An optimal sampling rule for nonintrusive polynomial chaos expansions of expensive models

Abstract

In this work we present the optimized stochastic collocation method (OSC). OSC is a new sampling rule that can be applied to polynomial chaos expansions (PCE) for uncertainty quantification. Given a model function, the goal of PCE is to find the polynomial from a given polynomial space that is closest to the model function with respect to the L2-norm induced by a given probability measure. Many PCE methods approximate the involved projection integral by discretization with a finite set of integration points. Our key idea is to choose these integration points through numerical optimization based on an operator norm derived from the discretized projection operator. OSC is a generalization of Gaussian quadrature: both methods coincide for one-dimensional integration and under appropriate problem settings in multidimensional problems. As opposed to many established integration rules, OSC does not generally lead to tensor grids in multidimensional problems. With OSC, the user can specify the number ofintegration points independently ofthe problem dimension and PCE expansion order. This allows one to reduce the number of model evaluations and still achieve a high accuracy. The input parameters can follow any kind of probability distribution, as long as the statistical moments up to a certain order are available. Even statistically dependent parameters can be handled in a straightforward and natural fashion. Moreover, OSC allows reusing integration points, ifresults from earlier model evaluations are available. Gauss-Kronrod and Stroud integration rules can be reproduced with OSC for the respective special cases.