Numerics and analysis of Cahn-Hilliard critical points

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

Instantons for rare events in heavy-tailed distributions

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

Sharp Asymptotic Estimates for Expectations, Probabilities, and Mean First Passage Times in Stochastic Systems with Small Noise

T. Grafke, T. Schäfer, and E. Vanden-Eijnden

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

A Large Deviation Theory Approach to Rogue Waves

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

Link

Samplepath large deviations: Numerics and Theory

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 occurrence can be 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 address 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

Details...

Samplepath Large Deviations: Theory and Numerical Tools

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. 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)

  2. 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", ArXiv (2021) (link)

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

  4. 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)

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

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

Multistable climate regimes

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. Concretely, 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. 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.

Approximate Optimal Controls via Instanton Expansion for Low Temperature Free Energy Computation

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

Rogue Waves: Freaks of Nature Studied with Math and Lasers

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

Link

Stability of Atmospheric Jets

Atmospheric flows on rotating planets are prone to form jets that are surprisingly stable in time. This behaviour can be predicted by simulating the planet's atmosphere numerically. In certain parameter regimes, the underlying dynamical system has multiple locally stable solutions, corresponding to atmosphere configurations with different numbers of jets. This project explores the mechanisms by which random turbulent fluctuations in the atmosphere drive the system to transition between these fixed points, effectively creating or destroying atmospheric jets in the process.