Andrew Leach's Abstracts

       Andrew Leach
       Ph.D. Student
       Applied Mathematics

       Conference Summary
       11th AIMS Conference on Dynamical Systems, Differential Equations, and Applications
       July 1 - July 5, 2016
       Orlando, FL

Abstract

Symmetrization of Rare Event Samplers for Stochastic Differential Equations

Andrew Leach, Kevin Lin, & Matthias Morzfeld

Many interesting behaviors in stochastic differential equations (SDE) occur infrequently and are difficult to observe through direct simulation. Recently, optimal control based sampling methods have been proposed for efficient sim- ulation of rare events in SDE. We analyze the performance of these techniques when the noise parameter is small, and show that the relative variance of such methods is order one in the noise parameter. Moreover, we show that this order can be improved by a symmetrization procedure akin to antithetic variates. We illustrate our small noise analysis with numerical examples, and compare the control based samplers to other methods.

Abstract for Lay Audience

Simulating Rare Events
Andrew Leach, Kevin Lin, & Matthias Morzfeld

Some of the most intriguing phenomena in nature are also among the least frequent to be observed. For example, geological evidence shows that Earth’s magnetic poles have undergone numerous reversals, where magnetic North be- comes magnetic South and visa versa. The reversals occur somewhat unpre- dictably, and analysis of data has offered little insight into what the indicators of an upcoming reversal are.

Cast as a mathematical model, phenomena like the magnetic pole reversal are known as rare events. Counterintuitively, the less likely a rare event is, the easier it is to understand just how it would occur. By studying the most likely way a rare event could occur, numerical techniques can be developed to better estimate their likelihood. These techniques borrow from the deep mathematical theory for rare events, and couple them with control theory techniques from the engineering discipline. My collaborators and I have been able to show that a modification of these techniques provides a significant improvement in accuracy with little additional computational cost.