Towards Goal-Oriented Stochastic Design Employing Adaptive Collocation Methods

Abstract

Non-intrusive polynomial chaos expansion (NIPCE) methods based on orthogonal polynomials and stochastic collocation (SC) methods based on Lagrange interpolation polynomials are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic dependence. Both techniques reside in the collocation family, in that they sample the response metrics of interest at selected locations within the random domain without intrusion to simulation software. In this work, we explore the use of polynomial order refinement (prefinement) approaches, both uniform and adaptive, in order to automate the assessment of UQ convergence and improve computational efficiency. In the first class of p-refinement approaches, we employ a general-purpose metric of response covariance to control the uniform and adaptive refinement processes. For the adaptive case, we detect anisotropy in the importance of the random variables as determined through variance-based decomposition and exploit this decomposition through anisotropic tensor-product and anisotropic sparse grid constructions. In the second class of p-refinement approaches, we move from anisotropic sparse grids to generalized sparse grids and employ a goal-oriented refinement process using statistical quantities of interest. Since these refinement goals can frequently involve metrics that are not analytic functions of the expansions (i.e., beyond low order response moments), we additionally explore efficient mechanisms for accurately and efficiently estimated tail probabilities from the expansions based on importance sampling.