Application of hierarchical matrices for computing the Karhunen- loève expansion

B. N. Khoromskij; A. Litvinenko; H. G. Matthies

doi:10.1007/s00607-008-0018-3

Application of hierarchical matrices for computing the Karhunen- loève expansion

B. N. Khoromskij, A. Litvinenko^*, H. G. Matthies

^*Corresponding author for this work

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

55 Scopus citations

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-Loève 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.

Original language	English (US)
Pages (from-to)	49-67
Number of pages	19
Journal	Computing (Vienna/New York)
Volume	84
Issue number	1-2
DOIs	https://doi.org/10.1007/s00607-008-0018-3
State	Published - Apr 2009
Externally published	Yes

Bibliographical note

Funding Information:
The authors are appreciative to Eveline Rosseel (K.U.Leuven, Department of Computer Science, Belgium) for valuable comments. We would like also to thank our student Jeremy Rodriguez for the help in providing Tables 1, 2, 5, 6 and 7. It is also acknowledged that this research has been conducted within the project MUNA under the framework of the German Luftfahrtforschungsprogramm funded by the Ministry of Economics (BMWA).

Keywords

Data-sparse approximation
Eigenvalue computation
Hierarchical matrix
Karhunen-Loève expansion
Random fields
Uncertainty quantification

ASJC Scopus subject areas

Software
Theoretical Computer Science
Numerical Analysis
Computer Science Applications
Computational Theory and Mathematics
Computational Mathematics

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10.1007/s00607-008-0018-3

Cite this

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title = "Application of hierarchical matrices for computing the Karhunen- lo{\`e}ve expansion",

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{\`e}ve 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.",

keywords = "Data-sparse approximation, Eigenvalue computation, Hierarchical matrix, Karhunen-Lo{\`e}ve expansion, Random fields, Uncertainty quantification",

author = "Khoromskij, {B. N.} and A. Litvinenko and Matthies, {H. G.}",

note = "Funding Information: The authors are appreciative to Eveline Rosseel (K.U.Leuven, Department of Computer Science, Belgium) for valuable comments. We would like also to thank our student Jeremy Rodriguez for the help in providing Tables 1, 2, 5, 6 and 7. It is also acknowledged that this research has been conducted within the project MUNA under the framework of the German Luftfahrtforschungsprogramm funded by the Ministry of Economics (BMWA). ",

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language = "English (US)",

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N2 - 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-Loève 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.

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Application of hierarchical matrices for computing the Karhunen- loève expansion

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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