Instanton based importance sampling for rare events in stochastic PDEs

Ebener, Lasse; Margazoglou, Georgios; Friedrich, Jan; Biferale, Luca; Grauer, Rainer

doi:10.1063/1.5085119

Cited by 27 publications

(32 citation statements)

References 74 publications

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“…In recent years there has been several attempts to apply these methods for geophysical applications. They have been applied to Lorenz models [275], partial differential equations [226], turbulence problems [77,113,160,164,165], geophysical fluid dynamics [28], heatwaves in general circulation models [217][218][219] and data-based stochastic weather generators [281], and tropical cyclones in regional climate models [212,272]. Here we give an overview of methods and their applications that have been used to study problems directly related to the dynamics of planetary atmospheres and making use of concepts from LDT.…”

Section: Rare Event Sampling Algorithms Based On Large Deviation Theorymentioning

confidence: 99%

Applications of large deviation theory in geophysical fluid dynamics and climate science

et al. 2021

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The climate is a complex, chaotic system with many degrees of freedom. Attaining a deeper level of understanding of climate dynamics is an urgent scientific challenge, given the evolving climate crisis. In statistical physics, many-particle systems are studied using Large Deviation Theory (LDT). A great potential exists for applying LDT to problems in geophysical fluid dynamics and climate science. In particular, LDT allows for understanding the properties of persistent deviations of climatic fields from long-term averages and for associating them to low-frequency, large-scale patterns. Additionally, LDT can be used in conjunction with rare event algorithms to explore rarely visited regions of the phase space. These applications are of key importance to improve our understanding of high-impact weather and climate events. Furthermore, LDT provides tools for evaluating the probability of noise-induced transitions between metastable climate states. This is, in turn, essential for understanding the global stability properties of the system. The goal of this review is manifold. First, we provide an introduction to LDT. We then present the existing literature. Finally, we propose possible lines of future investigations. We hope that this paper will prepare the ground for studies applying LDT to solve problems encountered in climate science and geophysical fluid dynamics.

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Section: Rare Event Sampling Algorithms Based On Large Deviation Theorymentioning

confidence: 99%

Applications of large deviation theory in geophysical fluid dynamics and climate science

et al. 2021

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“…The theory of Freidlin and Wentzell [1] gives asymptotic probability estimates of rare events in dynamical systems perturbed by small noise [2][3][4][5]. Specifically, Freidlin-Wentzell theory yields estimates of the stationary distributions and mean first-passage times.…”

Section: Introductionmentioning

confidence: 99%

Ritz method for transition paths and quasipotentials of rare diffusive events

Kikuchi

Singh

Cates

et al. 2020

Phys. Rev. Research

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The probability of trajectories of weakly diffusive processes to remain in the tubular neighborhood of a smooth path is given by the Freidlin-Wentzell-Graham theory of large deviations. The most probable path between two states (the instanton) and the leading term in the logarithm of the process transition density (the quasipotential) are obtained from the minimum of the Freidlin-Wentzell action functional. Here we present a Ritz method that searches for the minimum in a space of paths constructed from a global basis of Chebyshev polynomials. The action is reduced, thereby, to a multivariate function of the basis coefficients, whose minimum can be found by nonlinear optimization. For minimization regardless of path duration, this procedure is most effective when applied to a reparametrization-invariant "on-shell" action, which is obtained by exploiting a Noether symmetry and is a generalization of the scalar work [Olender and Elber, J. Mol. Struct: THEOCHEM 398, 63 (1997)] for gradient dynamics and the geometric action [Heyman and Vanden-Eijnden (2008)] for nongradient dynamics. Our approach provides an alternative to chain-of-states methods for minimum energy paths and saddle points of complex energy landscapes and to Hamilton-Jacobi methods for the stationary quasipotential of circulatory fields. We demonstrate spectral convergence for three benchmark problems involving the Müller-Brown potential, the Maier-Stein force field, and the Egger weather model.

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“…On the other hand, from a computational perspective, even though we could quantify the closeness of the expansion to the optimal control, it is yet difficult to assess that the tilted estimator indeed reduces the variance in simulations in general [56,25,2,33]. Our proposed control is therefore only the first step in the direction of rigorously establishing how and when the optimal control can be expanded around the instanton, improving on existing suggestions to use the instanton as an approximation for the optimal tilt as a mere heuristic in importance sampling, for instance in cloning algorithms [60,30] or in instanton biased importance sampling motivated from path integral techniques [19]. Our results suggest that the approximation is built in such a way that variance is indeed reduced in the small temperature limit for simple systems, but this remains to be proved rigorously, and under which precise conditions.…”

Section: Riccati Estimatormentioning

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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Instanton based importance sampling for rare events in stochastic PDEs

Cited by 27 publications

References 74 publications

Applications of large deviation theory in geophysical fluid dynamics and climate science

Applications of large deviation theory in geophysical fluid dynamics and climate science

Ritz method for transition paths and quasipotentials of rare diffusive events

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

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