Reduction techniques for the numerical solution to nonlinear stochastic PDEs

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).

This work proposes and analyzes a sparse grid stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms (input data of the model).
This method can be viewed as an extension of the Stochastic Collocation method proposed in [Babuska-Nobile-Tempone, Technical report, MOX, Dipartimento di Matematica, 2005] which consists of a Galerkin approximation in space and a collocation at the zeros of suitable tensor product orthogonal polynomials in probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. The full tensor product spaces suffer from the curse of dimensionality since the dimension of the approximating space grows exponentially fast in the number of random variables. If the number of random variables is moderately large, this work proposes the use of sparse tensor product spaces utilizing either Clenshaw-Curtis or Gaussian abscissas. For both situations this work provides rigorous convergence analysis of the fully discrete problem and demonstrates algebraic convergence of the ``probability error'' with respect of the number of points in each direction of the probability space. The problem setting in which this procedure is recommended as well as suggestions for future enhancements to the method are discussed.

The complexity of such problems may also be decreased by means of reduced order modeling (ROM) in physical space-time. 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 context of proper orthogonal decomposition-based ROMs, we explore the effects on the accuracy of statistical information about outputs determined from ensembles of such solutions. The coupling of this technique with sparse approximation is a tool with which we plan to devote further investigation in a higher dimensional probability space.

Numerical examples demonstrate the analytic results and show the effectiveness of such methods.