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We present a highly efficient proximal Markov chain Monte Carlo methodologyto perform Bayesian computation in imaging problems. Similarly to previousproximal Monte Carlo approaches, the proposed method is derived from anapproximation of the Langevin diffusion. However, instead of the conventionalEuler-Maruyama approximation that underpins existing proximal Monte Carlomethods, here we use a state-of-the-art orthogonal Runge-Kutta-Chebyshevstochastic approximation that combines several gradient evaluations tosignificantly accelerate its convergence speed, similarly to acceleratedgradient optimisation methods. The proposed methodology is demonstrated via arange of numerical experiments, including non-blind image deconvolution,hyperspectral unmixing, and tomographic reconstruction, with total-variationand $\ell_1$-type priors. Comparisons with Euler-type proximal Monte Carlomethods confirm that the Markov chains generated with our method exhibitsignificantly faster convergence speeds, achieve larger effective sample sizes,and produce lower mean square estimation errors at equal computational budget.

Accelerating proximal Markov chain Monte Carlo by using an explicit stabilised method