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Calculating the Malliavin derivative of some stochastic mechanics problems

Hauseux, Paul, Hale, Jack S. and Bordas, Stephane 2017. Calculating the Malliavin derivative of some stochastic mechanics problems. PLoS ONE 12 (12) , e0189994. 10.1371/journal.pone.0189994

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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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Engineering
Additional Information: This is an open access article distributed under the terms of the Creative Commons Attribution License
Publisher: Public Library of Science
ISSN: 1932-6203
Date of First Compliant Deposit: 29 August 2018
Date of Acceptance: 6 December 2017
Last Modified: 17 Jul 2019 11:16
URI: http://orca-mwe.cf.ac.uk/id/eprint/111996

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