Instantons for rare events in heavy-tailed distributions

Abstract

Large deviation theory and instanton calculus for stochastic systems are widely used to gain insight into the evolution and probability of rare events. At its core lies the fact 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 a 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.

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

“…3.3, act in some cases in which the standard large deviation scaling fails due to the presence of some form of long-term memory. In a very recent paper [6], a nonlinear reparametrisation of the scaled cumulant generating functions is proposed to properly compute instantons in case of observables with heavy tailed distributions.…”

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

“…In contrast to the optimisation approach by Chernykh and Stepanov [9] and others, where only a fixed Lagrange multiplier is specified without a priori knowledge of the resulting value of a, this method allows us to directly compute instantons for specified observable values a. This is convenient if a) there are multiple local minima, and the map F → a becomes multivalued, and b) there are observable regions where the action fails to be convex and the F -a-duality breaks down because the scaled cumulant-generating function of the process u diverges [30]. On the downside, we now have to solve multiple optimisation problems for each observable value that we want to compute an instanton for.…”

Extreme events and instantons in Lagrangian passive scalar turbulence models

“…The final time constraint on the gradient of passive scalar to attain z = ψ j (t f ) is implemented in (27) through a Lagrange multiplier λ ∈ R, [38]. The function F : R → R is a nonlinear reparametrization to ensure there is a unique λ for every large passive scalar gradients of interest [23].…”

“…The benefit of solving equations ( 28) is even more significant for extreme events since this factor of improvement grows as Re and/or z increase. To overcome the problem of heavy tails, we convexify the rate function with a reparametrization of the observable according to the scheme presented in [23]. Concretely, we choose F (z) = sign(z) log log |z|, to be inserted as boundary condition into (27).…”

“…Section V then analyses heavy-tailed PDFs of the passive scalar gradient. Such heavy-tailed distributions, associated with non-convex rate-functions, pose a particular difficulty for the application of sample path large deviations; thus, we apply in section V B a revised formalism based on nonlinear convexification of extreme event instantons [23]. Finally, we conclude in section VI.…”