Large Deviation Minimisers for Stochastic Partial Differential Equations with Degenerate Noise

TL;DR

This paper develops an adapted computational method for large deviation minimisers in stochastic PDEs with degenerate noise, enabling analysis of rare transitions in complex multistable systems.

Contribution

It introduces a novel approach combining optimal control, large deviation theory, and numerical optimization to handle degenerate noise in multistable stochastic PDEs.

Findings

Efficient computation of transition pathways in biological systems.

Application to transition to turbulence in pipe flow.

Demonstration of method's effectiveness in various complex systems.

Abstract

Noise-induced transitions between multistable states happen in a multitude of systems, such as species extinction in biology, protein folding, or tipping points in climate science. Large deviation theory is the rigorous language to describe such transitions for non-equilibrium systems in the small noise limit. At its core, it requires the computation of the most likely transition pathway, solution to a PDE constrained optimization problem. Standard methods struggle to compute the minimiser in the particular coexistence of (1) multistability, i.e. coexistence of multiple long-lived states, and (2) degenerate noise, i.e. stochastic forcing acting only on a small subset of the system's degrees of freedom. In this paper, we demonstrate how to adapt existing methods to compute the large deviation minimiser in this setting by combining ideas from optimal control, large deviation theory, and numerical optimisation. We show the efficiency of the introduced method in various applications in biology, medicine, and fluid dynamics, including the transition to turbulence in subcritical pipe flow.