  This paper considers the classical problem of sampling with Monte Carlo
methods a target probability distribution obtained by conditioning on a rare
event defined by the level set of a real-valued score function that is very
expensive to compute. We also consider a context where, with each new
evaluation of the true score function, a method that iteratively builds a
sequence of reduced scores is available; these reduced scores being moreover
certified with pointwise error bounds. This work proposes a fully adaptive
algorithm that iteratively: i) builds a sequence of proposal distributions
obtained by conditioning on the reduced score above an adaptively well-chosen
level, and ii) draws from the latter both for importance sampling of the true
target rare events, as well as for proposing relevant (expensive) updates to
the reduced score. An essential contribution consists in the adaptive choice of
the level in i) and ii). The latter is calculated solely from the reduced score
and its error bound, and is interpreted as the first non-achievable level as
quantified by a given cost (in a pessimistic scenario) of importance sampling
of the associated true target distribution. From a practical point of view,
sampling the proposal sequence is performed by extending the framework of the
popular Adaptive Multilevel Splitting (AMS) algorithm to the use of score
function reduction. Numerical experiments evaluate the proposed importance
sampling algorithm in terms of computational complexity versus squared error.
In particular, we investigate the performance of the algorithm when simulating
rare events related to the solution of a parametric PDE approximated by a
reduced basis.
