Abstract: Modeling the effective macroscopic dynamics of complex systems as noise-driven motion in a potential landscape has found its use for topics ranging from protein folding to the thermohaline ocean circulation. I will discuss the estimation of such models from timeseries, focussing on a methodology that makes use of the spectral properties (leading eigenpairs) of the Fokker-Planck operator associated with the diffusion process. This methodology is well suited to infer stochastic differential equations that give effective, coarse-grained descriptions of multiscale systems. I will discuss estimation of coordinate-dependent diffusion, subsampling, nonconstant sampling intervals and inference from non-equilibrium data.

Abstract: Much can be learned about systems exhibiting complex dynamics by studying the evolution of probability densities rather than single trajectories. In the stochastic case, this evolution is governed by the transfer semigroup which allows to connect the correlation functions and power spectra to the Fokker-Planck equation. Here, we propose to approximate the transfer operators of high-dimensional systems by Markov operators on a reduced space. While these Markov operators do not in general constitute a semigroup, rigorous results can be obtained regarding their spectral properties, in particular allowing to reconstruct correlation functions and quantify mixing in the reduced space. The approach is applied to the study of the variability and the stability stochastic nonlinear oscillators exhibiting resonant behavior. New analytical and numerical results are found for the mixing spectrum of the Hopf bifurcation with additive noise, bringing new insights on the phenomena of noise-induced oscillations and phase diffusion. These results allow to give new interpretations on the stochastic dynamics of high-dimensional climate models and support the applicability of the method to the study of stochastic bifurcations.

Abstract: In this talk I shall illustrate an approach to study the dynamics of stochastic PDEs (or more generally stochastic dynamical systems) with respect to parameters using deterministic continuation methods. In particular, I shall focus on the case of local fluctuations for the stochastic Allen-Cahn equation and explain the practical implementation as well as applications in various scientific disciplines.

Abstract: Linear matrix equations such as the Lyapunov and Sylvester equations play an important role in the analysis of dynamical systems, in control theory, in eigenvalue computation, and in other scientific and engineering application problems. A variety of robust numerical methods exists for the solution of small dimensional linear equations, whereas the large scale case still provides a great challenge. In this talk we review several available methods, from classical ADI to recently developed projection methods making use of ``second generation'' Krylov subspaces. All methods reply on the possible low-rank form of the given data. Both algebraic and computational aspects will be considered.

Abstract: Techniques from numerical bifurcation analysis are very useful when studying transitions between steady states of flows and the instabilities that are involved. In this presentation we discuss how we use parameter continuation in determining probability density functions of flows governed by stochastic partial differential equations near fixed points under small noise perturbations. We first discuss the traditional way of doing this by stochastically forced time forward simulation, and then show how this can also be done by solving generalized Lyapunov equations using a novel iterative method involving low-rank approximations. One of the advantages of this method is that preconditioning techniques that are known from iterative methods for linear systems can be used. We illustrate the capabilities of the method on a phenomenon in physical oceanography: the occurrence of multiple equilibria in the Atlantic Meridional Ocean Circulation.