Tobias Grafke

T. Schorlepp, and T. Grafke

Abstract

We show how to efficiently compute asymptotically sharp estimates of extreme event probabilities in stochastic differential equations (SDEs) with small multiplicative Brownian noise. The underlying approximation is known as sharp large deviation theory or precise Laplace asymptotics in mathematics, the second-order reliability method (SORM) in reliability engineering, and the instanton or optimal fluctuation method with 1-loop corrections in physics. It is based on approximating the tail probability in question with the most probable realization of the stochastic process, and local perturbations around this realization. We first recall and contextualize the relevant classical theoretical result on precise Laplace asymptotics of diffusion processes [Ben Arous (1988), Stochastics, 25(3), 125-153], and then show how to compute the involved infinite-dimensional quantities — operator traces and Carleman-Fredholm determinants — numerically in a way that is scalable with respect to the time discretization and remains feasible in high spatial dimensions. Using tools from automatic differentiation, we achieve a straightforward black-box numerical computation of the SORM estimates in JAX. The method is illustrated in examples of SDEs and stochastic partial differential equations, including a two-dimensional random advection-diffusion model of a passive scalar. We thereby demonstrate that it is possible to obtain efficient and accurate SORM estimates for very high-dimensional problems, as long as the infinite-dimensional structure of the problem is correctly taken into account. Our JAX implementation of the method is made publicly available.

arXiv

Summary: Earth's climate is a highly complex, non-equilibrium and chaotic stochastic system. In this project, we attempt to classify its chaotic attractors with methods from non-equilibrium statistical mechanics, large deviation theory and manifold learning. For example, due to the ice albedo feedback, the climate is known to exist in two locally stable states, the current (warm) climate, and a "snowball" state, where the globe is covered in ice. Some models even suggest additional metastable climate states, such as the slushball Earth. Similarly, the currently active Atlantic Meridional Overturning Circulation (AMOC, colloquially known as Gulf Steram), transports hot water to northern Europe, but is suspected to be only metastable: If perturbed the right way, the climate would persist with the AMOC non-existent. Transitions between these climate states, and their local stability, can in principle be analyzed in light of the non-equilibrium quasipotential, characterizing the expected transition times and most likely escape paths out of the current climate state.

Relevant publications

1. G. Margazoglou, T. Grafke, A. Laio, and V. Lucarini, "Dynamical Landscape and Multistability of a Climate Model", Proc. R. Soc. A 447 (2021), 2250 (link)

2. Jelle Soons, Tobias Grafke, Henk A. Dijkstra, "Optimal Transition Paths for AMOC Collapse and Recovery in a Stochastic Box Model", J Physical Oceanography 54 (2024), 2537 (link)

3. Jelle Soons, Tobias Grafke, Henk A. Dijkstra, "Most Likely Noise-Induced Overturning Circulation Collapse in a 2D Boussinesq Fluid Model", ArXiv (2024) (link)

Jelle Soons, Tobias Grafke, Henk A. Dijkstra

Abstract

The present-day Atlantic Meridional Overturning Circulation (AMOC) is considered to be in a bi-stable regime and hence it is important to determine probabilities and pathways for noise-induced transitions between its equilibrium states. Here, using Large Deviation Theory (LDT), the most probable transition pathways for the collapse and recovery of the AMOC are computed in a stochastic box model of the World Ocean. This allows us to determine the physical mechanisms of noise-induced AMOC transitions. We show that the most likely path of an AMOC collapse starts paradoxically with a strengthening of the AMOC followed by an immediate drop within a couple of years due to a short but relatively strong freshwater pulse. The recovery on the other hand is a slow process, where the North Atlantic needs to be gradually salinified over a course of 20 years. The proposed method provides several benefits, including an estimate of probability ratios of collapse between various freshwater noise scenarios, showing that the AMOC is most vulnerable to freshwater forcing into the Atlantic thermocline region.

doi:10.1175/JPO-D-23-0234.1

arXiv

Reyk Börner, Ryan Deeley, Raphael Römer, Tobias Grafke, Valerio Lucarini, Ulrike Feudel

Abstract

In multistable dynamical systems driven by weak Gaussian noise, transitions between competing states are often assumed to pass via a saddle on the separating basin boundary. In contrast, we show that timescale separation can cause saddle avoidance in non-gradient systems. Using toy models from neuroscience and ecology, we study cases where sample transitions deviate strongly from the instanton predicted by Large Deviation Theory, even for weak finite noise. We attribute this to a flat quasipotential and propose an approach based on the Onsager-Machlup action to aptly predict transition paths.

doi:10.1103/PhysRevResearch.6.L042053

arXiv

J. Liu, J. E. Sprittles, T. Grafke, J. Stat. Mech (2024) 103206

Abstract

Thermally activated phenomena in physics and chemistry, such as conformational changes in biomolecules, liquid film rupture, or ferromagnetic field reversal, are often associated with exponentially long transition times described by Arrhenius' law. The associated subexponential prefactor, given by the Eyring-Kramers formula, has recently been rigorously derived for systems in detailed balance, resulting in a sharp limiting estimate for transition times and reaction rates. Unfortunately, this formula does not trivially apply to systems with conserved quantities, which are ubiquitous in the sciences: The associated zeromodes lead to divergences in the prefactor. We demonstrate how a generalised formula can be derived, and show its applicability to a wide range of systems, including stochastic partial differential equations from fluctuating hydrodynamics, with applications in rupture of nanofilm coatings and social segregation in socioeconomics.

doi:10.1088/1742-5468/ad8075

arXiv

Paolo Bernuzzi, Tobias Grafke

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.

arXiv

Jelle Soons, Tobias Grafke, Henk A. Dijkstra

Abstract

There is a reasonable possibility that the present-day Atlantic Meridional Overturning Circulation is in a bi-stable regime and hence it is relevant to compute probabilities and pathways of noise-induced transitions between the stable equilibrium states. Here, the most probable transition pathway of a noise-induced collapse of the northern overturning circulation in a spatially-continuous two-dimensional model with surface temperature and stochastic salinity forcings is directly computed using Large Deviation Theory (LDT). This pathway reveals the fluid dynamical mechanisms of such a collapse. Paradoxically it starts off with a strengthening of the northern overturning circulation before a short but strong salinity pulse induces a second overturning cell. The increased atmospheric energy input of this two-cell configuration cannot be mixed away quickly enough, leading to the collapse of the northern overturning cell and finally resulting in a southern overturning circulation. Additionally, the approach allows us to compare the probability of this collapse under different parameters in the deterministic part of the salinity surface forcing, which quantifies the increase in collapse probability as the bifurcation point of the system is approached.

arXiv

Nayef Shkeir, Tobias Schäfer, Tobias Grafke

Abstract

Matrix evolution equations occur in many applications, such as dynamical Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic control, machine learning or data assimilation. In many cases, their tightest stability condition is coming from a linear term. Exponential time differencing (ETD) is known to produce highly stable numerical schemes by treating the linear term in an exact fashion. In particular, for stiff problems, ETD methods are a method of choice. We propose an extension of the class of ETD algorithms to matrix-valued dynamical equations. This allows us to produce highly efficient and stable integration schemes. We show their efficiency and applicability for a variety of real-world problems, from geophysical applications to dynamical problems in machine learning.

arXiv

T. Grafke, T. Schäfer, and E. Vanden-Eijnden, Commun. Pure Appl. Math. 77 (2024) 2268

Abstract

Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived for expectations, probabilities, and mean first passage times in a form that is geared towards numerical purposes: they require solving well-posed matrix Riccati equations involving the minimizer of the Freidlin-Wentzell action as input, either forward or backward in time with appropriate initial or final conditions tailored to the estimate at hand. The usefulness of our approach is illustrated on several examples. In particular, invariant measure probabilities and mean first passage times are calculated in models involving stochastic partial differential equations of reaction-advection-diffusion type.

arXiv

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