of: scholarly article
title: Recompression techniques for adaptive cross approximation
zbMATH Open document ID: 1180.65162
DOI: 10.1216/JIE-2009-21-3-331
published in: Journal of Integral Equations and Applications
publication date: 2009-10-09T00:00:00Z
review text: The adaptive cross approximation method (ACA) generates low-rank approximations to suitable \(m\times n\) sub-blocks of discrete integral formulations of elliptic boundary value problems. The discretization method by which the sub-blocks are obtained from the integral operator is either the Nyström, the collocation or the Galerkin method. ACA requires \(k(m+n)\), \(k\sim|\log \varepsilon|\) (\(\varepsilon \) represents the approximation accuracy) units of storage and is generated in a purely algebraic manner using only few of matrix entries. Also it is an adaptive algorithm, i.e. the rank of approximation is determined during the approximation. The authors present recompression techniques based on optimized singular value decomposition, Chebyshev approximation and discrete cosine transform. The required amount of storage is brought down to sublinear order \(kk'\), where \(k'\) depends logarithmically on \(\varepsilon \) but is independent of the matrix size. Numerical results support the derived estimates.
reviewed by: Adrian Carabineanu
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 5612798
zbMATH Keywords: adaptive cross approximation
zbMATH Keywords: integral equation method
zbMATH Keywords: hierarchical matrices
zbMATH Keywords: Nyström method
zbMATH Keywords: elliptic boundary value problems
zbMATH Keywords: collocation
zbMATH Keywords: Galerkin method
zbMATH Keywords: algorithm
zbMATH Keywords: singular value decomposition
zbMATH Keywords: Chebyshev approximation
zbMATH Keywords: discrete cosine transform
zbMATH Keywords: numerical results
author: Mario Bebendorf
author: Stefan Kunis
MaRDI profile type: Publication
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full work available at URL: jie-2009-21-3-331
OpenAlex ID: W2088961409
description: scientific article; zbMATH DE number 5612798