Analysis and simulation of rare events for SPDEs

Abstract

Abstract. In this work, we consider the numerical estimation of the probability for a stochastic process to hit a set B before reaching another set A. This event is assumed to be rare. We consider reactive trajectories of the stochastic Allen-Cahn partial differential evolution equation (with double well potential) in dimension 1. Reactive trajectories are defined as the probability distribution of the trajectories of a stochastic process, conditioned by the event of hitting B before A. We investigate the use of the so-called Adaptive Multilevel Splitting algorithm in order to estimate the rare event and simulate reactive trajectories. This algorithm uses a reaction coordinate (a real valued function of state space defining level sets), and is based on (i) the selection, among several replicas of the system having hit A before B, of those with maximal reaction coordinate; (ii) iteration of the latter step. We choose for the reaction coordinate the average magnetization, and for B the minimum of the well opposite to the initial condition. We discuss the context, prove that the algorithm has a sense in the usual functional setting, and numerically test the method (estimation of rare event, and transition state sampling).

Unbiasedness of some generalized adaptive multilevel splitting algorithms

“…This is why naive Monte Carlo methods will not give reliable estimates of (2). We refer for example to [3,10] for some examples in the context of molecular simulation.…”

Recent advances in various fields of numerical probability

“…Finally, unbiasedness of the unbiased AMS algorithm is not restricted to the estimation of the probability: it also provides information about the so-called reactive trajectories, see for instance [13] for preliminary simulations for the stochastic Allen-Cahn equation.…”

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

“…The stochastic Allen-Cahn equation is also a popular model for the study and simulation of rare events in infinite dimensional stochastic system, see for instance [8,47,58,59].…”

“…The definition of Method 2, given by the scheme (9), is motivated by [49]. Finally, the definition of Method 3 is motivated by [8]. We have checked that the three variants give consistent results.…”