  Parallel tempering (PT) methods are a popular class of Markov chain Monte
Carlo schemes used to sample complex high-dimensional probability
distributions. They rely on a collection of $N$ interacting auxiliary chains
targeting tempered versions of the target distribution to improve the
exploration of the state-space. We provide here a new perspective on these
highly parallel algorithms and their tuning by identifying and formalizing a
sharp divide in the behaviour and performance of reversible versus
non-reversible PT schemes. We show theoretically and empirically that a class
of non-reversible PT methods dominates its reversible counterparts and identify
distinct scaling limits for the non-reversible and reversible schemes, the
former being a piecewise-deterministic Markov process and the latter a
diffusion. These results are exploited to identify the optimal annealing
schedule for non-reversible PT and to develop an iterative scheme approximating
this schedule. We provide a wide range of numerical examples supporting our
theoretical and methodological contributions. The proposed methodology is
applicable to sample from a distribution $\pi$ with a density $L$ with respect
to a reference distribution $\pi_0$ and compute the normalizing constant. A
typical use case is when $\pi_0$ is a prior distribution, $L$ a likelihood
function and $\pi$ the corresponding posterior.
