Efficient estimation of polynomial chaos proxies using generalized sparse quadrature

We investigate the use of sparse grid methods in computing polynomial chaos (PC) proxies for forward stochastic problems associated with numerically-expensive simulators. These are problems where some input parameters are random with known distributions, and stochastic properties of the simulator output are desired. The bottleneck for PC proxy construction is the estimation of the coefficients, which typically require computationally intensive forward simulations and multi-dimensional integration. To minimize the number of simulations, we compare two methods for computing polynomial coefficients using sparse quadrature integration: generalized Fejér quadrature (FQ), and sparse reduced quadrature (RQ). We compare the efficiency (as determined by the number of quadrature points needed to accurately estimate coefficients) of these methods for a 5dimensional stochastic electromagnetic problem. Paradoxically, we find that for general weight functions, sparse FQ requires very high degree exactness to accurately estimate proxy coefficients, which makes this scheme very inefficient. In contrast, RQ requires the minimum number of quadrature points for a pre-defined polynomial exactness. By using the sparse reduced quadrature approach, PC can apply to problems with arbitrary input PDFs and high-dimensional spaces. The trade-off is that sparse FQ has nested abscissae allowing for adaptive refinement of integration degree, while RQ does not.