Collocation Least-squares Polynomial Chaos Method

摘要

The polynomial chaos (PC) method has been used in many engineering applications to replace the traditional Monte Carlo (MC) approach for uncertainty quantification (UQ) due to its better convergence properties. Many researchers seek to further improve the efficiency of PC, especially in higher dimensional space with more uncertainties. The intrusive PC Galerkin approach requires the modification of the deterministic system, which leads to a stochastic system with a much bigger size. The non-intrusive collocation approach imposes the system to be satisfied at a set of collocation points to form and solve the linear system equations. Compared with the intrusive approach, the collocation method is easy to implement, however, choosing an optimal set of the collocation points is still an open problem. In this paper, we first propose using the low-discrepancy Hammersley/Halton dataset and Smolyak datasets as the collocation points, then propose a least-squares (LS) collocation approach to use more collocation points than the required minimum to solve for the system coefficients. We prove that the PC coefficients computed with the collocation LS approach converges to the optimal coefficients. The numerical tests on a simple 2-dimensional problem show that PC collocation LS results using the Hammersley/Halton points approach to optimal result.