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.
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Acknowledgments
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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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Khoromskij, B.N., Litvinenko, A. & Matthies, H.G. Application of hierarchical matrices for computing the Karhunen–Loève expansion. Computing 84, 49–67 (2009). https://doi.org/10.1007/s00607-008-0018-3
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DOI: https://doi.org/10.1007/s00607-008-0018-3
Keywords
- Hierarchical matrix
- Data-sparse approximation
- Karhunen–Loève expansion
- Uncertainty quantification
- Random fields
- Eigenvalue computation