  Inverse problems are ubiquitous because they formalize the integration of
data with mathematical models. In many scientific applications the forward
model is expensive to evaluate, and adjoint computations are difficult to
employ; in this setting derivative-free methods which involve a small number of
forward model evaluations are an attractive proposition. Ensemble Kalman based
interacting particle systems (and variants such as consensus based and
unscented Kalman approaches) have proven empirically successful in this
context, but suffer from the fact that they cannot be systematically refined to
return the true solution, except in the setting of linear forward models. In
this paper, we propose a new derivative-free approach to Bayesian inversion,
which may be employed for posterior sampling or for maximum a posteriori
estimation, and may be systematically refined. The method relies on a fast/slow
system of stochastic differential equations for the local approximation of the
gradient of the log-likelihood appearing in a Langevin diffusion. Furthermore
the method may be preconditioned by use of information from ensemble Kalman
based methods (and variants), providing a methodology which leverages the
documented advantages of those methods, whilst also being provably refineable.
We define the methodology, highlighting its flexibility and many variants,
provide a theoretical analysis of the proposed approach, and demonstrate its
efficacy by means of numerical experiments.
