An important sampling method for certain rare event problems involving small noise diffusions is proposed. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small noise limit. Previous work in rare event simulation has focused on developing estimators with optimal exponential variance decay rates. This criterion still allows for exponential growth of the statistical relative error. We show that an estimator related to a deterministic control problem not only has an optimal variance decay rate but can have vanishingly small relative statistical error in the small noise limit. The sampling method based on this estimator can be seen as the limit of the zero variance importance sampling scheme, which uses the solution of the second-order partial differential equation (PDE) associated with the diffusion. In the scheme proposed here this PDE is replaced by a Hamilton-Jacobi equation whose solution is computed pointwise on the fly from its variational formulation, an operation that remains practical even in high-dimensional problems. We test the scheme on several simple illustrative examples as well as a stochastic PDE, the noisy Allen-Cahn equation. © 2012 Wiley Periodicals, Inc.
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