An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

Title: An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

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Abstract

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methodsmore »« less

Authors:

Xiang, Ma ^[1]; Zabaras, Nicholas ^[1]

Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering, 101 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801 (United States)

Publication Date:: 2009-05-01

OSTI Identifier:: 21167778

Resource Type:: Journal Article

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 228; Journal Issue: 8; Other Information: DOI: 10.1016/j.jcp.2009.01.006; PII: S0021-9991(09)00028-X; Copyright (c) 2009 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; COMPARATIVE EVALUATIONS; CONVERGENCE; DECOMPOSITION; FINITE ELEMENT METHOD; INTERPOLATION; MONTE CARLO METHOD; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; RANDOMNESS; STOCHASTIC PROCESSES

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                    Xiang, Ma, and Zabaras, Nicholas. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations.  United States: N. p., 2009. 
        Web.

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                    Xiang, Ma, & Zabaras, Nicholas. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations.  United States.

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                    Xiang, Ma, and Zabaras, Nicholas. Fri .  
        "An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations".  United States.

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@article{osti_21167778,

  title        = {An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations},

  author       = {Xiang, Ma and Zabaras, Nicholas},

  abstractNote = {In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.},

  doi          = {},

  url          = {https://www.osti.gov/biblio/21167778},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = 8,

  volume       = 228,

  place        = {United States},

  year         = {2009},

  month        = {5}

}

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