Stochastic Integration for Chaotic Dynamical Systems
Emily Williams, Massachusetts Institute of Technology
Stochastic processes are used often in mathematical modeling of phenomena that appear to vary chaotically or in a random manner. Stochastic differential equations are ubiquitous in the formulation of these models, including population dynamics, neuron activity, blood clotting, turbulent diffusion, and more. Stochastic processes may be defined for all time instants in a bounded interval or in an unbounded interval, in which case they are continuous time stochastic processes. We aim to incorporate stochastic modeling to improve the time integration schemes for chaotic systems. Many stochastic integration numerical schemes recursively update the state using the Brownian increment, taken as independent and identically distributed normal random variables with expected value zero and variance equal to the timestep. We propose that this increment should be sampled from some function that is representative of the subgrid (i.e., unresolved) dynamics of the chaotic system. Further, we want to explore the potential stabilizing effect that stochastic models could have on linearized sensitivity of chaotic systems. Initial efforts toward this investigation include developing a higher-order finite-element-based time-marching approach for stochastic initial value problems.
Abstract Author(s): Emily Williams, David Darmofal