## 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

## Extreme events and instantons in Lagrangian passive scalar turbulence models

M. Alqahtani, L. Grigorio, and T. Grafke

#### Abstract

The advection and mixing of a scalar quantity by fluid flow is an important problem in engineering and natural sciences. If the fluid is turbulent, the statistics of the passive scalar exhibit complex behavior. This paper is concerned with two Lagrangian scalar turbulence models based on the recent fluid deformation model that can be shown to reproduce the statistics of passive scalar turbulence for a range of Reynolds numbers. For these models, we demonstrate how events of extreme passive scalar gradients can be recovered by computing the instanton, i.e., the saddle-point configuration of the associated stochastic field theory. It allows us to both reproduce the heavy-tailed statistics associated with passive scalar turbulence, and recover the most likely mechanism leading to such extreme events. We further demonstrate that events of large negative strain in these models undergo spontaneous symmetry breaking.

arXiv

## Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D Navier-Stokes Equations

T. Schorlepp, T. Grafke, S. May, and R. Grauer

#### Abstract

We investigate the spatio-temporal structure of the most likely configurations realising extremely high vorticity or strain in the stochastically forced 3D incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realising an extreme constraint as solution of a large optimisation problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realisations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry broken vortex-sheet configuration.

arXiv

## Dynamical Landscape and Multistability of a Climate Model

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

## Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems

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

## A new stochastic framework for ship capsizing

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

## 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.