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Articles Published in SIAM J. Sci. Comput.

  1. Xiu, D. and Karniadakis, G.E.. "The Weiner-Askey Polynomial Chaos for stochastic differential equations" SIAM J. Sci. Comput.. 24 (2). 2002. pp. 619--644.

    We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos. Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error. Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs.


  2. Wan, X. and Karniadakis, G.E.. "Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures" SIAM J. Sci. Comput.. 28 (3). 2006. pp. 901--928.

    We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.


  3. Lucor, D. and Karniadakis, G.E.. "Adaptive Generalized Polynomial Chaos for Nonlinear Random Oscillators" SIAM J. Sci. Comput.. 26 (2). 2005. pp. 720--735.

    The solution of nonlinear random oscillators subject to stochastic forcing is investigated numerically. In particular, solutions to the random Duffing oscillator with random Gaussian and non-Gaussian excitations are obtained by means of the generalized polynomial chaos (GPC). Adaptive procedures are proposed to lower the increased computational cost of the GPC approach in large-dimensional spaces. Adaptive schemes combined with the use of an enriched representation of the system improve the accuracy of the GPC approach by reordering the random modes according to their magnification by the system.


  4. Debusschere, B.J., Najm, H.N., Pebay, P.P., Knio, O.M., Ghanem, R.G., and Le Maitre, O.P.. "Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes" SIAM J. Sci. Comput.. 26 (2). 2005. pp. 698--719.

    This paper gives an overview of the use of polynomial chaos (PC) expansions to represent stochastic processes in numerical simulations. Several methods are presented for performing arithmetic on, as well as for evaluating polynomial and nonpolynomial functions of variables represented by PC expansions. These methods include Taylor series, a newly developed integration method, as well as a sampling-based spectral projection method for nonpolynomial function evaluations. A detailed analysis of the accuracy of the PC representations, and of the different methods for nonpolynomial function evaluations, is performed. It is found that the integration method offers a robust and accurate approach for evaluating nonpolynomial functions, even when very high-order information is present in the PC expansions.


  5. Xiu, D. and Hesthaven, J.S.. "High-order collocation methods for differential equations with random inputs" SIAM J. Sci. Comput.. 27 (3). 2005. pp. 1118--1139.