Multilevel Sequential Monte Carlo Samplers

Abstract: 

The approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs) is considered herein; this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods and leading to a discretisation bias, with step-size level h_L. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multi-level Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretisation levels \infty>h_0>h_1\cdots>h_L. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence of probability distributions. A sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained in the SMC context. The approach is numerically illustrated on a Bayesian inverse problem. This is a joint work with Kody Law (ORNL), Raul Tempone (KAUST) and Alex Beskos (UCL).