Title: An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

Speaker: Clayton Webster, Florida State University

Date/Time: Tuesday, November 28, 2006, 1:00pm – 2:00pm

Location: CSRI Building/Room 90

Brief Abstract: This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). Here, we especially address the situation where the input data are assumed to depend on a moderately large number of random variables, where in general the curse of dimensionality is encountered. This method can be viewed as an extension of the Sparse Grid Stochastic Collocation method proposed in [Nobile-Tempone-Webster, Technical report #85, MOX, Dipartimento di Matematica, 2006] which consists of a Galerkin approximation in space and a collocation, in probability space, at the zeros of sparse tensor product spaces utilizing either Clenshaw-Curtis or Gaussian interpolants.

As a consequence of the collocation approach our techniques naturally lead to the solution of uncoupled deterministic problems as in the Monte Carlo method. Our previous sparse collocation procedure is very effective for problems whose input data depend on a moderate number of random variables, which “weigh equally” in the solution. For such an isotropic situation the displayed convergence is faster than standard collocation techniques built upon full tensor product spaces. On the other hand, the convergence rate deteriorates when we attempt to solve highly anisotropic problems, such as those appearing when the input random variables come e.g. from Karhunen-Lo`eve -type truncations of “smooth” random fields. In such cases, a full anisotropic tensor product approximation may still be more effective for a small or modest number of random variables. However, if the number of random variables is large, the construction of the full tensor product spaces becomes infeasible, since the dimension of the approximating space grows exponentially fast in the number of random variables. Instead, this work proposes the use of anisotropic sparse tensor product spaces constructed from the Smolyak algorithm utilizing suitable abscissas. This approach is particularly attractive in the case of truncated expansions of random fields, since the anisotropy can be tuned to the decay properties of the expansion. We will present a priori and a posteriori procedures for choosing the anisotropy of the sparse grids which are extremely effective for the problems under study.

This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence of the “probability error” in the asymptotic regime and algebraic convergence of the “probability error” in the pre-asymptotic regime, with respect to the total number of collocation points. Numerical examples exemplify the theoretical results and are used to compare this approach with several others, including standard Monte Carlo. In particular, for moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.

CSRI POC: Scott Collis, (505) 284-1123