Abstract
For hierarchical matrices, approximations of the matrix-matrix sum and product can be computed in almost linear complexity, and using these matrix operations it is possible to construct the matrix inverse, efficient preconditioners based on approximate factorizations or solutions of certain matrix equations.
-matrices are a variant of hierarchical matrices which allow us to perform certain operations, like the matrix-vector product, in ``true'' linear complexity, but until now it was not clear whether matrix arithmetic operations could also reach this, in some sense optimal, complexity.
We present algorithms that compute the best-approximation of the sum and product of two -matrices in a prescribed -matrix format, and we prove that these computations can be accomplished in linear complexity. Numerical experiments demonstrate that the new algorithms are more efficient than the well-known methods for hierarchical matrices.
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References
M. Bebendorf W. Hackbusch (2003) ArticleTitleExistence of
-matrix approximants to the inverse FE-matrix of elliptic operators with L∞-coefficients
Numer. Math.
95
1–28
Occurrence
Handle10.1007/s00211-002-0445-6
Occurrence
Handle2004e:65128
S. Börm (2004) ArticleTitle
-matrices – multilevel methods for the approximation of integral operators
Comput. Visual. Sci.
7
173–181
Occurrence
Handle1070.65124
S. Börm (2005) ArticleTitleApproximation of integral operators by
-matrices with adaptive bases
Computing
74
249–271
Occurrence
Handle02177726
Occurrence
Handle2006b:65200
Börm, S.: Data-sparse approximation of non-local operators by
-matrices. Preprint 44/2005, Max-Planck Institute for Mathematics in the Sciences, 2005.S. Börm L. Grasedyck (2005) ArticleTitleHybrid cross approximation of integral operators Numer. Math. 101 221–249
Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices. Lecture Note 21, Max-Planck Institute for Mathematics in the Sciences, 2003.
S. Börm W. Hackbusch (2002) ArticleTitleData-sparse approximation by adaptive
-matrices
Computing
69
1–35
Occurrence
Handle2004c:65155
S. Börm W. Hackbusch (2002) ArticleTitle
-matrix approximation of integral operators by interpolation
Appl. Numer. Math.
43
129–143
Occurrence
Handle1936106
S. Börm M. Löhndorf J. M. Melenk (2005) ArticleTitleApproximation of integral operators by variable-order interpolation Numer. Math. 99 605–643 Occurrence Handle2005j:65176
S. Börm S. A. Sauter (2005) ArticleTitleBEM with linear complexity for the classical boundary integral operators Math. Comput. 74 1139–1177
W. Dahmen R. Schneider (1999) ArticleTitleWavelets on manifolds I: construction and domain decomposition SIAM J. Math. Anal. 31 184–230 Occurrence Handle10.1137/S0036141098333451 Occurrence Handle2000k:65242
Grasedyck, L.: Theorie und Anwendungen hierarchischer Matrizen. PhD thesis, Universität Kiel, 2001.
L. Grasedyck (2004) ArticleTitleAdaptive recompression of
-matrices for BEM
Computing
74
205–223
Occurrence
Handle2139413
L. Grasedyck W. Hackbusch (2003) ArticleTitleConstruction and arithmetics of
-matrices
Computing
70
295–334
Occurrence
Handle2004i:65035
L. Greengard V. Rokhlin (1987) ArticleTitleA fast algorithm for particle simulations J. Comput. Phys. 73 325–348 Occurrence Handle10.1016/0021-9991(87)90140-9 Occurrence Handle88k:82007
Greengard, L., Rokhlin, V.: A new version of the fast multipole method for the Laplace in three dimensions. Acta Numerica 1997, Cambridge University Press, pp. 229–269.
W. Hackbusch (1999) ArticleTitleA sparse matrix arithmetic based on
-matrices. Part I : Introduction to -matrices
Computing
62
89–108
Occurrence
Handle10.1007/s006070050015
Occurrence
Handle0927.65063
Occurrence
Handle2000c:65039
Hackbusch, W.: Hierarchische Matrizen – Algorithmen und Analysis. Available online at http://www.mis.mpg.de/scicomp/fulltext/hmvorlesung.ps, 2004.
W. Hackbusch B. Khoromskij (2000) ArticleTitleA sparse matrix arithmetic based on
-matrices. Part II: Application to multi-dimensional problems
Computing
64
21–47
Occurrence
Handle2001i:65053
W. Hackbusch B. Khoromskij S. Sauter (2000) On
-matrices
H.
Bungartz
R.
Hoppe
C.
Zenger
(Eds)
Lectures on Applied Mathematics
Springer
Berlin
9–29
W. Hackbusch Z. P. Nowak (1989) ArticleTitleOn the fast matrix multiplication in the boundary element method by panel clustering Numer. Math. 54 463–491 Occurrence Handle10.1007/BF01396324 Occurrence Handle89k:65162
V. Rokhlin (1985) ArticleTitleRapid solution of integral equations of classical potential theory J. Comput. Phys. 60 187–207 Occurrence Handle10.1016/0021-9991(85)90002-6 Occurrence Handle0629.65122 Occurrence Handle86k:65120
Sauter, S.: Variable order panel clustering (extended version). Preprint 52/1999, Max-Planck-Institut für Mathematik, Leipzig, Germany, 1999.
S. Sauter (2000) ArticleTitleVariable order panel clustering Computing 64 223–261 Occurrence Handle10.1007/s006070050045 Occurrence Handle0959.65135 Occurrence Handle2001e:65197
J. Tausch J. White (2003) ArticleTitleMultiscale bases for the sparse representation of boundary integral operators on complex geometries SIAM J. Sci. Comput. 24 1610–1629 Occurrence Handle10.1137/S1064827500369451 Occurrence Handle2004d:65151
-matrix approximants to the inverse FE-matrix of elliptic operators with L∞-coefficients
Numer. Math.
95
1–28
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Börm, S. 
-Matrix Arithmetics in Linear Complexity.
Computing 77, 1–28 (2006). https://doi.org/10.1007/s00607-005-0146-y
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DOI: https://doi.org/10.1007/s00607-005-0146-y