  In this work, we propose a multi-agent actor-critic reinforcement learning
(RL) algorithm to accelerate the multi-level Monte Carlo Markov Chain (MCMC)
sampling algorithms. The policies (actors) of the agents are used to generate
the proposal in the MCMC steps; and the critic, which is centralized, is in
charge of estimating the long term reward. We verify our proposed algorithm by
solving an inverse problem with multiple scales. There are several difficulties
in the implementation of this problem by using traditional MCMC sampling.
Firstly, the computation of the posterior distribution involves evaluating the
forward solver, which is very time consuming for a problem with heterogeneous.
We hence propose to use the multi-level algorithm. More precisely, we use the
generalized multiscale finite element method (GMsFEM) as the forward solver in
evaluating a posterior distribution in the multi-level rejection procedure.
Secondly, it is hard to find a function which can generate samplings which are
meaningful. To solve this issue, we learn an RL policy as the proposal
generator. Our experiments show that the proposed method significantly improves
the sampling process
