Gel’fand–Yaglom type equations for calculating fluctuations around instantons in stochastic systems

Schorlepp, Timo; Grafke, Tobias; Grauer, Rainer

doi:10.1088/1751-8121/abfb26

Cited by 17 publications

(23 citation statements)

References 54 publications

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“…Improved estimates are possible in principle when taking into account the fluctuations around the instantons, as discussed e.g. in [23]. The computational cost of computing this fluctuation determinant is vastly bigger than the already large problem sizes encountered in the optimisation problem in this work.…”

Section: Discussionmentioning

confidence: 91%

“…the result of which is analysed in the main text for the three-dimensional NSE. Sharper asymptotics in the sense of a prefactor analysis as in [23] are beyond the scope of the current work.…”

Section: Setup Of the Problemmentioning

confidence: 99%

“…In contrast to this setup, we are interested in flows at a given and possibly large Reynolds number. This can be achieved by realising that the small noise (small Re) limit can be exchanged for an extreme event limit (see remark 1 in [23]): If the length and time scales are chosen such that ε = χ 0 /(νa 2 0 ) → 0, and we focus on an event with |O[u(•, 0)]| = a 0 0 (for an observable with dimension velocity over length), for a given Reynolds number, the instanton estimate for the typical event itself and its probability will be accurate for sufficiently large a 0 or sufficiently extreme events. For high Re, these observables must take very extreme values for the scaling limit to apply, making it very hard to observe in DNS.…”

Section: Stochastic Action and Minimisersmentioning

confidence: 99%

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

Schorlepp,

Grafke,

May

et al. 2021

Preprint

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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.

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Section: Discussionmentioning

confidence: 91%

Section: Setup Of the Problemmentioning

confidence: 99%

Section: Stochastic Action and Minimisersmentioning

confidence: 99%

See 1 more Smart Citation

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

Schorlepp,

Grafke,

May

et al. 2021

Preprint

Self Cite

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“…It is common to express the first correction to the Freidlin--Wentzell small temperature asymptotics as the determinant of a Hessian matrix. As a result, the perturbative formula (3.10) can be understood as a prefactor analysis, and the integral of the Riccati matrix as a continuous version of the determinant prefactor that arises for instance in the Eyring--Kramers formula (see, e.g., [4] and references therein, as well as [29,57]). Following Remark 3.1, our methodology allows one to compute…”

Section: 2mentioning

confidence: 99%

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

Ferré¹,

Grafke²

2021

Multiscale Model. Simul.

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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.

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“…In contrast, surprisingly little work has been done on the numerical side of prefactor calculations (see however [38]). The main objective of this paper is to show how to extend methods such as MAM or gMAM to efficiently estimate prefactors in the context of the calculations of expectations, probabilities, and mean first passage times.…”

Section: Introductionmentioning

confidence: 99%

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

Grafke,

Schäfer,

Vanden-Eijnden

2021

Preprint

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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.

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Gel’fand–Yaglom type equations for calculating fluctuations around instantons in stochastic systems

Cited by 17 publications

References 54 publications

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

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

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

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

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