Accelerating proximal Markov chain Monte Carlo by using explicit stabilised methods
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We present a highly efficient proximal Markov chain Monte Carlo methodology to perform Bayesian computation in imaging problems. Similarly to previous proximal Monte Carlo approaches, the proposed method is derived from an approximation of the Langevin diffusion. However, instead of the conventional Euler-Maruyama approximation that underpins existing proximal Monte Carlo methods, here we use a state-of-the-art orthogonal Runge-Kutta-Chebyshev stochastic approximation that combines several gradient evaluations to significantly accelerate its convergence speed, similarly to accelerated gradient optimization methods. For Gaussian models, we prove rigorously the acceleration of the Markov chains in the 2-Wasserstein distance as a function of the condition number κ. The performance of the proposed method is further demonstrated with a range of numerical experiments, including non-blind image deconvolution, hyperspectral unmixing, and tomographic reconstruction, with total-variation and ℓ_1-type priors. Comparisons with Euler-type proximal Monte Carlo methods confirm that the Markov chains generated with our method exhibit significantly faster convergence speeds, achieve larger effective sample sizes, and produce lower mean square estimation errors at equal computational budget.
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