Comparison of Non-Intrusive Polynomial Chaos and Stochastic Collocation Methods for Uncertainty Quantification

Abstract

Non-intrusive polynomial chaos expansion (PCE) and stochastic collocation (SC) methods are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. PCE estimates coefficients for known orthogonal polynomial basis functions based on a set of response function evaluations, using sampling, linear regression, tensor-product quadrature, or Smolyak sparse grid approaches. SC, on the other hand, forms interpolation functions for known coefficients, and requires the use of structured collocation point sets derived from tensor-products or sparse grids. When tailoring the basis functions or interpolation grids to match the forms of the input uncertainties, exponential convergence rates can be achieved with both techniques for general probabilistic analysis problems. In this paper, we explore relative performance of these methods using a number of simple algebraic test problems, and analyze observed differences. In these computational experiments, performance of PCE and SC is shown to be very similar, although when differences are evident, SC is the consistent winner over traditional PCE formulations. This stems from the practical difficulty of optimally synchronizing the formof the PCE with the integration approach being employed, resulting in slight over- or under-integration of prescribed expansion form. With additional nontraditional tailoring of PCE form, it is shown that this performance gap can be reduced, and in some cases, eliminated.