Andrew Leach Abstract
Andrew Leach
Ph.D. Candidate
Applied Mathematics GIDP
SIAM* Conference on Application of Dynamical Systems 2015
Snowbird, UT
May 17-21, 2015
*Society for Industrial and Applied Mathematics
ABSTRACT
Coupling Methods as a Tool for Sensitivity Analysis of Stochastic Differential Equations
Abstract:
Under appropriate assumptions of Ergodicity, the expected value of observables for dynamical systems can be approximated by time averages. The introduction of stochastic forcing in the form of additive white noise can result in greatly increased varience of the observable estimates, making sensitivity analysis a delicate task. In this setting, coupling methods can be employed as a tool for varience reduction. The methods and their applications to Stochastic Differential Equations will be discussed
Abstract (for Lay Audience)
Differential Equations describe how the state of a physical system evolves over time, and are the underpinning of many models in fields ranging from Physics to Biology. However, they rarely provide a complete description, and often ignore the more subtle influences on the state of the system. For example, a model for the concentrations of chemicals in a closed system may focus on only a handful of compounds, while ignoring those that appear in lesser concentration. Stochastic Differential Equations accommodate for this lack of accuracy by including the action of a random forcing term, referred to as white noise. While this helps to resolve some of the unaccounted for influences on the system, it also can make both the mathematical and computational analysis of the model more difficult.
To validate the study of a Differential Equations model, it is necessary to establish a robustness of results with respect to the model parameters. If these parameters are varied slightly, the results should only vary slightly in response. In this setting, the \Sensitivity Analysis" techniques to accomplish this are well developed, and are standard practice in the engineering industry. While Stochastic Differential Equation models have become more commonplace, the theory of Sensitivity Analysis has not been adapted adequately. Estimates of the dependence on parameters becomes inaccurate due to the noise in the model, often to the point of being unusable.
The inaccuracy of the estimates used in Sensitivity Analysis can be reduced by employing a technique referred to as \coupling". While it is a standard tool in the Theory of Probability, its use in computation is not well documented. The formalism of Coupling Methods will be presented, and the success of their application to Sensitivity Analysis will be demonstrated through computational experiments.