Continuation of Probability Density Functions using a Generalized Lyapunov Approach

TL;DR

This paper introduces a computational method leveraging a generalized Lyapunov approach to efficiently determine probability density functions of stochastic PDEs near fixed points, demonstrated on ocean circulation models.

Contribution

It presents a novel iterative low-rank solution technique for a generalized Lyapunov equation applied to stochastic PDEs, enhancing analysis of fluid flow transitions.

Findings

Successfully applied to ocean circulation models with multiple steady states.

Demonstrated efficient computation of probability densities near fixed points.

Provides a new tool for studying stochastic fluid dynamics transitions.

Abstract

Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.