Local polynomial chaos expansion method for high dimensional stochastic differential equations

Yi Chen, Purdue University


Polynomial chaos expansion is a widely adopted method to determine evolution of uncertainty in dynamical system with probabilistic uncertainties in parameters. In particular, we focus on linear stochastic problems with high dimensional random inputs. Most of the existing methods enjoyed the efficiency brought by PC expansion compared to sampling-based Monte Carlo experiments, but still suffered from relatively high simulation cost when facing high dimensional random inputs. We propose a localized polynomial chaos expansion method that employs a domain decomposition technique to approximate the stochastic solution locally. In a relatively lower dimensional random space, we are able to solve subdomain problems individually within the accuracy restrictions. Sampling processes are delayed to the last step of the coupling of local solutions to help reduce computational cost in linear systems. We perform a further theoretical analysis on combining a domain decomposition technique with a numerical strategy of epistemic uncertainty to approximate the stochastic solution locally. An establishment is made between Schur complement in traditional domain decomposition setting and the local PCE method at the coupling stage. A further branch of discussion on the topic of decoupling strategy is presented at the end to propose some of the intuitive possibilities of future work. Both the general mathematical framework of the methodology and a collection of numerical examples are presented to demonstrate the validity and efficiency of the method.




Dong, Purdue University.

Subject Area

Applied Mathematics

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