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International Journal for Uncertainty Quantification
IF: 0.967 5-Year IF: 1.301 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2012004727
pages 37-61

A HYBRID GENERALIZED POLYNOMIAL CHAOS METHOD FOR STOCHASTIC DYNAMICAL SYSTEMS

Vincent Heuveline
Engineering Mathematics and Computing Lab (EMCL), Fritz-Erler-Str. 23, Karlsruhe Institute of Technology, Karlsruhe, 76133, Germany
Michael Schick
Engineering Mathematics and Computing Lab (EMCL), Fritz-Erler-Str. 23, Karlsruhe Institute of Technology, Karlsruhe, 76133, Germany

ABSTRACT

Generalized Polynomial Chaos (gPC) is known to exhibit a convergence breakdown for problems involving strong nonlinear dependencies on stochastic inputs, which especially arise in the context of long term integration or stochastic discontinuities. In the literature there are various attempts which address these difficulties, such as the time−dependent generalized Polynomial Chaos (TD-gPC) and the multielement generalized Polynomial Chaos (ME-gPC), both leading to higher accuracies but higher numerical costs in comparison to the standard gPC approach. A combination of these methods is introduced, which allows utilizing parallel computation to solve independent subproblems. However, to be able to apply the hybrid method to all types of ordinary differential equations subject to random inputs, new modifications with respect to TD-gPC are carried out by creating an orthogonal tensor basis consisting of the random input variable as well as the solution itself. Such modifications allow TD-gPC to capture the dynamics of the solution by increasing the approximation quality of its time derivatives.