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

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

doi:10.1016/j.amc.2015.09.042

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

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

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

34 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 language	English
Pages (from-to)	604-623
Number of pages	20
Journal	Applied Mathematics and Computation
Volume	273
Early online date	12 Nov 2015
DOIs	https://doi.org/10.1016/j.amc.2015.09.042
Publication status	Published - 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.

Funding

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.

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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10.1016/j.amc.2015.09.042

Cite this

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title = "Fast tensor product solvers for optimization problems with fractional differential equations as constraints",

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.",

keywords = "Fractional calculus, Iterative solvers, Low-rank methods, Preconditioning, Schur complement, Sylvester equations, Tensor equations",

author = "Sergey Dolgov and Pearson, {John W.} and Savostyanov, {Dmitry V.} and Martin Stoll",

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: {\textcopyright} 2015 Elsevier Inc. All rights reserved.",

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AU - Pearson, John W.

AU - Savostyanov, Dmitry V.

AU - Stoll, Martin

N1 - 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.

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N2 - 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.

AB - 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.

KW - Fractional calculus

KW - Iterative solvers

KW - Low-rank methods

KW - Preconditioning

KW - Schur complement

KW - Sylvester equations

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ER -

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

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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