Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis

TL;DR

This paper introduces an adaptive stochastic collocation method with mesh refinement and dimensionality reduction for efficient uncertainty analysis in high-dimensional systems with discontinuities, demonstrated on complex examples.

Contribution

It presents a novel SCAMR approach combining adaptive mesh refinement, gPC expansion, and dimensionality reduction for high-dimensional stochastic problems with discontinuities.

Findings

Effective in high-dimensional problems with up to 300 input dimensions.

Accurately captures discontinuities and non-smooth features.

Outperforms existing algorithms in efficiency and accuracy.

Abstract

Presence of a high-dimensional stochastic parameter space with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on the second order gPC approximation error to track discontinuities or non-smoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the full-dimensional problem to a number of lower-dimensional subproblems. Specifically, this criterion checks all the existing interactions between input dimensions of a specific problem based on the high-dimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting dimensions. The efficiency of the approach is demonstrated using both smooth and non-smooth function examples with input dimensions up to 300, and the approach is compared against other existing algorithms.