  It is well known in many settings that reversible Langevin diffusions in
confining potentials converge to equilibrium exponentially fast. Adding
irreversible perturbations to the drift of a Langevin diffusion that maintain
the same invariant measure accelerates its convergence to stationarity. Many
existing works thus advocate the use of such non-reversible dynamics for
sampling. When implementing Markov Chain Monte Carlo algorithms (MCMC) using
time discretisations of such Stochastic Differential Equations (SDEs), one can
append the discretization with the usual Metropolis-Hastings accept-reject step
and this is often done in practice because the accept--reject step eliminates
bias. On the other hand, such a step makes the resulting chain reversible. It
is not known whether adding the accept-reject step preserves the faster mixing
properties of the non-reversible dynamics. In this paper, we address this gap
between theory and practice by analyzing the optimal scaling of MCMC algorithms
constructed from proposal moves that are time-step Euler discretisations of an
irreversible SDE, for high dimensional Gaussian target measures. We call the
resulting algorithm the \imala, in comparison to the classical MALA algorithm
(here {\em ip} is for irreversible proposal). In order to quantify how the cost
of the algorithm scales with the dimension $N$, we prove invariance principles
for the appropriately rescaled chain. In contrast to the usual MALA algorithm,
we show that there could be two regimes asymptotically: (i) a diffusive regime,
as in the MALA algorithm and (ii) a ``fluid" regime where the limit is an
ordinary differential equation. We provide concrete examples where the limit is
a diffusion, as in the standard MALA, but with provably higher limiting
acceptance probabilities. Numerical results are also given corroborating the
theory.
