We introduce Markov chain Monte Carlo (MCMC) algorithms based on numerical approximations of piecewise-deterministic Markov processes obtained with the framework of splitting schemes. We present unadjusted as well as adjusted algorithms, for which the asymptotic bias due to the discretisation error is removed applying a non-reversible Metropolis-Hastings filter. In a general framework we demonstrate that the unadjusted schemes have weak error of second order in the step size, while typically maintaining a computational cost of only one gradient evaluation of the negative log-target function per iteration. Focusing then on unadjusted schemes based on the Bouncy Particle and Zig-Zag samplers, we provide conditions ensuring geometric ergodicity and consider the expansion of the invariant measure in terms of the step size. We analyse the dependence of the leading term in this expansion on the refreshment rate and on the structure of the splitting scheme, giving a guideline on which structure is best. Finally, we illustrate promising results for our samplers with numerical experiments on a Bayesian imaging inverse problem and a system of interacting particles.

Abstract. 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 regularized by Moreau–Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretization of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretization is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is [Formula: see text]-strongly log-concave (i.e., requiring in the order of [Formula: see text] iterations to converge, similar to accelerated optimization schemes), comparing favorably to Pereyra, Vargas Mieles, and Zygalakis [ SIAM J. Imaging Sci., 13 (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 discretization 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 discretization. For targets that are [Formula: see text]-strongly log-concave, the provided nonasymptotic 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 .

Abstract. This paper studies a new and highly efficient Markov chain Monte Carlo (MCMC) methodology to perform Bayesian inference in low-photon imaging problems, with particular attention given to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric, and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low-photon numbers lead to severe identifiability issues, poor stability, and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult, e.g., hard nonnegativity constraints, nonsmooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE), as both the SDE and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularized Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in low-photon imaging problems. The proposed approach is demonstrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric, and Poisson noise.

On the basis of our research, the GDV does not seem to possess a specific medical diagnostic capability, however, the GDV does display very significant change with interventions designed to reduce stress; therefore, the GDV technique can potentially be used in tracking the effectiveness of physical and mental fitness programmes. We also concluded that the level of GDV-entropy is an important indicator of mental health and can help to identify which individuals are more likely to suffer from stress and its consequences.