Efficient approximation of random fields for numerical applications

Abstract

We consider the rapid computation of separable expansions for the approximation of random fields. We compare approaches based on techniques from the approximation of non-local operators on the one hand and based on the pivoted Cholesky decomposition on the other hand. We provide an a-posteriori error estimate for the pivoted Cholesky decomposition in terms of the trace. Numerical examples validate and quantify the considered methods. The Karhunen-Loève expansionLet (Ω, F, P) be a separable, complete probability space and let D ⊂ R d be a Lipschitz domain. In the sequel, we consider random fields a ∈ L 2 P Ω; L 2 (D) . For a given random field a, we denote the related centered field according to a 0 (ω, x) := a(ω, x) − E[a](x).Moreover, we define the Hilbert-Schmidt operator associated to a 0 asOne can show that the covariance operator C := SS is of trace-class, i.e. Tr C :One readily verifiesAdditionally, C is symmetric and positive semi-definite. Therefore, by the spectral theorem, C exhibits a representation of the formwhere {(λ i , φ i )} i∈I denote the corresponding eigen-pairs.is called Karhunen-Loève expansion with respect to a. Herein, the random variables {X m } m∈I are given according to Finite element approximationFor the approximation of spatial functions in L 2 (D), we employ (parametric) finite elements of order s. To that end, we introduce a family of quasi-uniform triangulations T h for D with mesh width h and define the spacesThen, given a function v ∈ H t (D) with 0 ≤ t ≤ s, it holds due to the Bramble-Hilbert lemma the approximation estimateuniformly in h.In the sequel, we assume that the random field a exhibits additional spatial regularity, i.e. a ∈ L 2 P (Ω; H p (D) for some p > 0. Then, we may consider the spatial approximationIn terms of the trace, we obtain the following approximation result in V s h . Theorem 2.1. Let a ∈ L 2 P Ω; H p (D) . Then, the spatially approximated, centered random field Q h a 0 satisfies the error estimateBy the application of the theorem and the approximation estimate (1) it is straightforward to show the following Corollary 2.2. The trace error satisfiesand its covariance according to C h,M := P h C h P h . We arrive at the subsequent approximation result., then the random field a h,M given by (2) satisfies the error estimateThe theorem indicates that, after fixing the ansatz space V s h , the approximation error of the stochastic field is controllable in terms of the discretized operators C h and C h,M . The optimal choice of P h in terms of minimizing the trace error is the orthogonal projection onto the dominant invariant subspace of C h , i.e. U M,h := span{φ 1,h , . . . , φ M,h } corresponding to the M dominant eigenvalues of C h . The related Karhunen-Loève expansion then readswhere the random variables are given according toIn this setting, the discretization of the stochastic field implies a change of the stochastic model induced by (5). The pivoted Cholesky decompositionBased on the observation in Theorem 2.3 and the subsequent discussion, we ...

Analysis of the domain mapping method for elliptic diffusion problems on random domains

“…Especially, we introduce here the related function spaces which are used in the rest of this article. For further details on the KarhunenLoève expansion in general and also on computational aspects, we refer to [10,11,17,29]. …”

“…In both examples, we employ the pivoted Cholesky decomposition, cf. [15,17], in order to approximate the Karhunen-Loève expansion of V. The spatial discretization is performed by using piecewise linear parametric finite elements on the mapped domain V(D ref , y i ) for each sample y i . It would of course be also possible to perform the computations on the reference domain.…”

An interpolation‐based fast multipole method for higher‐order boundary elements on parametric surfaces

“…The first example shall be concerned with the boundary value problems (20) and (21) in the unit ball with a prescribed analytical solution based on the spherical harmonic Y 2 0 ,…”

“…The Dirichlet and Neumann data for (20) and (21) are then given by Furthermore, on the sphere, the spherical harmonic is an eigenfunction to all integral operators under consideration, i.e. there holds…”

“…In particular, no specific properties of the integral kernel are required except for its smoothness apart from the diagonal. For example, in [20], a lowest order black-box FMM on parametric surfaces has been employed to discretize correlation kernels.…”