### Abstract

Original language | English |
---|---|

Pages (from-to) | 260-291 |

Number of pages | 32 |

Journal | SIAM/ASA Journal on Uncertainty Quantification |

Volume | 7 |

Issue number | 1 |

Early online date | 7 Mar 2019 |

DOIs | |

Publication status | Published - 2019 |

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### Keywords

- math.NA
- 65F10, 65F30, 65N22, 65N30, 65N35

### Cite this

**A hybrid Alternating Least Squares - TT Cross algorithm for parametric PDEs.** / Dolgov, Sergey; Scheichl, Robert.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - A hybrid Alternating Least Squares - TT Cross algorithm for parametric PDEs

AU - Dolgov, Sergey

AU - Scheichl, Robert

PY - 2019

Y1 - 2019

N2 - We consider the approximate solution of parametric PDEs using the low-rank Tensor Train (TT) decomposition. Such parametric PDEs arise for example in uncertainty quantification problems in engineering applications. We propose an algorithm that is a hybrid of the alternating least squares and the TT cross methods. It computes a TT approximation of the whole solution, which is beneficial when multiple quantities of interest are sought. This might be needed, for example, for the computation of the probability density function (PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm., 1986]. The new algorithm exploits and preserves the block diagonal structure of the discretized operator in stochastic collocation schemes. This disentangles computations of the spatial and parametric degrees of freedom in the TT representation. In particular, it only requires solving independent PDEs at a few parameter values, thus allowing the use of existing high performance PDE solvers. In our numerical experiments, we apply the new algorithm to the stochastic diffusion equation and compare it with preconditioned steepest descent in the TT format, as well as with (multilevel) quasi-Monte Carlo and dimension-adaptive sparse grids methods. For sufficiently smooth random fields the new approach is orders of magnitude faster.

AB - We consider the approximate solution of parametric PDEs using the low-rank Tensor Train (TT) decomposition. Such parametric PDEs arise for example in uncertainty quantification problems in engineering applications. We propose an algorithm that is a hybrid of the alternating least squares and the TT cross methods. It computes a TT approximation of the whole solution, which is beneficial when multiple quantities of interest are sought. This might be needed, for example, for the computation of the probability density function (PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm., 1986]. The new algorithm exploits and preserves the block diagonal structure of the discretized operator in stochastic collocation schemes. This disentangles computations of the spatial and parametric degrees of freedom in the TT representation. In particular, it only requires solving independent PDEs at a few parameter values, thus allowing the use of existing high performance PDE solvers. In our numerical experiments, we apply the new algorithm to the stochastic diffusion equation and compare it with preconditioned steepest descent in the TT format, as well as with (multilevel) quasi-Monte Carlo and dimension-adaptive sparse grids methods. For sufficiently smooth random fields the new approach is orders of magnitude faster.

KW - math.NA

KW - 65F10, 65F30, 65N22, 65N30, 65N35

U2 - 10.1137/17M1138881

DO - 10.1137/17M1138881

M3 - Article

VL - 7

SP - 260

EP - 291

JO - SIAM/ASA Journal on Uncertainty Quantification

JF - SIAM/ASA Journal on Uncertainty Quantification

SN - 2166-2525

IS - 1

ER -