Tobias Grafke

G. Margazoglou, T. Grafke, A. Laio, and V. Lucarini, Proc. R. Soc. A 447 (2021) 2250

Abstract

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyze their interplay. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize the data landscape of the simulation output to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate.

doi:10.1098/rspa.2021.0019

arXiv

T. Schorlepp, T. Grafke, and R. Grauer, J. Phys. A: Math. Theor. 54 (2021) 235003

Abstract

In recent years, instanton calculus has successfully been employed to estimate tail probabilities of rare events in various stochastic dynamical systems. Without further corrections, however, these estimates can only capture the exponential scaling. In this paper, we derive a general, closed form expression for the leading prefactor contribution of the fluctuations around the instanton trajectory for the computation of probability density functions of general observables. The key technique is applying the Gel'fand-Yaglom recursive evaluation method to the suitably discretized Gaussian path integral of the fluctuations, in order to obtain matrix evolution equations that yield the fluctuation determinant. We demonstrate agreement between these predictions and direct sampling for examples motivated from turbulence theory.

doi:10.1088/1751-8121/abfb26

arXiv

M.L. Bujorianu, R.S. MacKay, T. Grafke, S. Naik, E. Boulougouris

Abstract

We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extensi on of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs.

arXiv

T. Grafke, S. Scholtes, A. Wagner, M. Westdickenberg

Abstract

We explore recent progress and open questions concerning local minima and saddle points of the Cahn-Hilliard energy in $d\ge 2$ and the critical parameter regime of large system size and mean value close to $-1$. We employ the String Method of E, Ren, and Vanden-Eijnden — a numerical algorithm for computing transition pathways in complex systems — in $d=2$ to gain additional insight into the properties of the minima and saddle point. Motivated by the numerical observations, we adapt a method of Caffarelli and Spruck to study convexity of level sets in $d\ge2$.

arXiv

M. Alqahtani, and T. Grafke, J. Phys. A: Math. Theor. 54 (2021), 175001

Abstract

Large deviation theory and instanton calculus for stochastic systems is widely used to gain insight into the evolution and probability of rare events. At its core lies the realization that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by "convexifying" the rate function through nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation.

doi:10.1088/1751-8121/abe67b

arXiv

Using experimental data and instanton theory to model rogue waves as extreme events at SIAM CSE21.

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Summary: Rare but extreme events in complex systems can often efficiently be described by samplepath large deviations: In the limit of some smallness-parameter approaching zero (such as temperature for chemical reactions, inverse number of particles for thermodynamic limits, or inverse timescale separation for multiscale systems), probabilities and most likely pathways of occurence can are readily accessible. For large and strongly coupled stochastic systems, such as climate, atmosphere, or ocean, the corresponding computations pose a huge numerical challenge. These method borrow heavily from field theory, and represent the rare probability as a path integral, necessitating the computation of instantons and fluctuation determinants. In this project, we adress these challenges, including (1) how to compute the large deviation minimizer (instanton) for large systems, (2) how to compute next-order prefactor corrections, and (3) how to deal with heavy-tailed distributions

Relevant publications

1. T. Grafke, T. Schäfer, and E. Vanden-Eijnden, "Sharp Asymptotic Estimates for Expectations, Probabilities, and Mean First Passage Times in Stochastic Systems with Small Noise", Communications on Pure and Applied Mathematics 77 (2024), 2268 (link)

2. T. Schorlepp, S. Tong, T. Grafke, and G. Stadler, "Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems", Statistics and Computing 33 (2023), 137 (link)

3. T. Schorlepp, T. Grafke, and R. Grauer, "Symmetries and Zero Modes in Sample Path Large Deviations", J Stat Phys 190 (2023), 50 (link)

4. T. Schorlepp, T. Grafke, and R. Grauer, "Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems", J. Phys. A: Math. Theor. 54 (2021), 235003 (link)

5. M. Alqahtani, and T. Grafke, "Instantons for rare events in heavy-tailed distributions", J. Phys. A: Math. Theor. 54 (2021), 175001 (link)

6. G. Ferré and T. Grafke, "Approximate Optimal Controls via Instanton Expansion for Low Temperature Free Energy Computation", SIAM Multiscale Model. Simul. 19(3) (2021), 1310 (link)

7. T. Grafke, and E. Vanden-Eijnden, "Numerical computation of rare events via large deviation theory", Chaos 29 (2019), 063118 (link)

8. T. Grafke, "String Method for Generalized Gradient Flows: Computation of Rare Events in Reversible Stochastic Processes", J. Stat. Mech. 2019/4 (2019), 043206 (link)

G. Ferré and T. Grafke, SIAM Multiscale Model. Simul. 19(3) (2021) 1310–1332

Abstract

The computation of free energies is a common issue in statistical physics. A natural technique to compute such high- dimensional integrals is to resort to Monte Carlo simulations. However, these techniques generally suffer from a high variance in the low temperature regime, because the expectation is often dominated by high values corresponding to rare system trajectories. A standard way to reduce the variance of the estimator is to modify the drift of the dynamics with a control enhancing the probability of rare events, leading to so-called importance sampling estimators. In theory, the optimal control leads to a zero-variance estimator; it is, however, defined implicitly and computing it is of the same difficulty as the original problem. We propose here a general strategy to build approximate optimal controls in the small temperature limit for diffusion processes, with the first goal to reduce the variance of free energy Monte Carlo estimators. Our construction builds upon low noise asymptotics by expanding the optimal control around the instanton, which is the path describing most likely fluctuations at low temperature. This technique not only helps reducing variance, but it is also interesting as a theoretical tool since it differs from usual small temperature expansions (WKB ansatz). As a complementary consequence of our expansion, we provide a perturbative formula for computing the free energy in the small temperature regime, which refines the now standard Freidlin-Wentzell asymptotics. We compute this expansion explicitly for lower orders, and explain how our strategy can be extended to an arbitrary order of accuracy. We support our findings with illustrative numerical examples.

doi:10.1137/20M1385809

arXiv

The elusive waves, once thought to be myths, are explained by the same math that's found in a wide range of settings.

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