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A hybrid collocation-perturbation approach for PDEs with random domains

Abstract

Consider a linear elliptic PDE defined over a stochastic stochastic geometry a function of N random variables. In many application, quantify the uncertainty propagated to a quantity of interest (QoI) is an important problem. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method and added as a correction term to the large variation sparse grid component. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem making it suitable for large dimensional problems. The computational cost of the correction term increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.

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Funding

This material is based upon work supported by the National Science Foundation under Grant No. 1736392. Research reported in this technical report was supported in part by the National Institute of General Medical Sciences (NIGMS) of the National Institutes of Health under award number 5R01GM131409-03.

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Correspondence to Julio E. Castrillón-Candás.

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Communicated by: Anthony Nouy

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Castrillón-Candás, J.E., Nobile, F. & Tempone, R.F. A hybrid collocation-perturbation approach for PDEs with random domains. Adv Comput Math 47, 40 (2021). https://doi.org/10.1007/s10444-021-09859-6

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  • DOI: https://doi.org/10.1007/s10444-021-09859-6

Keywords

  • Uncertainty quantification
  • Stochastic collocation
  • Perturbation
  • Stochastic PDEs
  • Finite elements
  • Complex analysis
  • Smolyak sparse grids