Tobias Grafke

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

doi

T. Grafke, A. Laio, Multiscale Modeling & Simulation, Vol. 22, Iss. 1 (2024)

Abstract

Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows to estimate rigorously the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection.

doi:10.1137/23M1563025

arXiv

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.

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.

arXiv

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T. Schorlepp, S. Tong, T. Grafke, and G. Stadler, Statistics and Computing 33 (2023), 137

Abstract

We introduce and compare computational techniques for sharp extreme event probability estimates in stochastic differential equations with small additive Gaussian noise. In particular, we focus on strategies that are scalable, i.e. their efficiency does not degrade upon spatial and temporal refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensions. The method estimates the limiting exponential scaling using a single realization of the random variable, the large deviation minimizer. Finding this minimizer amounts to solving an optimization problem governed by a differential equation. The probability estimate becomes sharp when it additionally includes prefactor information, which necessitates computing the determinant of a second derivative operator to evaluate a Gaussian integral around the minimizer. We present an approach in infinite dimensions based on Fredholm determinants, and develop numerical algorithms to compute these determinants efficiently for the high-dimensional systems that arise upon discretization. We also give an interpretation of this approach using Gaussian process covariances and transition tubes. An example model problem, for which we also provide an open-source python implementation, is used throughout the paper to illustrate all methods discussed. To study the performance of the methods, we consider examples of stochastic differential and stochastic partial differential equations, including the randomly forced incompressible three-dimensional Navier-Stokes equations.

doi:10.1007/s11222-023-10307-2

arXiv