We consider a generalization of the discrete-time Self Healing UmbrellaSampling method, which is an adaptive importance technique useful to samplemultimodal target distributions. The importance function is based on theweights (namely the relative probabilities) of disjoint sets which form apartition of the space. These weights are unknown but are learnt on the flyyielding an adaptive algorithm. In the context of computational statisticalphysics, the logarithm of these weights is, up to a multiplicative constant,the free energy, and the discrete valued function defining the partition iscalled the collective variable. The algorithm falls into the general class ofWang-Landau type methods, and is a generalization of the original Self HealingUmbrella Sampling method in two ways: (i) the updating strategy leads to alarger penalization strength of already visited sets in order to escape morequickly from metastable states, and (ii) the target distribution is biasedusing only a fraction of the free energy, in order to increase the effectivesample size and reduce the variance of importance sampling estimators. Thealgorithm can also be seen as a generalization of well-tempered metadynamics.We prove the convergence of the algorithm and analyze numerically itsefficiency on a toy example.
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