Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

H. Bin Zubair; C. C.W. Leentvaar; C. W. Oosterlee

doi:10.1080/00207160701356365

Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

H. Bin Zubair^*, C. C.W. Leentvaar, C. W. Oosterlee

^*Corresponding author for this work

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

Abstract

Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method.

Original language	English
Pages (from-to)	1131-1149
Number of pages	19
Journal	International Journal of Computer Mathematics
Volume	84
Issue number	8
DOIs	https://doi.org/10.1080/00207160701356365
Publication status	Published - Aug 2007
Externally published	Yes

Bibliographical note

Funding Information:
This research has been partially supported by the Dutch government through the national program BSIK: knowledge and research capacity, in the ICT project BRICKS (http://www.bsik-bricks.nl), theme MSV1, partially by the Government of Pakistan through The HEC-Pakistan research grant, contract Ref: 1-3/PM-OVER/Neth/SPMU/2004 and partially by the Dutch Technology foundation STW. We would like to express our thanks to these sponsors.

Funding

This research has been partially supported by the Dutch government through the national program BSIK: knowledge and research capacity, in the ICT project BRICKS (http://www.bsik-bricks.nl), theme MSV1, partially by the Government of Pakistan through The HEC-Pakistan research grant, contract Ref: 1-3/PM-OVER/Neth/SPMU/2004 and partially by the Dutch Technology foundation STW. We would like to express our thanks to these sponsors.

Keywords

Anisotropic diffusion equation
Coarsening strategies
High-dimensional PDE
Multigrid preconditioned Bi-CGSTAB
Sparse grids

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10.1080/00207160701356365

Cite this

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title = "Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations",

abstract = "Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method.",

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author = "Zubair, {H. Bin} and Leentvaar, {C. C.W.} and Oosterlee, {C. W.}",

note = "Funding Information: This research has been partially supported by the Dutch government through the national program BSIK: knowledge and research capacity, in the ICT project BRICKS (http://www.bsik-bricks.nl), theme MSV1, partially by the Government of Pakistan through The HEC-Pakistan research grant, contract Ref: 1-3/PM-OVER/Neth/SPMU/2004 and partially by the Dutch Technology foundation STW. We would like to express our thanks to these sponsors.",

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AU - Oosterlee, C. W.

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AB - Fast and efficient solution techniques are developed for high-dimensional parabolic partial differential equations (PDEs). In this paper we present a robust solver based on the Krylov subspace method Bi-CGSTAB combined with a powerful, and efficient, multigrid preconditioner. Instead of developing the perfect multigrid method, as a stand-alone solver for a single problem discretized on a certain grid, we aim for a method that converges well for a wide class of discrete problems arising from discretization on various anisotropic grids. This is exactly what we encounter during a sparse grid computation of a high-dimensional problem. Different multigrid components are discussed and presented with operator construction formulae. An option-pricing application is focused and presented with results computed with this method.

KW - Anisotropic diffusion equation

KW - Coarsening strategies

KW - High-dimensional PDE

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KW - Sparse grids

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Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations

Abstract

Bibliographical note

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Keywords

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