  This paper introduces the Boomerang Sampler as a novel class of
continuous-time non-reversible Markov chain Monte Carlo algorithms. The
methodology begins by representing the target density as a density, $e^{-U}$,
with respect to a prescribed (usually) Gaussian measure and constructs a
continuous trajectory consisting of a piecewise elliptical path. The method
moves from one elliptical orbit to another according to a rate function which
can be written in terms of $U$. We demonstrate that the method is easy to
implement and demonstrate empirically that it can out-perform existing
benchmark piecewise deterministic Markov processes such as the bouncy particle
sampler and the Zig-Zag. In the Bayesian statistics context, these competitor
algorithms are of substantial interest in the large data context due to the
fact that they can adopt data subsampling techniques which are exact (ie induce
no error in the stationary distribution). We demonstrate theoretically and
empirically that we can also construct a control-variate subsampling boomerang
sampler which is also exact, and which possesses remarkable scaling properties
in the large data limit. We furthermore illustrate a factorised version on the
simulation of diffusion bridges.
