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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Volume 13, Issue 2

Issues

Hierarchical Kronecker tensor-product approximations

W. Hackbusch
  • Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ B. N. Khoromskij
  • Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ E. E. Tyrtyshnikov
  • Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow GSP-1, 119991, Russia
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar

The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integro-differential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices (cf. [37,38]). In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called 'profile-low-rank matrices' or hierarchical matrices (cf. [18,19]). We give a proof for the existence of such formats and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.

Key Words: integral equations,; BEM,; low-rank matrices,; hierarchical matrices,; Kronecker products,; multi-dimensional matrices,; tensors

About the article

Published in Print: 2005-06-01


Citation Information: Journal of Numerical Mathematics jnma, Volume 13, Issue 2, Pages 119–156, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/1569395054012767.

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