  In this paper we propose and analyze a novel multilevel version of Stein
variational gradient descent (SVGD). SVGD is a recent particle based
variational inference method. For Bayesian inverse problems with
computationally expensive likelihood evaluations, the method can become
prohibitive as it requires to evolve a discrete dynamical system over many time
steps, each of which requires likelihood evaluations at all particle locations.
To address this, we introduce a multilevel variant that involves running
several interacting particle dynamics in parallel corresponding to different
approximation levels of the likelihood. By carefully tuning the number of
particles at each level, we prove that a significant reduction in computational
complexity can be achieved. As an application we provide a numerical experiment
for a PDE driven inverse problem, which confirms the speed up suggested by our
theoretical results.
