Collaborative Research: Computation of instantons in complex nonlinear systems

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A wide variety of systems exhibit rare events -- events far from the average system behavior with low probability of occurring. Rare events can have significant consequences, and improved understanding of their occurrence can aid in the design or management of such systems. The characterization of the likelihood of rare events is essential in all systems in which stochasticity plays an important role, as it allows us to take advantage of such events if they are desirable and to avoid them if they present a threat. The outcomes of this project will contribute to the understanding of rare events in complex systems, in particular, fluid dynamics and related geophysical systems. Further applications include the characterization of extreme events in the context of epidemics, population dynamics, and molecular biology. The goal of this project is to develop efficient computational methods to characterize the most likely way rare events occur in complex stochastic systems and to estimate the tails of their probability distributions. For this purpose, the investigators will develop efficient algorithms to compute the so-called "instantons" that are minimizers of the action functional that large deviation theory associates with the stochastic differential equation describing the system's evolution. Numerical methods to calculate instantons will first be developed in the context of turbulence (in particular Burgers equation, magneto-hydrodynamics (MHD), Navier-Stokes equations, and the surface-quasi-geostrophic (SQG) equation) driven by diffusive processes. Then the investigators will extend the methods to stochastic equations that are driven by non-Markovian noise (fractional Brownian motion) or jump-processes, which play an important role in physics, biology, and chemistry. Finally, the investigators will calculate fluctuations around the instantons to get finer estimates of their probability of occurrence via prefactor calculations.