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

Sergey Dolgov, John Pearson, Dmitry Savostyanov, Martin Stoll

Research output: Contribution to journalArticle

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
DOIs
Publication statusPublished - 12 Nov 2015

Bibliographical note

© 2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Keywords

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

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