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Original Articles

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

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Pages 1131-1149 | Received 29 Dec 2006, Accepted 04 Feb 2007, Published online: 28 Aug 2007
 

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.

Acknowledgements

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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