Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

Abstract

A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, ω) in a bounded domain D ⊂ R d is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x ∈ D) and stochastic (ω ∈ Ω) variables in a(x, ω) via Karhúnen-Loève or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, ω) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.

Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients

“…This is the well known Stochastic Galerkin formulation (see e.g. [1,[7][8][9][10]). In this respect a Tensor Product polynomial space that contains all the N-variate polynomials with maximum degree in each variable lower than a given w ∈ N is not a good choice.…”

“…Two relevant polynomial approximation strategies that can be conveniently applied to the problem at hand are the Stochastic Galerkin [1,[7][8][9][10] and the Stochastic Collocation methods [2,[11][12][13], which are a projection technique and an interpolation technique, respectively. In this work, we reconsider the quasi-optimal Stochastic Galerkin method proposed in the previous work [3], and provide rigorous convergence results in the special case in which the analyticity region contains a polydisc in the complex plane C N .…”

Adaptive wavelet methods for elliptic partial differential equations with random operators

“…Sparse tensor product constructions have been shown to be highly effective in [35,6,5,30]. Given sufficient prior knowledge on the regularity of the solution, these methods can be tuned to achieve nearly optimal complexity.…”

“…[5,6,11,35], we consider random components that are linear in y ∈ Γ, 8) with R m ∈ L(V, W * ) for all m. Such operators arise e.g. if A is a differential operator that depends affinely on a random field and this fields is expanded in a series.…”

A multiscale preconditioner for stochastic mortar mixed finite elements

“…However, the stochastic collocation method shares the approximation properties of the stochastic finite element method [6,41,15], making it more efficient than MCS. Choices of collocation points include tensor product of zeros of orthogonal polynomials [5,45], sparse grid approximations [17,32,40,45], and probabilistic collocation [26].…”