Tobias Grafke

T. Schorlepp, S. Tong, T. Grafke, and G. Stadler

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.

arXiv

T. Schorlepp, T. Grafke, and R. Grauer, J Stat Phys 190 (2023), 50

Abstract

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin-Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman's theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar-Parisi-Zhang equation with flat initial profile.

doi:10.1007/s10955-022-03051-w

arXiv

T. Grafke

Abstract

Metadynamics allow to efficiently sample probability landscapes with multiple maxima, in conditions where transitions between them are rare. It was formulated for sampling processes in detailed balance, with an underlying Gibbs measure. Here, we generalize this procedure to transition pathways in arbitrary dynamics, including irreversible and time-dependent ones. One key realization is that every stochastic diffusion induces a gradient flow in path space, namely for the gradient of its corresponding Onsager-Machlup functional. Using this, we can apply metadynamics in path space for transition path sampling in stochastic systems, including stochastic differential equations and stochastic partial differential equations.

doi:10.48550/arXiv.2211.09476

arXiv

A. Frishman and T.Grafke, Proc. R. Soc. A 478 (2022) 2265

Abstract

The subcritical transition to turbulence, as occurs in pipe flow, is believed to generically be a phase transition in the directed percolation universality class. At its heart is a balance between the decay rate and proliferation rate of localized turbulent structures, called puffs in pipe flow. Here we propose the first-ever dynamical mechanism for puff proliferation---the process by which a puff splits into two. In the first stage of our mechanism, a puff expands into a slug. In the second stage, a laminar gap is formed within the turbulent core. The notion of a split-edge state, mediating the transition from a single puff to a two puff state, is introduced and its form is predicted. The role of fluctuations in the two stages of the transition, and how splits could be suppressed with increasing Reynolds number, are discussed. Using numerical simulations, the mechanism is validated within the stochastic Barkley model. Concrete predictions to test the proposed mechanism in pipe and other wall bounded flows, and implications for the universality of the directed percolation picture, are discussed.

doi:https://doi.org/10.1098/rspa.2022.0218

arXiv

M. Alqahtani, L. Grigorio, and T. Grafke, Phys. Rev. E 106 (2022) 015101

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.

doi:10.1103/PhysRevE.106.015101

arXiv

Once considered a maritime myth, these towering waves can pose serious risks to ships in the open sea. Now scientists are developing ways to predict them before they strike.

Link

T. Schorlepp, T. Grafke, S. May, and R. Grauer, Phil Trans Roy Soc A 380 (2022), 2226

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.

doi:10.1098/rsta.2021.0051

arXiv

A. Frishman and T.Grafke, Phys. Rev. E 105 (2022), 045108

Abstract

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge---states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge.

doi:10.1103/PhysRevE.105.045108

arXiv

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