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.
Highlights
The classical derivative is a fundamental tool of calculus that is widely used across every field of mathematics, science and engineering
The contribution of this paper is as follows; we show the application of the Malliavin Weight Sampling method [15] to four archetypal problems in mechanics
In this paper we deal with random noise and we show numerical results of stochastic mechanics problems where models are defined as partial differential equations (PDEs)
Summary
The classical derivative is a fundamental tool of calculus that is widely used across every field of mathematics, science and engineering. The Malliavin Weight Sampling method (MWS) [16] allows the evaluation of the sensitivity of the expected value of the quantity of interest with respect to the mean value of the stochastic parameter m as [9, 11, 16, 18]:. Under certain condition of regularity [11, 20, 21] when the probability density function (PDF) of the parameter m is known, the Malliavin weight qm associated can be computed directly from the PDF of m This approach can be viewed as an integration by parts, and is a direct result of Malliavin calculus where we take the derivative of random functions rather than the classical derivative. Through a simple practical examples, we will explain how to use the MWS method, determine the specification of the weights for both Gaussian and non-Gaussian distributions on the parameter m, and calculate the Malliavin derivative Eq (3).
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