New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

Venturi, D.

doi:10.1016/J.JCP.2012.07.013

Title: New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

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Abstract

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection-reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.

Authors:

Venturi, D. ^[1]

Division of Applied Mathematics, Brown University, Providence, RI 02912 (United States)

Publication Date:: Thu Aug 30 00:00:00 EDT 2012

OSTI Identifier:: 22192340

Resource Type:: Journal Article

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 231; Journal Issue: 21; Other Information: Copyright (c) 2012 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; ADVECTION; BOUNDARY CONDITIONS; COMPARATIVE EVALUATIONS; EVOLUTION; EXCITATION; INTEGRALS; NONLINEAR PROBLEMS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILISTIC ESTIMATION; PROBABILITY DENSITY FUNCTIONS; RANDOMNESS; SCALARS; STOCHASTIC PROCESSES

Citation Formats


                    Venturi, D., and Karniadakis, G.E., E-mail: george_karniadakis@brown.edu. New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs.  United States: N. p., 2012. 
        Web.  doi:10.1016/J.JCP.2012.07.013.

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                    Venturi, D., & Karniadakis, G.E., E-mail: george_karniadakis@brown.edu. New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs.  United States.  https://doi.org/10.1016/J.JCP.2012.07.013

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                    Venturi, D., and Karniadakis, G.E., E-mail: george_karniadakis@brown.edu. 2012.  
        "New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs".  United States.  https://doi.org/10.1016/J.JCP.2012.07.013.

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@article{osti_22192340,

  title        = {New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs},

  author       = {Venturi, D. and Karniadakis, G.E., E-mail: george_karniadakis@brown.edu},

  abstractNote = {By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection-reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.},

  doi          = {10.1016/J.JCP.2012.07.013},

  url          = {https://www.osti.gov/biblio/22192340},
  journal      = {Journal of Computational Physics},

  issn         = {0021-9991},

  number       = 21,

  volume       = 231,

  place        = {United States},

  year         = {Thu Aug 30 00:00:00 EDT 2012},

  month        = {Thu Aug 30 00:00:00 EDT 2012}

}

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