Title: Seminar: Reduction techniques for the numerical solution to nonlinear stochastic PDEs

Date/Time: Wednesday, April 19, 2006, 1:00 – 2:00 pm

Location: Building 980, Room 95 (Sandia NM)

Brief Abstract: The objective of this work is the development of novel, efficient and reliable reduction techniques, in both physical space-time and multi-dimensional probability spaces, for the numerical solution to various types of nonlinear Stochastic Partial Differential Equations (SPDEs). We propose and analyze a Stochastic-Collocation method to solve some nonlinear elliptic Partial Differential Equations (PDEs) with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials in the probability space. This work is an extension of [Babuska, Nobile and Tempone, Submitted SIAM J. Num. Anal. (2005)] a collocation method for linear elliptic PDEs with random input data. Our methods allow one to treat easily a wider range of situations, such as: input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, random variables that are correlated or have unbounded support. This method seems reasonable if the dimension of the problem is small, since, in the multivariate case we use a full tensor product of collocation points. Therefore, for higher dimensions, we suggest the use interpolation based on the Smolyak sparse algorithm. The use of such sparse approximation techniques is an adequate tool for breaking the complexity of the problem. The complexity of such problems may also be decreased by means of reduced order modeling (ROM) in physical space. We also explore the use of ROMs for determining outputs that depend on solutions of nonlinear parabolic SPDEs driven by space-time white noise. In the contexts of proper orthogonal decomposition-based ROMs, we explore the counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions. The coupling of this technique with sparse approximation is a tool with which we plan to devote further investigation.

CSRI POC: Richard Lehoucq, (505) 845-8929