Efficient Numerical Approximations of Tracking Statistical Quantities of Interest From the Solution of High-Dimensional Stochastic Partial Differential Equations

Abstract

Abstract : Mathematical modeling and computer simulations are nowadays widely used tools to predict the behavior of problems in engineering and in the natural and social sciences. All such predictions are obtained by formulating mathematical models and then using computational methods to solve the corresponding problems. We use a probability theory approach for uncertainty quantification (UQ) since it is particularly well suited for SPDE models, and focus on the broad research areas of algorithmic development and numerical analysis for the discretization of systems of linear or nonlinear SPDEs, building upon and significantly extending our previous successful work. We conduct comprehensive theoretical and computational comparison of the efficiency, accuracy, and range of applicability of non-intrusive methods, such as stochastic collocation methods, and intrusive techniques, such as stochastic Galerkin methods, for solving SPDEs and for UQ applications. We extend the algorithmic and analysis advances wrought by these efforts to the even more challenging settings of optimal control and parameter identification problems for SPDEs. The parameter identification problem is especially important in the SPDE setting since it provides a very useful mechanism for determining statistical information about the input parameters from, e.g., measurements of output quantities. This effort builds on our previous work on adjoint and sensitivity-based methods for deterministic optimal control and parameter identification problems to develop similar methods for tracking statistical quantities of interest from the computational solutions of linear and nonlinear SPDEs driven by high-dimensional random inputs.