Importance sampling for metastable and multiscale dynamical systems

In this article, we address the issues that come up in the design of importance sampling schemes for rare events associated to stochastic dynamical systems. We focus on the issue of metastability and on the effect of multiple scales. We discuss why seemingly reasonable schemes that follow large deviations optimal paths may perform poorly in practice, even though they are asymptotically optimal. Pre-asymptotic optimality is important when one deals with metastable dynamics and we discuss possible ways as to how to address this issue. Moreover, we discuss how the effect of the multiple scales (either in periodic or random environments) on the efficient design of importance sampling should be addressed. We discuss the mathematical and practical issues that come up, how to overcome some of the issues and discuss future challenges.