The analysis and applications of an anisotropic sparse grid stochastic collocation technique for PDEs with random input data

This talk will consist of two parts: In the first part we propose and analyze an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on anisotropic sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the stochastic collocation approach leads to the solution of uncoupled deterministic problems, similar to sampling-based methods, such as Monte Carlo. This talk includes both a priori and a posteriori approaches to adapt the anisotropy of the sparse grids to each given problem. Our rigorously derived error estimates, for the fully discrete problem, will be described and used to compare the efficiency of the method with several other ensemble-based methods.
These methods have proven to have dramatic impact on several application areas, including statistical mechanics, financial mathematics, bioinformatics, and other fields that must properly predict certain model behaviors. However, in many of these fields it is often the case that not all random input coefficients can be fully realized, and therefore, in the second part of this talk we will provide a mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This parameter identification algorithm couples an adjoint-based deterministic algorithm with the sparse grid stochastic collocation approach discussed in the first part, for tracking statistical quantities of interest from the computational solutions of PDEs driven by random inputs.
Numerical examples illustrate the theoretical results and are used to show that, for moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very efficient and superior to all examined methods, including Monte Carlo.