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Calculating the Malliavin derivative of some stochastic mechanics problems

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  • Paul Hauseux
  • Jack S Hale
  • Stéphane P A Bordas

Abstract

The Malliavin calculus is an extension of the classical calculus of variations from deterministic functions to stochastic processes. In this paper we aim to show in a practical and didactic way how to calculate the Malliavin derivative, the derivative of the expectation of a quantity of interest of a model with respect to its underlying stochastic parameters, for four problems found in mechanics. The non-intrusive approach uses the Malliavin Weight Sampling (MWS) method in conjunction with a standard Monte Carlo method. The models are expressed as ODEs or PDEs and discretised using the finite difference or finite element methods. Specifically, we consider stochastic extensions of; a 1D Kelvin-Voigt viscoelastic model discretised with finite differences, a 1D linear elastic bar, a hyperelastic bar undergoing buckling, and incompressible Navier-Stokes flow around a cylinder, all discretised with finite elements. A further contribution of this paper is an extension of the MWS method to the more difficult case of non-Gaussian random variables and the calculation of second-order derivatives. We provide open-source code for the numerical examples in this paper.

Suggested Citation

  • Paul Hauseux & Jack S Hale & Stéphane P A Bordas, 2017. "Calculating the Malliavin derivative of some stochastic mechanics problems," PLOS ONE, Public Library of Science, vol. 12(12), pages 1-18, December.
    • Handle: RePEc:plo:pone00:0189994
    DOI: 10.1371/journal.pone.0189994
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    References listed on IDEAS

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    1. Yang, Xiao-Jun & Machado, J.A. Tenreiro, 2017. "A new fractional operator of variable order: Application in the description of anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 481(C), pages 276-283.
    2. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "A new derivative with normal distribution kernel: Theory, methods and applications," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 476(C), pages 1-14.
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