Skip to main content
SpringerLink
Log in

Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators

  • Published:
Computing Aims and scope Submit manuscript

Abstract

This article is the second part continuing Part I [16]. We apply the -matrix techniques combined with the Kronecker tensor-product approximation to represent integral operators as well as certain functions F(A) of a discrete elliptic operator A in a hypercube (0,1)d ∈ ℝd in the case of a high spatial dimension d. We focus on the approximation of the operator-valued functions A − σ, σ>0, and sign (A) for a class of finite difference discretisations A ∈ ℝN × N. The asymptotic complexity of our data-sparse representations can be estimated by (n p log q n), p = 1, 2, with q independent of d, where n=N 1/ d is the dimension of the discrete problem in one space direction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • G. Beylkin M. J. Mohlenkamp (2002) ArticleTitleNumerical operator calculus in higher dimensions Proc. Natl. Acad. Sci. USA 99 10246–10251 Occurrence Handle10.1073/pnas.112329799 Occurrence Handle2003h:65071

    Article  MathSciNet  Google Scholar 

  • Flad, H. -J., Hackbusch, W., Khoromskij, B. N., Schneider, R.: Concept of data-sparse tensor- product approximation in many-particle models (in preparation).

  • H.-J. Flad W. Hackbusch D. Kolb R. Schneider (2002) ArticleTitleWavelet approximation of correlated wavefunctions. I. Basics J. Chem. Phys. 116 9641–9657 Occurrence Handle10.1063/1.1476008

    Article  Google Scholar 

  • I. P. Gavrilyuk W. Hackbusch B. N. Khoromskij (2002) ArticleTitle -matrix approximation for the operator exponential with applications Numer. Math. 92 83–111 Occurrence Handle10.1007/s002110100360 Occurrence Handle2003g:65061

    Article  MathSciNet  Google Scholar 

  • I. P. Gavrilyuk W. Hackbusch B. N. Khoromskij (2004) ArticleTitleData-sparse approximation to operator-valued functions of elliptic operators Math. Comp. 73 1297–1324 Occurrence Handle10.1090/S0025-5718-03-01590-4 Occurrence Handle2005b:47086

    Article  MathSciNet  Google Scholar 

  • I. P. Gavrilyuk W. Hackbusch B. N. Khoromskij (2005) ArticleTitleData-sparse approximation to a class of operator-valued functions Math. Comp. 74 681–708 Occurrence Handle2005i:65068

    MathSciNet  Google Scholar 

  • I. P. Gavrilyuk W. Hackbusch B. N. Khoromskij (2005) ArticleTitleTensor-product approximation to elliptic and parabolic solution operators in higher dimensions Computing 74 131–157 Occurrence Handle10.1007/s00607-004-0086-y Occurrence Handle2133692

    Article  MathSciNet  Google Scholar 

  • I. S. Gradshteyn I. M. Ryzhik (2000) Table of integrals, series and products EditionNumber6 Academic Press San Diego

    Google Scholar 

  • L. Grasedyck (2004) ArticleTitleExistence and computation of a low Kronecker-rank approximation to the solution of a tensor system with tensor right-hand side Computing 72 247–265 Occurrence Handle10.1007/s00607-003-0037-z Occurrence Handle1058.65036 Occurrence Handle2005j:65130

    Article  MATH  MathSciNet  Google Scholar 

  • L. Grasedyck W. Hackbusch (2003) ArticleTitleConstruction and arithmetics of -matrices Computing 70 295–334 Occurrence Handle2004i:65035

    MathSciNet  Google Scholar 

  • L. Grasedyck W. Hackbusch B. N. Khoromskij (2003) ArticleTitleSolution of large scale algebraic matrix Riccati equations by use of hierarchical matrices Computing 70 121–165 Occurrence Handle2004f:65056

    MathSciNet  Google Scholar 

  • M. Griebel S. Knapek (2000) ArticleTitleOptimized tensor-product approximation spaces Constr. Approx. 16 303–332 Occurrence Handle10.1007/s003650010010 Occurrence Handle2001g:41025

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MATH  MathSciNet  Google Scholar 

  • W. Hackbusch B. N. Khoromskij (2000) ArticleTitleA sparse -matrix arithmetic. Part II Application to multi-dimensional problems Computing 64 21–47 Occurrence Handle2001i:65053

    MathSciNet  Google Scholar 

  • Hackbusch, W., Khoromskij, B. N.: Towards -matrix approximation of linear complexity. Operator theory: Advances and Applications, vol. 121. Basel: Birkhäuser 2001, pp. 194–220.

  • W. Hackbusch B. N. Khoromskij (2006) ArticleTitleLow-rank Kronecker product approximation to multi-dimensional nonlocal operators. Part I. Separable approximation of multi-variate functions Computing 76 177–202

    Google Scholar 

  • W. Hackbusch B. N. Khoromskij E. Tyrtyshnikov (2005) ArticleTitleHierarchical Kronecker tensor-product approximation J. Numer. Math. 13 207–243 Occurrence Handle10.1163/1569395054012767 Occurrence Handle2006d:65152

    Article  MathSciNet  Google Scholar 

  • H. Luo D. Kolb H.-J. Flad W. Hackbusch T. Koprucki (2002) ArticleTitleWavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes J. Chem. Phys. 117 3625–3638

    Google Scholar 

  • V. N. Temlyakov (1989) ArticleTitleApproximation of functions with bounded mixed derivative Proc. Steklov Inst. Math. 178 IssueID1 275–293 Occurrence Handle90e:00007

    MathSciNet  Google Scholar 

  • Tyrtyshnikov, E. E.: Tensor approximations of matrices generated by asymptotically smooth functions. Mat. Sb. 194(6), 147–160 (2003) (in Russian; translation in Sb. Math. 194(5–6), 941–954(2003)).

  • T. Zhang G. H. Golub (2001) ArticleTitleRank-one approximation to high order tensors SIAM J. Matrix Anal. Appl. 23 534–550 Occurrence Handle10.1137/S0895479899352045 Occurrence Handle2002j:15039

    Article  MathSciNet  Google Scholar 

  • L. Lathauwer ParticleDe B. Moor ParticleDe J. Vandewalle (2000) ArticleTitleOn the best rank-1 and rank-(R 1, . . . , R N) approximation of higher-order tensors SIAM J. Matrix Anal. Appl. 21 1324–1342 Occurrence Handle2001i:15034

    MathSciNet  Google Scholar 

  • L. Lathauwer ParticleDe B. Moor ParticleDe J. Vandewalle (2004) ArticleTitleComputation of the canonical decomposition by means of a simultaneous generalized Schur decomposition SIAM J. Matrix Anal. Appl. 26 295–327 Occurrence Handle2006c:15020

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22-26, 04103, Leipzig, Germany

    W. Hackbusch & B. N. Khoromskij

Authors
  1. W. Hackbusch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. N. Khoromskij
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. Hackbusch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hackbusch, W., Khoromskij, B.N. Low-rank Kronecker-product Approximation to Multi-dimensional Nonlocal Operators. Part II. HKT Representation of Certain Operators. Computing 76, 203–225 (2006). https://doi.org/10.1007/s00607-005-0145-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-005-0145-z

AMS Subject Classifications

Keywords

Search

Navigation