  The increasing availability of data presents an opportunity to calibrate
unknown parameters which appear in complex models of phenomena in the
biomedical, physical and social sciences. However, model complexity often leads
to parameter-to-data maps which are expensive to evaluate and are only
available through noisy approximations. This paper is concerned with the use of
interacting particle systems for the solution of the resulting inverse problems
for parameters. Of particular interest is the case where the available forward
model evaluations are subject to rapid fluctuations, in parameter space,
superimposed on the smoothly varying large scale parametric structure of
interest. Multiscale analysis is used to study the behaviour of interacting
particle system algorithms when such rapid fluctuations, which we refer to as
noise, pollute the large scale parametric dependence of the parameter-to-data
map. Ensemble Kalman methods (which are derivative-free) and Langevin-based
methods (which use the derivative of the parameter-to-data map) are compared in
this light. The ensemble Kalman methods are shown to behave favourably in the
presence of noise in the parameter-to-data map, whereas Langevin methods are
adversely affected. On the other hand, Langevin methods have the correct
equilibrium distribution in the setting of noise-free forward models, whilst
ensemble Kalman methods only provide an uncontrolled approximation, except in
the linear case. Therefore a new class of algorithms, ensemble Gaussian process
samplers, which combine the benefits of both ensemble Kalman and Langevin
methods, are introduced and shown to perform favourably.
