  This paper presents a new accelerated proximal Markov chain Monte Carlo
methodology to perform Bayesian inference in imaging inverse problems with an
underlying convex geometry. The proposed strategy takes the form of a
stochastic relaxed proximal-point iteration that admits two complementary
interpretations. For models that are smooth or regularised by Moreau-Yosida
smoothing, the algorithm is equivalent to an implicit midpoint discretisation
of an overdamped Langevin diffusion targeting the posterior distribution of
interest. This discretisation is asymptotically unbiased for Gaussian targets
and shown to converge in an accelerated manner for any target that is
$\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$
iterations to converge, similarly to accelerated optimisation schemes),
comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J.
Imaging Sciences, 13,2 (2020), pp. 905-935] which is only provably accelerated
for Gaussian targets and has bias. For models that are not smooth, the
algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin
diffusion targeting a Moreau-Yosida approximation of the posterior distribution
of interest, and hence achieves a significantly lower bias than conventional
unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For
targets that are $\kappa$-strongly log-concave, the provided non-asymptotic
convergence analysis also identifies the optimal time step which maximizes the
convergence speed. The proposed methodology is demonstrated through a range of
experiments related to image deconvolution with Gaussian and Poisson noise,
with assumption-driven and data-driven convex priors. Source codes for the
numerical experiments of this paper are available from
https://github.com/MI2G/accelerated-langevin-imla.
