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MiS Preprint
81/2008

Application of hierarchical matrices for computing the Karhunen-Loève expansion

Boris N. Khoromskij, Alexander Litvinenko and Hermann G. Matthies

Abstract

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right hand side or the coefficients are represented as random fields.

To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo\`eve expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem.

Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.

Received:
Nov 11, 2008
Published:
Nov 21, 2008
MSC Codes:
65F30, 65F50, 65N35
Keywords:
hierarchical matrices, random fields, Karhunen-Lo\'eve expansion

Related publications

inJournal
2009 Journal Open Access
Boris N. Khoromskij, Alexander Litvinenko and Hermann G. Matthies

Application of hierarchical matrices for computing the Karhunen-Loève expansion

In: Computing, 84 (2009) 1/2, pp. 49-67