Adaptive Reduced Multilevel Splitting

This paper considers the classical problem of sampling with Monte Carlo methods a target rare event distribution of the form $\eta^\star_{l_{max}} \propto \mathbb{1}_{S^\star> l_{max}} d\pi$, with $\pi$ a reference probability distribution, $S^\star$ a real-valued function that is very expensive to compute and $l_{max} \in \mathbb{R}$ some level of interest. We assume we can iteratively build a sequence of reduced models $\{S^{(k)}\}_k$ of the scores, which are increasingly refined approximations of $S^\star$ as new score values are computed; these reduced models being moreover certified with error bounds. This work proposes a fully adaptive algorithm to iteratively build a sequence of proposal distributions with increasing target levels $l^{(k)}$ of the form $\propto \mathbb{1}_{S^{(k)} > l^{(k)}} d\pi$ and draw from them, the objective being importance sampling of the target rare events, as well as proposing relevant updates to the reduced score. An essential contribution consists in {adapting} the target level to the reduced score: the latter is defined as the first non-achievable level $l^{(k)}$ corresponding to a cost, in a pessimistic scenario, for importance sampling matching an acceptable budget; it is calculated solely from the reduced score function and its error bound. From a practical point of view, sampling the proposal sequence is performed by extending the framework of the popular adaptive multilevel splitting algorithm to the use of reduced score approximations. 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, which is approximated by a reduced basis.