Rare-Event Simulation for the Stochastic Korteweg-de Vries Equation

An asymptotic analysis of the tail probabilities for the dynamics of a soliton wave U(x, t) under a stochastic time-dependent force is developed. The dynamics of the soliton wave U(x, t) is described by the Korteweg-de Vries (KdV) equation with homogeneous Dirichlet boundary conditions under a stochastic time-dependent force, which is modeled as a time-dependent Gaussian noise with amplitude epsilon. The tail probability considered is w(b) :- P(sup(t is an element of[0,T]) U(x, t) > b), as b -> infinity, for some constant T > 0 and a fixed x, which can be interpreted as tail probability of the amplitude of a water wave on the shallow surface of a fluid or long internal wave in a density-stratified ocean. Our goal is to characterize the asymptotic behaviors of w(b) and evaluate the tail probability of the event that the soliton wave exceeds a certain threshold value under a random force term. Such rare-event calculation of w(b) is especially useful for fast estimation of the risk of the potential damage that could be caused by the water wave in a density-stratified ocean modeled by the stochastic KdV equation. In this work, the asymptotic approximation of the probability that the soliton wave exceeds a high-level b is derived. In addition, we develop a provably efficient rare-event simulation algorithm to compute w(b) that runs in polynomial time of log b with a prescribed relative accuracy.