“Fast Numerical Methods for Stochastic Computations: A Review” is written by Dongbin Xiu, and published in “Communications in computational physics at Purdue University, Department of mathematics” (Xiu, 2009). This paper provides an examination of the emerging state of the art computational techniques for stochastic calculations. The main focus is on effective high-order, functional methods with an emphasis on methods focused on generalized polynomial chaos (gPC). The gPC’s structure and collocation methods for solving stochastic equations are examined with an emphasis on fast algorithms that are suited to complex large-scale problems. The focus is on effective, and realistic high-order methods with a special importance on those focused on generalized polynomial chaos (gPC). Extensive attempts are being made to create precise numerical algorithms such that forecasts for simulations are consistent in that numerical errors are well controlled and understood. Because of the unpredictable existence of the uncertainty, the most dominant approach is to deal with information uncertainty as random variables or random processes.
With gPC, stochastic equations are expressed as orthogonal polynomials of the randomized input variables and various orthogonal polynomials can be selected in order to obtain improved convergence. The two main deployment methods, Galerkin and collocation, will be addressed with the introduction of the gPC framework. The benefit of stochastic collocation is simple – it only involves repeated performance of existing deterministic solvers. On the other hand, the stochastic Galerkin approach is comparatively more cumbersome to apply, particularly as the expansion coefficient equations are almost always coupled. Both methods can achieve rapid integration and high precision and are highly effective in functional calculations when correctly applied. This is because the gPC structure is a natural extension of the multi-dimensional random space of spectral methods. With the science progressing at such a high rate, it is anticipated that new findings will emerge constantly to help us learn and improve the techniques.
Xiu, D. (2009). Fast numerical methods for stochastic computations: a review. Communications in computational physics, 5(2-4), 242-272.