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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles and news from researchers in related subjects, suggested using machine learning.Avoid common mistakes on your manuscript.
References
Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge monographs on applied and computational mathematics, vol 4. Cambridge University Press, Cambridge
Babuška I, Tempone R, Zouraris GE (2004) Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J Numer Anal 42(2):800–825 (electronic)
Babuška I, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005–1034 (electronic)
Bai Z, Demmel J, Dongarra J, Ruhe A, van der Vorst H (eds) (2000) Templates for the solution of algebraic eigenvalue problems— practical guide, vol 11 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
Bebendorf M (2000) Approximation of boundary element matrices. Numer Math 86(4): 565–589
Bebendorf M, Hackbusch W (2007) Stabilized rounded addition of hierarchical matrices. Numer Linear Algebra Appl 14(5): 407–423
Bebendorf M, Rjasanow S (2003) Adaptive low-rank approximation of collocation matrices. Computing 70(1): 1–24
Börm S, Grasedyck L (2005) Hybrid cross approximation of integral operators. Numer Math 101(2): 221–249
Börm S, Grasedyck L, Hackbusch W (2003) Hierarchical Matrices, Lecture Note. Max-Planck Institute for Mathematics, Leipzig
Eiermann M, Ernst OG, Ullmann E (2007) Computational aspects of the stochastic finite element method. Comput Vis Sci 10(1): 3–15
Elman HC, Ernst OG, O’Leary DP, Stewart M (2005) Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering. Comput Methods Appl Mech Eng 194(9–11): 1037–1055
Frigo M, Johnson S (1998) FFTW: An adaptive software architecture for the FFT. In: Proc. ICASSP, IEEE, Seattle, WA, (3):1381–1384. http://www.fftw.org
Ghanem R, Spanos D (1991) Spectral stochastic finite-element formulation for reliability analysis. J Eng Mech 117(10): 2351–2370
Ghanem R, Spanos P (1991) Stochastic finite elements: a spectral approach. Springer, New York
Goreinov SA, Tyrtyshnikov EE, Zamarashkin NL (1997) A theory of pseudoskeleton approximations. Linear Algebra Appl 261: 1–21
Grasedyck L, Börm S: \({\mathcal{H}}\) -matrix library. http://www.hlib.org
Grasedyck L, Hackbusch W (2003) Construction arithmetics of \({\mathcal{H}}\)-matrices. Computing 70(4): 295–334
Hackbusch W, Khoromskij BN (2000) A sparse \({\mathcal{H}}\)-matrix arithmetic. II. Application to multi-dimensional problems. Computing 64(1): 21–47
Hackbusch W (1995) Integral equations, vol 120 of International Series of Numerical Mathematics. Birkhäuser Verlag, Basel, 1995. Theory and numerical treatment, Translated and revised by the author from the 1989 German original
Hackbusch W (1999) A sparse matrix arithmetic based on \({\mathcal{H}}\)-matrices. I. Introduction to \({\mathcal{H}}\)-matrices. Computing 62(2): 89–108
Hackbusch W (2004) Hierarchische Matrizen—Algorithmen und Analysis. Max-Planck-Institut für Mathematik, Leipzig
Hackbusch W, Khoromskij BN, Kriemann R (2004) Hierarchical matrices based on a weak admissibility criterion. Computing 73(3): 207–243
Hackbusch W, Khoromskij BN, Tyrtyshnikov EE (2005) Hierarchical Kronecker tensor-product approximations. J Numer Math 13(2): 119–156
Hida T, Kuo H-H, Potthoff J, Streit L (1993) White noise—an infinite-dimensional calculus, vol 253 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1993)
Keese A (2004) Numerical solution of systems with stochastic uncertainties. A general purpose framework for stochastic finite elements. Ph.D. Thesis, TU Braunschweig, Germany
Lanczos C (1995) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J Res Nat Bur Standards 45: 255–282
Lehoucq RB, Sorensen DC, Yang C (1998) ARPACK users’ guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, vol 6 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA
Loève, M (1977) Probability theory I. Graduate Texts in Mathematics, vol 45, 6, 4th edn. Springer, New York
Matthies HG (2007) Uncertainty quantification with stochastic finite elements. Part 1. Fundamentals. Encyclopedia of Computational Mechanics. Wiley, New York
Matthies HG, Keese A (2005) Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput Methods Appl Mech Eng 194(12–16): 1295–1331
Pellissetti MF, Ghanem R (2000) Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv Eng Softw 31(8–9): 607–616
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C—the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge
Riesz F, Sz.-Nagy B (1990) Functional analysis. Dover Books on Advanced Mathematics. Dover Publications Inc., New York, 1990. Translated from the second French edition by Leo F. Boron, Reprint of the 1955 original
Saad Y (1992) Numerical methods for large eigenvalue problems. Manchester University Press, Manchester
Schwab C, Todor RA (2003) Sparse finite elements for elliptic problems with stochastic loading. Numer Math 95(4): 707–734
Schwab C, Todor RA (2003) Sparse finite elements for stochastic elliptic problems—higher order moments. Computing 71: 43–63
Schwab C, Todor RA (2006) Karhunen–Loève approximation of random fields by generalized fast multipole methods. J Comput Phys 217(1): 100–122
Seynaeve B, Rosseel E, Nicolaï B, Vandewalle S (2007) Fourier mode analysis of multigrid methods for partial differential equations with random coefficients. J Comput Phys 224(1): 132–
Todor RA, Schwab C (2007) Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J Numer Anal 27(2): 232–261
Werner D (2000) Funktionalanalysis, extended edn. Springer, Berlin
Wiener N (1938) The homogeneous chaos. Am J Math 60: 897–936
Wu K, Simon H (2000) Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J Matrix Anal Appl 22(2):602–616 (electronic)
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).
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
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