Type of Document	Dissertation
Author	Webster, Clayton Garrett
Author's Email Address	webster@scs.fsu.edu
URN	etd-03302007-154630
Title	Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Parial Differential Equations with Random Input Data
Degree	Doctor of Philosophy
Department	Mathematics, Department of
Advisory Committee	Advisor Name Title Max Gunzburger Committee Chair Janet Peterson Committee Member Kyle Gallivan Committee Member Raul Tempone Committee Member
Keywords	Multivariate Polynomial Interpolation Smolyak Algorithm Anisotropic Sparse Grids Uncertainty Quantification Finite Elements Differential Equations Pdes With Random Data Collocation Techniques
Date of Defense	2007-02-15
Availability	unrestricted
Abstract The objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian abscissas. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just 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 in the number of random variables. When this number is moderately large, we combine the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh equally in the solution; the latter approach is ideal when solving highly anisotropic problems depending on a relatively small number of random variables. We also include a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each problem. These procedures are very effective for the problems under study. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence in the asymptotic regime and algebraic convergence in the pre-asymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and compare this approach with several others, including the standard Monte Carlo. For moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very efficient and superior to all examined methods. Due to the high cost of effecting each realization of the PDE this work also proposes the use of reduced-order models (ROMs) that assist in minimizing the cost of determining accurate statistical information about outputs from ensembles of realizations. We explore the use of ROMs, that greatly reduce the cost of determining approximate solutions, for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decomposition-based ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
Files	Filename       Size       Approximate Download Time (Hours:Minutes:Seconds)     28.8 Modem   56K Modem   ISDN (64 Kb)   ISDN (128 Kb)   Higher-speed Access    phd_thesis.pdf 7.08 Mb 00:32:46 00:16:51 00:14:45 00:07:22 00:00:37