Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

Abstract

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.

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…In Table 1 below, we provide for various N the estimated expectation from standard Monte Carlo δ N and importance sampling δN (14), the empirical relative error from standard Monte Carlo ρ(δ N ) and for importance sampling ρ( δN ), and the exact value of the expectation (3) from ( 50) and (51). Here ρ( δN ) is as in ( 16) but where M = 1 and the expectation and second moment are computed empirically.…”

“…In Figure 1(a) we plot an average of trajectories of the sum (54) for the uncontrolled and controlled particles, respectively, computed using the same noise. In Figure 1(b), we plot the analytical relative error for the Monte Carlo scheme (ρ(δ N ) from ( 16) with v N i ≡ 0) on a log scale for various values of N , again as computed via (50) and (51). One can see that the first few values of N considered in Table 1 are in a region where the growth for the standard Monte Carlo relative error has not yet shifted from linear to exponential, where importance sampling becomes even more valuable.…”

“…As far as we are aware, this paper also represents the first time the problem of estimating (3) with an asymptotically optimal importance sampling scheme has been considered, though many other versions of the small-noise importance sampling problem have appeared in the literature. These include, for example, discrete-time Markov chains [37,38], diffusions with multiscale structure [36,69], and stochastic partial differential equations [40,51]. See in addition [11,35,39,53,57,79] for a diverse, but by no means comprehensive, collection of small-noise importance sampling research in a variety of settings.…”

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…In Figure 1(a) we plot a trajectory of the sum (49) for the uncontrolled and controlled particles, respectively, computed using the same noise. In Figure 1(b), we plot the expected relative error for the Monte-Carlo scheme for various values of N , again as computed via Equations ( 44) and (45).…”

“…It may also prove useful for some target statistics to design an importance sampling scheme using the moderate deviations principle for the empirical measure [10,18]. This is known to aid with the problems discussed above in the small noise setting due to the linearization of the HJB Equation under the Moderate Deviations scaling [49,66]. As discussed in Remark 3.4, this can likely also be supplemented via use of an importance sampling scheme arising from large deviations of the empirical measure in the joint small noise and large N limit as derived in [16,69].…”