Polynomial Chaos Using Transformed Sparse Grids

The topic of general polynomial chaos has received significant attention in the last few years as a means to efficiently estimate model outcomes based on known stochastic processes. The method requires numerical integrations in order to evaluate the expectation integrals that are the coefficients of the stochastic polynomial. The key concern is that these numerical integrations are very time consuming when applied to demanding computational problems such as flow simulation. Therefore, methods which can perform this integration with a minimum number of integration points are highly desirable. An obvious choice is a sparse-grid method based on a 1D Gauss-quadrature rule, because this allows highly accurate integration rules to be designed for arbitrary PDFs in the expectation integral. Unfortunately, Gauss quadrature is a very poor choice for sparse-grid integration, because the corresponding integration rules are weakly nested, and thus do not allow reuse of integration points from one sparse-grid level to the next. Alternatively, Clenshaw-Curtis integration produces very strongly nested quadrature rules, but these are designed for uniform distributions and are inefficient for many PDFs, e.g., Gaussians. In this work, we present a nonlinear transformation of Fejer type 2 quadrature rules that realizes the benefits of having both a high degree of sparsity and being tailored to specific PDFs. We demonstrate that this method has the potential to integrate arbitrary functions in high stochastic dimensions (>8) with orders of magnitude fewer integration points than are required for similar Gauss-quadrature rules.