  This work presents a multilevel variant of Stein variational gradient descent
to more efficiently sample from target distributions. The key ingredient is a
sequence of distributions with growing fidelity and costs that converges to the
target distribution of interest. For example, such a sequence of distributions
is given by a hierarchy of ever finer discretization levels of the forward
model in Bayesian inverse problems. The proposed multilevel Stein variational
gradient descent moves most of the iterations to lower, cheaper levels with the
aim of requiring only a few iterations on the higher, more expensive levels
when compared to the traditional, single-level Stein variational gradient
descent variant that uses the highest-level distribution only. Under certain
assumptions, in the mean-field limit, the error of the proposed multilevel
Stein method decays by a log factor faster than the error of the single-level
counterpart with respect to computational costs. Numerical experiments with
Bayesian inverse problems show speedups of more than one order of magnitude of
the proposed multilevel Stein method compared to the single-level variant that
uses the highest level only.
