Estimation of rare event statistics plays an enormous role in research fields like materials science, molecular dynamics or climate modelling. Most estimators suffer from large variance such that the precise characterization of exponentially long transition times or exponentially small probabilities is challenging. We will discuss why the construction of zero or very low variance estimators is connected on HJB PDE theory.
Related results are based on the insight that certain equilibrium expectation values can be equivalently expressed in terms of (1) the solution of certain stochastic optimal control problem, and simultaneously as (2) optimal change of measure problems. This insight allows for designing low variance estimators by using optimal protocols, the downside being that one has to solve an optimal control or change of measure problem. We will discuss how this can be done efficiently and demonstrate the performance of the resulting algorithms for computing rare events in molecular systems.
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