An efficient and accurate method for the identification of the most influential random parameters appearing in the input data for pdes

Abstract

Cut-HDMR expansions (also referred to as anchored-ANOVA expansions) have often been used to represent multivariate functions in high dimensions because they can be used to identify unimportant variables. Past efforts in this direction have examined only the separate influence of each variable. However, simple examples show that variables that have small separate influences can have large interactions, and thus those variables should not be ignored. In this paper, a methodology is developed for determining the importance of variables by examining both their separate and pairwise effects. As a result, not only are all unimportant terms omitted from the cut-HDMR expansion, but all important terms are retained. This is in contrast to existing methods that can omit important pairwise interaction terms. The application and effectiveness of the new methodology are demonstrated for a nonlinear system of partial differential equations having random inputs; specifically, we consider a magnetohydrodynamics setting which also serves to illustrate that a realistic problem can indeed involve random input parameters that have small individual influences but large pairwise influences. In such settings, the new method is not only computationally attractive, but it is the only method that can correctly identify all important individual effects and pairwise interactions. Also discussed is a possible further application of the new methodology, namely reducing the cost of sparse-grid approximations of quantities of interest that depend on the solution of a partial differential equation.