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Rare events of significant importance arise in several scientific fields including climate modeling, chemistry, material science and quantum mechanics. Related physical phenomena range from heat waves and tsunamis to phase separation and transitions in composite materials, quantum tunelling and magnetization reversal. We aim at the development of theoretical tools and numerical methods for the analysis and simulation of rare events for stochastic reaction-diffusion equations in the range of moderate deviations. The latter describes an asymptotic behavior that interpolates between the Central Limit Theorem and Large Deviation Principle and allows us to answer questions that are still open in the range of large deviations. In the first part of this work we prove a Moderate Deviation Principle for a multiscale system of stochastic reaction-diffusion equations with small noise in the slow component. Using weak convergence methods in infinite dimensions we obtain an exact form for the action functional in different regimes as the small noise and time-scale separation parameters vanish. In the second part 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. Based in stochastic control and variational methods we show that our scheme performs well both in the zero-noise limit and pre-asymptotically. Examples and simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results. |
