  We consider a generalization of the discrete-time Self Healing Umbrella
Sampling method, which is an adaptive importance technique useful to sample
multimodal target distributions. The importance function is based on the
weights (namely the relative probabilities) of disjoint sets which form a
partition of the space. These weights are unknown but are learnt on the fly
yielding an adaptive algorithm. In the context of computational statistical
physics, the logarithm of these weights is, up to a multiplicative constant,
the free energy, and the discrete valued function defining the partition is
called the collective variable. The algorithm falls into the general class of
Wang-Landau type methods, and is a generalization of the original Self Healing
Umbrella Sampling method in two ways: (i) the updating strategy leads to a
larger penalization strength of already visited sets in order to escape more
quickly from metastable states, and (ii) the target distribution is biased
using only a fraction of the free energy, in order to increase the effective
sample size and reduce the variance of importance sampling estimators. The
algorithm can also be seen as a generalization of well-tempered metadynamics.
We prove the convergence of the algorithm and analyze numerically its
efficiency on a toy example.
