Extending bluff-and-fix estimates for polynomial chaos expansions

Polynomial chaos methods, which are part of a broader class known as stochastic Galerkin schemes, can be used to approximate the solution to a PDE with uncertainties represented by stochastic inputs or parameters. The stochastic solution is expressed as an infinite polynomial expansion truncated to M + 1 terms. The approach is then to derive a resulting system of coupled, deterministic PDEs and solve this system with standard numerical techniques. Some challenges with conventional numeric techniques applied in this context are as follows: (1) the solution to a polynomial chaos M system cannot easily reuse an already existing computer solution to an M-0 system for some M-0 < M, and (2) there is no flexibility around choosing which variables in an M system are more important or advantageous to estimate accurately. This latter point is especially relevant when, rather than focusing on the PDE solution itself, the objective is to approximate some function of the PDE solution that weights the solution variables with relative levels of importance. In Lyman and Iaccarino (2020) [1], we first present a promising iterative strategy (bluff-and-fix) to address challenge (1); we find that numerical estimates of the accuracy and efficiency demonstrate that bluff-and-fix can be more effective than using monolithic methods to solve a whole M system directly. This paper is an extended version of Lyman and Iaccarino (2020) [1] that showcases how bluff-and-fix successfully addresses challenge (2) as well by allowing for choice in which variables are approximated more accurately, in particular when estimating statistical properties such as the mean and variance of an M system solution.

Automated summary

Polynomial chaos methods are used to approximate solutions to PDEs with stochastic inputs, where an infinite polynomial expansion is truncated to a certain number of terms. Challenges include the inability to reuse solutions to previous systems and the lack of flexibility in choosing important variables. The bluff-and-fix iterative strategy is proposed to address these challenges effectively.