ANOVA Expansions and Efficient Sampling Methods for Parameter Dependent Nonlinear PDEs

Abstract

The impact of parameter dependent boundary conditions on solutions of a class of nonlinear partial differential equations and on optimization problems constrained by such equations is considered. The tools used to gain insights about these issues are the Analysis of Variance (ANOVA) expansion of functions and the related notion of the effective dimension of a function; both concepts are reviewed. The effective dimension is then used to study the accuracy of truncated ANOVA expansions. Then, based on the ANOVA expansions of functionals of the solutions, the effects of different parameter sampling methods on the accuracy of surrogate optimization approaches to constrained optimization problems are considered. Demonstrations are given to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. Although the results are presented and discussed in the context of surrogate optimization problems, most also apply to other settings such as stochastic ensemble methods and reduced-order modeling for nonlinear partial differential equations.