Adaptive reduced tempering For Bayesian inverse problems and rare event simulation

This work proposes an adaptive sequential Monte Carlo sampling algorithm to solve Bayesian inverse problems in scenarios where likelihood evaluations are costly but can be approximated using a surrogate model built from previous evaluations of the true likelihood. A rough estimate of the surrogate error is required. The method relies on an adaptive SMC framework that simultaneously adjusts both the likelihood approximations and a standard tempering scheme of the target posterior distribution. This algorithm is well-suited for cases where the posterior is concentrated in a rare and unknown region of the prior. It is also suitable for solving low-temperature and rare event simulation problems. The main contribution is to propose an entropy criterion that relates the accuracy of the current surrogate to a maximum inverse temperature for the likelihood approximation. The latter is instrumental to sample a so-called snapshot, on which is performed an exact likelihood evaluation, used to update the surrogate and its error quantification. Some consistency results are presented in an idealized framework for the proposed algorithm. Our numerical experiments use in particular a reduced basis approach to construct approximate parametric solutions to a partially observed solution of an elliptic partial differential equation. They demonstrate the convergence of the algorithm and show a significant cost reduction (close to a factor of $10$) for comparable accuracy.