### Abstract

We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen–Loève expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor train and quantized tensor train formats. We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.

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

Pages (from-to) | 136-155 |

Number of pages | 20 |

Journal | Mathematics and Computers in Simulation |

Volume | 145 |

Early online date | 27 Oct 2017 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

### Fingerprint

### Keywords

- Elliptic equations
- Parametric problems
- Preconditioning
- Sherman–Morrison correction
- Tensor formats

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics

### Cite this

*Mathematics and Computers in Simulation*,

*145*, 136-155. https://doi.org/10.1016/j.matcom.2017.10.009

**Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients.** / Dolgov, Sergey V.; Kazeev, Vladimir A.; Khoromskij, Boris N.

Research output: Contribution to journal › Article

*Mathematics and Computers in Simulation*, vol. 145, pp. 136-155. https://doi.org/10.1016/j.matcom.2017.10.009

}

TY - JOUR

T1 - Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients

AU - Dolgov, Sergey V.

AU - Kazeev, Vladimir A.

AU - Khoromskij, Boris N.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen–Loève expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor train and quantized tensor train formats. We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.

AB - We consider a one-dimensional second-order elliptic equation with a high-dimensional parameter in a hypercube as a parametric domain. Such a problem arises, for example, from the Karhunen–Loève expansion of a stochastic PDE posed in a one-dimensional physical domain. For the discretization in the parametric domain we use the collocation on a tensor-product grid. The paper is focused on the tensor-structured solution of the resulting multiparametric problem, which allows to avoid the curse of dimensionality owing to the use of the separation of parametric variables in the tensor train and quantized tensor train formats. We suggest an efficient tensor-structured preconditioning of the entire multiparametric family of one-dimensional elliptic problems and arrive at a direct solution formula. We compare this method to a tensor-structured preconditioned GMRES solver in a series of numerical experiments.

KW - Elliptic equations

KW - Parametric problems

KW - Preconditioning

KW - Sherman–Morrison correction

KW - Tensor formats

UR - http://www.scopus.com/inward/record.url?scp=85034781533&partnerID=8YFLogxK

U2 - 10.1016/j.matcom.2017.10.009

DO - 10.1016/j.matcom.2017.10.009

M3 - Article

AN - SCOPUS:85034781533

VL - 145

SP - 136

EP - 155

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

SN - 0378-4754

ER -