Application of hierarchical matrices for computing the Karhunen–Loève 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-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.

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

“…The kernel is chosen to be a member of the Matérn covariance family, the exponential covariance kernel κ(x, y) = exp(− x − y /l), where l is the integral length scale and is chosen to be 1. It can be shown that this kernel satisfies the requirements of asymptotic smoothness [33] as stated in Section 2. We also make the following choices for the parameters: we pick η = 0.75 and n min = 32.…”

“…In terms of applications, it is useful to consider three kinds of covariance kernels [33]: 1. isotropic and translation invariant, i.e., κ(x, y) = κ(|x−y|); 2. stationary and anisotropic, i.e., κ(x, y) = κ(x − y); 3. non-stationary. Some possible choices for the covariance kernel κ(·, ·) arise from the Matérn family of covariance kernels [34], corresponding to a stationary and isotropic stochastic process.…”

Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen–Loève expansion

“…However, in the presence of round-off errors, these quantities are not numerically zero. We compare the results for three different covariance kernels defined in (15) and we have A = M QM and B = M . The results are summarized in Table II.…”

“…To demonstrate the effect of correlation length l on the accuracy of the randomized calculations, we consider the following numerical experiment. The eigenvalues are computed for the KLE using the covariance kernel κ ν=5/2 as defined in Equation (15). The domain for the computations is [−1, 1] and the number of grid points are 501.…”

“…Here, f k is the actual error in the low-rank representation and the random estimator e k from Proposition 1. We use κ 3/2 defined in Equation (15).…”