We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.

翻译：我们提出了一种可证明高效的重要性采样方案，用于估计小噪声随机反应扩散方程的解从稳定平衡点的缩放邻域中逃逸的概率。中等偏差标度允许通过线性化版本对非线性动力学进行局部近似。此外，我们确定了一个有限维子空间，在该子空间中逃逸以高概率发生。利用随机控制和变分方法，我们证明了该方案在零噪声极限和预渐近条件下均表现良好。针对随机扰动双稳态动力学的仿真研究验证了理论结果。