Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Sergey Dolgov, John W. Pearson, Dmitry V. Savostyanov, Martin Stoll

Research output: Contribution to journalArticlepeer-review

31 Citations (SciVal)

Abstract

Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.

Original languageEnglish
Pages (from-to)604-623
Number of pages20
JournalApplied Mathematics and Computation
Volume273
Early online date12 Nov 2015
DOIs
Publication statusPublished - 15 Jan 2016

Bibliographical note

Funding Information:
The authors would like to thank two anonymous referees for their useful comments. JWP gratefully acknowledges funding from the Engineering and Physical Sciences Research Council (EPSRC) Fellowship EP/M018857/1. SD and DVS were partially supported by RSCF grants 14-11-00806 , 14-11-00659 and RFBR 13-01-12061_ofi_m , 14-01-00804_A during their visits to Institute of Numerical Mathematics, Moscow, Russia. DVS is also thankful to the MPI Magdeburg for their hospitality.

Publisher Copyright:
© 2015 Elsevier Inc. All rights reserved.

Keywords

  • Fractional calculus
  • Iterative solvers
  • Low-rank methods
  • Preconditioning
  • Schur complement
  • Sylvester equations
  • Tensor equations

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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