A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes

Choi, Minseok; Sapsis, Themistoklis P.

doi:10.1016/J.JCP.2013.02.047

Title: A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes

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Abstract

We study the convergence properties of the recently developed Dynamically Orthogonal (DO) field equations [1] in comparison with the Polynomial Chaos (PC) method. To this end, we consider a series of one-dimensional prototype SPDEs, whose solution can be expressed analytically, and which are associated with both linear (advection equation) and nonlinear (Burgers equation) problems with excitations that lead to unimodal and strongly bi-modal distributions. We also propose a hybrid approach to tackle the singular limit of the DO equations for the case of deterministic initial conditions. The results reveal that the DO method converges exponentially fast with respect to the number of modes (for the problems considered) giving same levels of computational accuracy comparable with the PC method but (in many cases) with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional.

Authors:

Choi, Minseok ^[1]; Sapsis, Themistoklis P. ^[2]

Division of Applied Mathematics, Brown University, Providence, RI 02912 (United States)
Courant Institute of Mathematical Sciences, New York University, NY 10012 (United States)

Publication Date:: Mon Jul 15 00:00:00 EDT 2013

OSTI Identifier:: 22233608

Resource Type:: Journal Article

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 245; Other Information: Copyright (c) 2013 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:: 97 MATHEMATICAL METHODS AND COMPUTING; ADVECTION; CHAOS THEORY; COMPARATIVE EVALUATIONS; CONVERGENCE; EXCITATION; FIELD EQUATIONS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PARTIAL DIFFERENTIAL EQUATIONS; POLYNOMIALS; STOCHASTIC PROCESSES

Citation Formats


                    Choi, Minseok, Sapsis, Themistoklis P., and Karniadakis, George Em, E-mail: george_karniadakis@brown.edu. A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes.  United States: N. p., 2013. 
        Web.  doi:10.1016/J.JCP.2013.02.047.

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                    Choi, Minseok, Sapsis, Themistoklis P., & Karniadakis, George Em, E-mail: george_karniadakis@brown.edu. A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes.  United States.  https://doi.org/10.1016/J.JCP.2013.02.047

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                    Choi, Minseok, Sapsis, Themistoklis P., and Karniadakis, George Em, E-mail: george_karniadakis@brown.edu. 2013.  
        "A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes".  United States.  https://doi.org/10.1016/J.JCP.2013.02.047.

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@article{osti_22233608,

  title        = {A convergence study for SPDEs using combined Polynomial Chaos and Dynamically-Orthogonal schemes},

  author       = {Choi, Minseok and Sapsis, Themistoklis P. and Karniadakis, George Em, E-mail: george_karniadakis@brown.edu},

  abstractNote = {We study the convergence properties of the recently developed Dynamically Orthogonal (DO) field equations [1] in comparison with the Polynomial Chaos (PC) method. To this end, we consider a series of one-dimensional prototype SPDEs, whose solution can be expressed analytically, and which are associated with both linear (advection equation) and nonlinear (Burgers equation) problems with excitations that lead to unimodal and strongly bi-modal distributions. We also propose a hybrid approach to tackle the singular limit of the DO equations for the case of deterministic initial conditions. The results reveal that the DO method converges exponentially fast with respect to the number of modes (for the problems considered) giving same levels of computational accuracy comparable with the PC method but (in many cases) with substantially smaller computational cost compared to stochastic collocation, especially when the involved parametric space is high-dimensional.},

  doi          = {10.1016/J.JCP.2013.02.047},

  url          = {https://www.osti.gov/biblio/22233608},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = ,

  volume       = 245,

  place        = {United States},

  year         = {Mon Jul 15 00:00:00 EDT 2013},

  month        = {Mon Jul 15 00:00:00 EDT 2013}

}

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