Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Low-rank tensor Krylov subspace methods for parametrized linear systems

by D. Kressner and Ch. Tobler

(Report number 2010-16)

Abstract
We consider linear systems depending on possibly many parameters. Solving these systems simultaneously for a standard discretization of the parameter space would require a computational effort growing exponentially in the number of parameters. We show that this curse of dimensionality can be avoided for sufficiently smooth parameter dependencies. For this purpose, computational methods are developed that benefit from the fact that $x(\alpha)$ can be well approximated by tensor of low rank. In particular, low-rank tensor variants of short-recurrence Krylov subspace methods are presented. Numerical experiments for deterministic PDEs with parametrized coefficients and stochastic elliptic PDEs demonstrate the effectiveness of our approach.

Keywords:

@Techreport{KT10_55,
  author = {D. Kressner and Ch. Tobler},
  title = {Low-rank tensor Krylov subspace methods for parametrized linear systems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-16.pdf },
  year = {2010}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).