In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loève expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.

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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 ...

This article is dedicated to the anisotropic sparse grid quadrature for functions which are analytically extendable into an anisotropic tensor product domain. Taking into account this anisotropy, we end up with a dimension independent error versus cost estimate of the proposed quadrature. In addition, we provide a novel and improved estimate for the cardinality of the underlying anisotropic index set. To validate the theoretical findings, we present several examples ranging from simple quadrature problems to diffusion problems on random domains. These examples demonstrate the remarkable convergence behaviour of the anisotropic sparse grid quadrature in applications.2000 Mathematics Subject Classification. 65D30, 65C30, 60H25.

Abstract. This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normally distributed diffusion coefficient. The key idea of such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments.Key words. multilevel quadrature, PDEs with stochastic data, log-normal diffusion, Karhunen-Loève expansion, finite element method AMS subject classifications. 65N30, 65D32, 60H15, 60H351. Introduction. In this article, we consider multilevel quadrature methods to compute the moments of the solution to elliptic partial differential equations with log-normally distributed diffusion coefficient. The basic idea of the multilevel quadrature is a sparse-grid-like discretization of the underlying Bochner space L 2 P Ω; H 1 0 (D) . The spatial variable is discretized by a classical finite element method whereas the stochastic variable is treated by an appropriately chosen quadrature rule, which naturally leads to a non-intrusive method. Since the problem's solution provides the necessary mixed Sobolev regularity, the approximation errors on the different levels of resolution can be equilibrated in a sparse-grid-like fashion, cf. [8,20,42]. This idea has already been proposed for different quadrature strategies in case of uniformly elliptic diffusion coefficients in [23]. The well-known Multilevel Monte Carlo Method (MLMC), as introduced in [3,15,16,26,27], and also the Randomized Multilevel Quasi-Monte Carlo Method, as introduced in [30], only provide probabilistic error estimates in the mean-square sense. To avoid this drawback, two fully deterministic methods have been proposed in [23], namely the Multilevel Quasi-Monte Carlo Method (MLQMC) and the Multilevel Polynomial Chaos Method (MLPC).The multilevel Monte Carlo method has been considered at first for a log-normal diffusion coefficient in [12] and further been analyzed in [11,40]. However, for deterministic quadrature methods, the log-normal case is much more involved due to the unboundedness of the domain of integration, i.e. R m for some m ∈ N, in combination with the stronger regularity requirements on the integrand. This makes the analysis of the quadrature error difficult. In particular, special regularity results are required which extend those of [2,10,29].For a finite stochastic dimension, we show that the multilevel quasi-Monte Carlo quadrature is feasible also for a log-normal diffusion coefficients if an auxiliary density is introduced.