Article (Scientific journals)
Calculating the Malliavin derivative of some stochastic mechanics problems
HAUSEUX, Paul; HALE, Jack; BORDAS, Stéphane
2017In PLoS ONE, 12 (12), p. 0189994
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Abstract :
[en] 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.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others
Mathematics
Author, co-author :
HAUSEUX, Paul ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Engineering Research Unit
HALE, Jack  ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Engineering Research Unit
BORDAS, Stéphane ;  University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Engineering Research Unit
External co-authors :
no
Language :
English
Title :
Calculating the Malliavin derivative of some stochastic mechanics problems
Publication date :
20 December 2017
Journal title :
PLoS ONE
ISSN :
1932-6203
Publisher :
Public Library of Science, San Franscisco, United States - California
Volume :
12
Issue :
12
Pages :
e0189994
Peer reviewed :
Peer Reviewed verified by ORBi
Focus Area :
Computational Sciences
European Projects :
FP7 - 279578 - REALTCUT - Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery
FnR Project :
FNR6693582 - Advanced Computational Methods For The Simulation Of Cutting In Surgery, 2013 (01/01/2014-31/12/2015) - Jack Samuel Hale
Name of the research project :
R-AGR-0515 - IRP15 - MOmENTUM (20150101-20180731) - BORDAS Stéphane
Funders :
University of Luxembourg - UL
CE - Commission Européenne [BE]
Available on ORBilu :
since 25 December 2017

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