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.
We define the higher order moments associated to the stochastic solution of an elliptic BVP in D & R d with stochastic input data. We prove that the k-th moment solves a deterministic problem in D k & R dk , for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting system.
Abstract.We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form −a :where a ∈ R d×d is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse finite element approximation u h on a partition of Ω of mesh size h = hL = 2 −L satisfies the following bound in the streamline-diffusion norm ||| · |||SD, provided u belongs to the space H k+1 (Ω) of functions with square-integrable mixed (k + 1)st derivatives: Mathematics Subject Classification. 65N30. Received March 7, 2007. Received February 11, 2008. Published online July 30, 2008 Dedicated to Henryk Woźniakowski, on the occasion of his 60th birthday.
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