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

by R. A. Todor and Ch. Schwab

(Report number 2006-05)

Abstract
A scalar, elliptic boundary value problem in divergence form with stochastic diffusion coefficient a(x,U) 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 e.g. Karhunen-Loève or Legendre polynomial chaos expansion in the sense of N. Wiener [Wie38]. Deterministic, approximate solvers are based on projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M dimensional parametric problem. Under regularity assumptions on the fluctuation of $a(x, omega)$ in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analyzed in terms of the chaos dimension M and of the number N of deterministic problems to be solved as both, dimension M and the multiresolution level of the sparse discretization resp. the degree of the polynomial chaos expansion increase simultaneously. Based on analytic regularity estimates of the solution to the truncated parametric deterministic problem, new sparse FE spaces for the discretization in the parametric variable are proposed. Optimal convergence rates of the semi-discrete solution to the stochastic problem, in terms of the number N of deterministic problems to be solved, are proved for these spaces when the dimension M of the parameter space increases simultaneously with the multiresolution level in the sparse approximation resp. the spectral order in the polynomial chaos approximation.

Keywords: PDE's with Stochastic Data, Karhunen-Loève Expansion, Polynomial Chaos, Sparse Tensor Product Approximation

@Techreport{TS06_354,
  author = {R. A. Todor and Ch. Schwab},
  title = {Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2006-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2006/2006-05.pdf },
  year = {2006}
}

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