Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. (English) Zbl 1120.65004

Todor, Radu Alexandru; Schwab, Christoph

IMA J. Numer. Anal. 27, No. 2, 232-261 (2007).

Authors’ summary: A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient \(a(x,\omega)\) in a bounded domain \(D\subset\mathbb R^d\) is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic \((x\in D)\) and stochastic \((\omega\in\Omega)\) variables in \(a(x,\omega)\) 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 \(a(x,\omega)\) 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.

Reviewer: Grigori N. Milstein (Ekaterinburg)

MSC:

65C30	Numerical solutions to stochastic differential and integral equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs

Keywords:

partial differential equations with stochastic coefficients; Karhúnen-Loève expansion; polynomial chaos; sparse tensor-product approximation; Galerkin method; elliptic boundary-value problem; Legendre expansions; collocation; convergence; algorithm