Jump to ContentJump to Main Navigation
Show Summary Details

Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year

IMPACT FACTOR 2015: 0.673

SCImago Journal Rank (SJR) 2015: 0.997
Source Normalized Impact per Paper (SNIP) 2015: 0.683
Impact per Publication (IPP) 2015: 0.765

Mathematical Citation Quotient (MCQ) 2015: 0.68

See all formats and pricing
Volume 11, Issue 3 (Jan 2011)


Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems

Daniel Kressner
  • Chair of Numerical Algorithms and HPC, MATHICSE, EPF Lausanne, CH-1015 Lausanne, Switzerland.
  • Email:
Christine Tobler
  • Email:


We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax=λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice.

Keywords: high-dimensional PDE eigenvalue problems; low-rank tensor methods; LOBPCG; ALS; DMRG

About the article

Received: 2011-07-23

Revised: 2011-08-16

Accepted: 2011-09-10

Published in Print:

Citation Information: Computational Methods in Applied Mathematics Comput. Methods Appl. Math., ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.2478/cmam-2011-0020. Export Citation

© Institute of Mathematics, NAS of Belarus. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY-NC-ND 4.0)

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Daniel Kressner and André Uschmajew
Linear Algebra and its Applications, 2016, Volume 493, Page 556
Sergey Dolgov, Boris N. Khoromskij, Alexander Litvinenko, and Hermann G. Matthies
SIAM/ASA Journal on Uncertainty Quantification, 2015, Volume 3, Number 1, Page 1109
Markus Bachmayr and Wolfgang Dahmen
Foundations of Computational Mathematics, 2015, Volume 15, Number 4, Page 839
Sergey Dolgov and Boris Khoromskij
SIAM Journal on Matrix Analysis and Applications, 2013, Volume 34, Number 2, Page 593
Christian Lubich, Thorsten Rohwedder, Reinhold Schneider, and Bart Vandereycken
SIAM Journal on Matrix Analysis and Applications, 2013, Volume 34, Number 2, Page 470
André Uschmajew and Bart Vandereycken
Linear Algebra and its Applications, 2013, Volume 439, Number 1, Page 133
Vladimir Kazeev, Oleg Reichmann, and Christoph Schwab
Linear Algebra and its Applications, 2013, Volume 438, Number 11, Page 4204
Daniel Kressner, Michael Steinlechner, and André Uschmajew
SIAM Journal on Scientific Computing, 2014, Volume 36, Number 5, Page A2346
Sergey V. Dolgov and Dmitry V. Savostyanov
SIAM Journal on Scientific Computing, 2014, Volume 36, Number 5, Page A2248
L. Giraldi, A. Nouy, and G. Legrain
SIAM Journal on Scientific Computing, 2014, Volume 36, Number 4, Page A1850
Dmitry V. Savostyanov
Linear Algebra and its Applications, 2014, Volume 458, Page 217
Arnaud Leclerc and Tucker Carrington
The Journal of Chemical Physics, 2014, Volume 140, Number 17, Page 174111
Wan Chen, Radek Erban, and S. Jonathan Chapman
SIAM Journal on Applied Mathematics, 2014, Volume 74, Number 1, Page 208
Bernhard Beckermann, Daniel Kressner, and Christine Tobler
SIAM Journal on Numerical Analysis, 2013, Volume 51, Number 6, Page 3307
Peter Benner and Tobias Breiten
Numerische Mathematik, 2013, Volume 124, Number 3, Page 441

Comments (0)

Please log in or register to comment.
Log in