J. Sci. Comput. X (2015)

Sparse Pseudo Spectral Projection Methods with Directional Adaptation for Uncertainty Quantification

J Winokur, D Kim, F Bisetti, O P Le Maitre, and O M Knio. “Sparse Pseudo Spectral Projection Methods with Directional Adaptation for Uncertainty Quantification”. J. Sci. Comput. 1 (2015). In press, pp. 1–28.
J Winokur, D Kim, F Bisetti, O Le Maitre, and O Knio
UQ, spectral methods, polynomial chaos
2015
​​We investigate two methods to build a polynomial approximation of a model output depending on some parameters. The two approaches are based on pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids, and aim at providing a finer control of the resolution along two distinct subsets of model parameters. The control of the error along different subsets of parameters may be needed for instance in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid PSP is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projec- tion surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube​.