Iterative Polynomial Approximation Adapting to Arbitrary Probability Distribution

Abstract

In this paper, we perform the numerical analysis of a new moment/generalized polynomial chaos (gPC) based approximation method under finite numerical integration. The paper addresses the impact of this constraint on the method, in particular analyzing the interplay between aliasing and truncation errors, depending on the type of functional to be represented. We set a new theoretical result defining conditions under which the iterative procedure ensures a gain after each step, putting forward the existence of a balance between aliasing (i.e., integration accuracy) and truncation errors. We demonstrate that the iterative process is viable in this context. We emphasize the existence of two regimes, an ideal one and another one for which we suggest alternatives. The method is applied to uncertainty propagation problems, i.e., here, nonlinear mappings of a single- (or multiple-)input random variables to a single-output random variable. The originality of the approach is to automatically and iteratively adapt the stochastic approximation space in order to build an accurate recursive representation of the solution.

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