### 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 |
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Pages (from-to) | 604-623 |

Number of pages | 20 |

Journal | Applied Mathematics and Computation |

Volume | 273 |

DOIs | |

Publication status | Published - 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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## Cite this

Dolgov, S., Pearson, J., Savostyanov, D., & Stoll, M. (2015). Fast tensor product solvers for optimization problems with fractional differential equations as constraints.

*Applied Mathematics and Computation*,*273*, 604-623. https://doi.org/10.1016/j.amc.2015.09.042