POLYNOMIAL CHAOS

Quantifying Uncertainty

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Tutorial Menu

• Introduction
• Problem Definition
• Representing Stochastic Processes
  • Polynomial Chaos Expansions
  • Generalised Polynomial Chaos Expansions
  • Karhunen-Loeve Expansions
• Stochastic Galerkin Methods
  • General Procedure
  • Example
  • Results
• Stochastic Collocation Methods

INTRODUCTION

Polynomial Chaos (PC) expansions [Weiner38] have risen as efficient means of representing stochastic processes with the intention of quantifying uncertainty in differential equations. PC expansions are based on a probabilistic framework and represent stochastic quantities as spectral expansions of orthogonal polynomials. PC expansions are described here.

Stochastic Galerkin (SG) methods employ PC expansions to represent the solution and inputs to stochastic differential equations [Ghanem 1991, Xiu 2002, Babuska 2004]. A Galerkin projection is used to minimise the error of the truncated expansion and the resulting set of coupled equations can be solved to obtain the expansion coefficients. SG methods are highly suited to dealing with ordinary and partial differential equations and have the ability to deal with steep non-linear dependence of the solution on random model data [Knio 2006]. Provided sufficient smoothness conditions are met, PC estimates of uncertainty converge exponentially with the order of the expansion and, for low dimensions, come with a small computational cost. The general procedure of Stochastic Galekin methods and its well used variants are described here.

SG necessitates the solution of a system of coupled equations that require efficient and robust solvers and the modification of existing deterministic code. Often the forms of the governing equations and/or the deterministic code used to solve the equations is complicated and makes implementing PC difficult or even impossible. A non-intrusive method referred to as Stochastic Collocation (SC) [Tatang 1997, Mathelin 2005, Xiu 2005] has arisen to address this limitation. Instead of constructing prior expansions of the sources of uncertainty and propagating the corresponding coefficients by solving the expanded governing equations, SC methods utilise interpolation methods and project a set of deterministic simulations, determined using carefully chosen sampled parameter sets, onto a polynomial basis. This approach is very useful when endeavouring to quantify uncertainty in models implemented with complex deterministic code which cannot be easily modified. Similar to SG methods SC methods achieve fast convergence when the solutions possess sufficient smoothness in random space. SC methods are discussed here.

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© 2008 John Jakeman