Calculating the Malliavin derivative of some stochastic mechanics problems

• Published in: PloS one Vol. 12; no. 12; p. e0189994
• Main Authors: Hauseux, Paul, Hale, Jack S, Bordas, Stéphane P A
• Format: Journal Article
• Language: English
• Published: United States Public Library of Science 20.12.2017; Public Library of Science (PLoS)
• Subjects: Algorithms; Analysis; Applied mathematics; Automation; Calculi; Calculus; Calculus (Mathematics); Calculus of variations; Computational fluid dynamics; Computer simulation; Cylinders; Discretization; Elastic Modulus; Engineering; Expected values; Finite difference method; Finite element analysis; Finite element method; Fluid flow; Incompressible flow; Mathematical models; Mechanical Phenomena; Mechanics; Mechanics (physics); Methods; Models, Theoretical; Monte Carlo methods; Monte Carlo simulation; Navier-Stokes equations; Partial differential equations; Physical Sciences; Random variables; Stochastic models; Stochastic Processes; Stochasticity; Stokes flow; Stress, Mechanical; Viscoelasticity; Viscosity; United Kingdom
• ISSN: 1932-6203; 1932-6203
• DOI: 10.1371/journal.pone.0189994

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.