Sparse Data Representation of Random Fields

Mathematical models with uncertainties are often described by stochastic partial differential equations (SPDEs) with multiplicative noise. The coefficients, the right-hand side, the boundary conditions are modelled by random fields. As a result the solution is also a random field. We offer to use the Karhunen-Loève expansion (KLE) to compute a sparse data format for the fast generation and representation of these random fields. 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 both, the sparse hierarchical matrix format as well as the low-rank Kronecker tensor format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)