Uncertainty Analysis
Differential equations are used to model a wide range of systems and processes in engineering, physics, biology, chemistry and the environmental sciences. These systems are subject to a wide range of uncertainty in initial and boundary conditions, model coefficients, forcing terms and geometry. The effects of such uncertainty should be traced through the system thoroughly enough to allow one to evaluate their effects on prediction of model outputs.
Polynomial Chaos
Polynomial Chaos (PC) expansions are an efficient means of representing random processes in stochastic in differential equations to help quantify uncertainty. Stochastic Galerkin (SG) methods based on PC expansions have a number of advantages over traditional uncertainty techniques. SG methods exhibit much faster rates of convergence than traditional Monte-Carlo methods and unlike perturbation methods and second-moment analysis SG is able to deal with highly non-linear systems with large uncertainties in the random inputs.
SG necessitates the solution of a system of coupled equations that require
efficient and robust solvers and the modification of existing deterministic
code. A non-intrusive method referred to as Stochastic Collocation (SC) addresses this limitation.
SC methods utilise interpolation methods and project a set of deterministic simulations, evaluated
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 much faster convergence than Monte-Carlo methods
The objective of this website is not to provide an in-depth review of the mathematical theory of SG and SC methods, but rather provide a concise summary of the strengths and limitations of these methods. A tutorial is available to provide a brief introduction to Polynomial Chaos expansions, Stochastic Galerkin and Stochastic Collocation methods and their use in quantifying uncertainty in differential equations. For those more interested an extensive list of publications is available. If you are wondering if Polynomial chaos is for you check out the applications page which gives a list of the problems for which Polynomial Chaos has already been used.