Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures

Xiaoliang Wan · George Em Karniadakis

498 citations0/0 files storedSIAM Journal on Scientific Computing

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Probabilistic and Robust Engineering Design Mathematical functions and polynomials Model Reduction and Neural Networks

Abstract

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.

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References

1991Stochastic Finite Elements: A Spectral Approach 2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2003Modeling uncertainty in flow simulations via generalized polynomial chaos 2004Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations 2005An adaptive multi-element generalized polynomial chaos method for stochastic differential equations 2004Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations

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2012Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion 2009An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations 2010A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2008A comparative study of uncertainty propagation methods for black-box-type problems 2008The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications 2009Multi-element probabilistic collocation method in high dimensions

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