  The estimation of the probability of rare events is an important task in
reliability and risk assessment. We consider failure events that are expressed
in terms of a limit state function, which depends on the solution of a partial
differential equation (PDE). Since numerical evaluations of PDEs are
computationally expensive, estimating such probabilities of failure by Monte
Carlo sampling is intractable. More efficient sampling methods from reliability
analysis, such as Subset Simulation, are popular, but can still be
impracticable if the PDE evaluations are very costly. In this article, we
develop a novel, highly efficient estimator for probabilities of rare events.
Our method is based on a Sequential Importance sampler using discretizations of
PDE-based limit state functions with different accuracies. A twofold adaptive
algorithm ensures that we obtain an estimate based on the desired
discretization accuracy. In contrast to the Multilevel Subset Simulation
estimator of [Ullmann, Papaioannou 2015; SIAM/ASA J. Uncertain. Quantif.
3(1):922-953], our estimator overcomes the nestedness problem. Of particular
importance in Sequential Importance sampling algorithms is the correct choice
of the MCMC kernel. Instead of the popular adaptive conditional sampling
method, we propose a new algorithm that uses independent proposals from an
adaptively constructed von Mises-Fischer-Nakagami distribution. The proposed
algorithm is applied to test problems in 1D and 2D space, respectively, and is
compared to the Multilevel Subset Simulation estimator and to single-level
versions of Sequential Importance Sampling and Subset Simulation.
