Continuation of probability density functions using a generalized Lyapunov approach

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.

Numerical bifurcation methods applied to climate models: analysis beyond simulation

“…The Gaussian probability density function at the fixed point can then be computed directly from C. When M is singular, special methods have been devised to cope with the singular part (Baars et al, 2017); also in this case, a generalized Lyapunov equation determines the covariance matrix C. Efficient solution methods for high-dimensional versions of Eq. 8are presented in Baars et al (2017).…”

Numerical Bifurcation Methods applied to Climate Models: Analysis beyond Simulation

“…The Gaussian probability density function at the fixed point can then be computed directly from C. When M is singular, special methods have been devised to cope with the singular part (Baars et al, 2017); also in this case, a generalized Lyapunov equation determines the covariance matrix C. Efficient solution methods for high-dimensional versions of (8) are presented in Baars et al (2017).…”

“…For a spatially two-dimensional ocean-only model, the covariance matrices C were determined from solving a Lyapunov equation 8in Baars et al (2017) for the case of noise in the freshwater forcing. While here it served only to test the new Lyapunov equation solver (RAILS), the methodology was extended recently to compute (noise-induced) transition probabilities of the AMOC and to relate that probability to the stability indicator Σ (Castellana et al, 2019).…”

Residual-based iterations for the generalized Lyapunov equation

“…To give further motivation to the suggested subspace in (5.1), we draw parallels with the (standard) rational Krylov subspace for the standard Lyapunov equation. The reasoning in this section can be compared to [4,Section 2]. We consider the equation…”