Article2014

Computational Aspects of Stochastic Collocation with Multifidelity Models

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

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Probabilistic and Robust Engineering Design Model Reduction and Neural Networks Mathematical Approximation and Integration

Abstract

In this paper we discuss a numerical approach for the stochastic collocation method with multifidelity simulation models. The method we consider was recently proposed in [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495--A521] to combine the computational efficiency of low-fidelity models with the high accuracy of high-fidelity models. This method is able to produce more accurate results at a much reduced simulation cost. The purpose of this paper includes (1) a presentation of the detailed implementation of the method developed by [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495--A521], which is largely theoretical; (2) an adaptation of that method to handle multifidelity scenarios that are of more practical interest; (3) a closer examination of the method via a set of more comprehensive benchmark examples including several two-dimensional stochastic PDEs with high-dimensional random parameters; and (4) a more detailed investigation of the strengths of the multifidelity approach in [A. Narayan, C. Gittelson, and D. Xiu, SIAM J. Sci. Comput., 36 (2014), pp. A495--A521]. Specifically, we present a numerical algorithm to construct accurate simulation results when multiple low-fidelity models are available. We suggest that trifidelity simulations with a low-fidelity, a medium-fidelity, and a high-fidelity model would be sufficient for most practical problems. The availability of more simulation models (i.e., more than three) would not present much improvement under the current framework.

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References

1991Stochastic Finite Elements: A Spectral Approach 2002The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2008A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2009An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations 2011A non-adapted sparse approximation of PDEs with stochastic inputs 2007Sparse grid collocation schemes for stochastic natural convection problems

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2017A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems 2019Multifidelity Monte Carlo Estimation with Adaptive Low-Fidelity Models 2017Multi-fidelity stochastic collocation method for computation of statistical moments 2015Nonhierarchical multi‐model fusion using spatial random processes 2019Bi-fidelity approximation for uncertainty quantification and sensitivity analysis of irradiated particle-laden turbulence

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