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Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems

Maarten Arnst · Christian Soize · Roger Ghanem

21 citations0/0 files storedSIAM/ASA Journal on Uncertainty Quantification

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Probabilistic and Robust Engineering Design Model Reduction and Neural Networks Gaussian Processes and Bayesian Inference

Abstract

In this paper, we present a hybrid method that combines Monte Carlo sampling and spectral methods for solving stochastic coupled problems. After partitioning the stochastic coupled problem into subsidiary subproblems, the proposed hybrid method entails iterating between these subproblems in a way that enables the use of the Monte Carlo sampling method for subproblems that depend on a very large number of uncertain parameters and the use of spectral methods for subproblems that depend on only a small or moderate number of uncertain parameters. To facilitate communication between the subproblems, the proposed hybrid method shares between the subproblems a reference representation of all the solution random variables in the form of an ensemble of samples; for each subproblem solved by a spectral method, it uses a dimension-reduction technique to transform this reference representation into a subproblem-specific reduced-dimensional representation to facilitate a computationally efficient solution in a reduced-dimensional space. After laying out the theoretical framework, we provide an example relevant to microelectomechanical systems.

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References

1991Stochastic Finite Elements: A Spectral Approach 2010Numerical Methods for Stochastic Computations: A Spectral Method Approach 2004Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure 2010Spectral methods for uncertainty quantification : with applications to computational fluid dynamics 2009A least-squares approximation of partial differential equations with high-dimensional random inputs 2012On the Statistical Dependence for the Components of Random Elasticity Tensors Exhibiting Material Symmetry Properties 2010Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data 2011A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension

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