The Self-Healing Umbrella Sampling (SHUS) algorithm is an adaptive biasing algorithm which has been proposed to efficiently sample a multimodal probability measure. We show that this method can be seen as a variant of the well-known Wang-Landau algor
ithm. Adapting results on the convergence of the Wang-Landau algorithm, we prove the convergence of the SHUS algorithm. We also compare the two methods in terms of efficiency. We finally propose a modification of the SHUS algorithm in order to increase its efficiency, and exhibit some similarities of SHUS with the well-tempered metadynamics method.
We analyze the efficiency of the Wang-Landau algorithm to sample a multimodal distribution on a prototypical simple test case. We show that the exit time from a metastable state is much smaller for the Wang Landau dynamics than for the original stand
ard Metropolis-Hastings algorithm, in some asymptotic regime. Our results are confirmed by numerical experiments on a more realistic test case.
