In computing, it is a challenge to obtain reliable information with uncertainties in inputs. To model uncertainty in physical applications, polynomial chaos is studied. The authors of this paper propose a multielement generalized polynomial chaos (ME-gPC) method that deals with stochastic processes (represented by random variables of arbitrary distributions in the random space). After discussions on the existence and convergence, in the L-2 sense, of the space orthogonal decomposition, the authors decompose random spaces with beta and Gaussian distributions, as samples, into smaller elements, with respect to a conditional probability density function (PDF) and a set of orthogonal polynomials. Their numerical experiments on a 1.5GH AMD CPU indicate that the ME-gPC method is efficient and effective. They further investigate the accuracy and convergence of ME-gPC in adaptively solving sample stochastic differential equations (both ordinary differential equations and partial differential equations, even with stochastic discontinuity). The computational results suggest that the performance of the ME-gPC method is comparable with that of the standard Monte Carlo method.

Among the ever-increasing number of theories and technologies in computing, one often finds astonishing results. This paper primarily reports computational results of numerical solutions on stochastic differential equations with arbitrary probability distribution. However, the general idea of orthogonal polynomial decomposition has broader applications in computing, including dealing with uncertainty. For this reason, readers in other areas of computing will benefit from this paper by seeing one more application of orthogonal decomposition and approximation.