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

Sergey V. Dolgov, Vladimir A. Kazeev, Boris N. Khoromskij

Research output: Contribution to journalArticle

1 Citation (Scopus)
4 Downloads (Pure)

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 languageEnglish
Pages (from-to)136-155
Number of pages20
JournalMathematics and Computers in Simulation
Volume145
Early online date27 Oct 2017
DOIs
Publication statusPublished - 1 Mar 2018

Fingerprint

Direct Product
Elliptic Equations
Tensor Product
Tensors
Tensor
Dependent
Coefficient
Stochastic PDEs
Karhunen-Loève Expansion
Second Order Elliptic Equations
GMRES
Curse of Dimensionality
Preconditioning
Collocation
Hypercube
Elliptic Problems
High-dimensional
Discretization
Numerical Experiment
Entire

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

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

In: Mathematics and Computers in Simulation, Vol. 145, 01.03.2018, p. 136-155.

Research output: Contribution to journalArticle

@article{120cafe9437e474aadb26d7c8c534206,
title = "Direct tensor-product solution of one-dimensional elliptic equations with parameter-dependent coefficients",
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–Lo{\`e}ve 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.",
keywords = "Elliptic equations, Parametric problems, Preconditioning, Sherman–Morrison correction, Tensor formats",
author = "Dolgov, {Sergey V.} and Kazeev, {Vladimir A.} and Khoromskij, {Boris N.}",
year = "2018",
month = "3",
day = "1",
doi = "10.1016/j.matcom.2017.10.009",
language = "English",
volume = "145",
pages = "136--155",
journal = "Mathematics and Computers in Simulation",
issn = "0378-4754",
publisher = "Elsevier",

}

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 -