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

Ferré, Grégoire; Grafke, Tobias

doi:10.1137/20m1385809

Cited by 5 publications

(20 citation statements)

References 50 publications

(46 reference statements)

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“…Importantly, one obtains the full limiting rare event probability or probability density instead of merely its exponential scaling, in regimes where direct sampling methods are completely intractable. In this paper, we have first set out to rederive such prefactor formulas at leading order for unique instantons [30][31][32][33], expressed in terms of Riccati matrices, explicitly using tools from field theory, i.e. by evaluating the appearing functional determinants using Forman's theorem [40].…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…Example B. 4 Since previous papers [30][31][32][33] have mostly dealt with additive noise, we test the more general case of multiplicative noise that is included here in a simple toy example. Consider the one-dimensional Itô SDE…”

Section: Data Availabilitymentioning

confidence: 99%

“…As an additional, concrete motivation to go beyond such rough estimates in practical applications, it has been pointed out very recently that for assessing the relative importance of different instantonic transition paths, knowledge of the LDT prefactor at leading order may be vital even at comparably small noise strengths [29]. In the last year, there has been a lot of activity to provide generic numerical tools that also allow for the computation of the leading order term of the large deviation prefactor for the statistics of final time observables of small noise ordinary SDEs using symmetric Riccati matrix differential equations, either forward or backward in time [30][31][32][33]. In an abstract setting, expressions for prefactors in this context, even at arbitrarily high order, have already been known rigorously since the 1980's [2,[34][35][36] and are, not surprisingly, related to a certain operator determinant at the leading order.…”

Section: Introductionmentioning

confidence: 99%

“…Relevant examples of this phenomenon in the context of sample path LDT include the one-dimensional Kardar-Parisi-Zhang (KPZ) equation [43][44][45][46] for the surface height at one point in space and with two-sided Brownian motion initial condition (leading to discrete mirror symmetry breaking), the two-dimensional [47] and three-dimensional [48] incompressible Navier-Stokes equations and a Lagrangian turbulence model [49] (all with rotational symmetry breaking). In all of these cases, due to the underlying symmetries, it turns out that it suffices to integrate a single Riccati equation, corresponding to a single reduced functional determinant evaluation, which thereby allows for a generalization of earlier results [30][31][32][33] without increasing the computational costs. In addition to the examples listed above, further systems where the methods and results of this paper could be applied are those within the scope of the macroscopic fluctuation theory [50], e.g.…”

Section: Introductionmentioning

confidence: 99%

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Symmetries and Zero Modes in Sample Path Large Deviations

2023

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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.

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Section: Discussion and Outlookmentioning

confidence: 99%

Section: Data Availabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetries and Zero Modes in Sample Path Large Deviations

2023

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“…We prove, under geometric conditions, that the estimator proposed in this work is a stochastic viscosity approximation of order 1, hence it is 1-log efficient. We then turn to the next order approximation presented in [12], showing that under similar conditions it provides a stochastic viscosity approximation of order 2, hence it is 2-log efficient and with logarithmic relative variance decaying linearly to zero. A simple numerical application confirms our theoretical findings.…”

Section: Introductionmentioning

confidence: 97%

Stochastic viscosity approximations of Hamilton–Jacobi equations and variance reduction

Ferré

2023

ESAIM: M2AN

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We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which order. We rely for this on a recently introduced notion of stochastic solution for Hamilton–Jacobi–Bellman (HJB) equations. Based on this tool, we introduce the notion of k -stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k-1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.

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Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

Schorlepp,

Tong,

Grafke

et al. 2023

Stat Comput

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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 temporal and possibly spatial refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensional path space. 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 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.

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

Cited by 5 publications

References 50 publications

Symmetries and Zero Modes in Sample Path Large Deviations

Symmetries and Zero Modes in Sample Path Large Deviations

Stochastic viscosity approximations of Hamilton–Jacobi equations and variance reduction

Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

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