SIAM Journal on Imaging Sciences, Volume 13, Issue 2, Page 905-935, January 2020.
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 [A. Abdulle, I. Aimuslimani, and G. Vilmart, SIAM/ASA J. Uncertain. Quantif., 6 (2018), pp. 937--964] that combines several gradient evaluations to significantly accelerate its convergence speed, similarly to accelerated gradient optimization methods. The proposed methodology is demonstrated via a range of numerical experiments, including non-blind image deconvolution, hyperspectral unmixing, and tomographic reconstruction, with total-variation and $\ell_1$-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.

中文翻译：

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使用显式稳定方法加速近邻马尔可夫链蒙特卡罗

SIAM影像科学杂志，第13卷，第2期，第905-935页，2020年1月。
我们提出了一种高效的近端马尔可夫链蒙特卡罗方法，可以在成像问题中进行贝叶斯计算。与先前的近端蒙特卡洛方法相似，所提出的方法是从朗格文扩散的近似推导而来的。但是，不是使用支持现有近端蒙特卡罗方法的常规欧拉-丸山近似，而是使用最新的正交朗格-库塔-切比雪夫随机近似[A. Abdulle，I.Aimuslimani和G.Vilmart，SIAM / ASA J.不确定。Quantif。，6（2018），pp。937--964]，它结合了多个梯度评估以显着加快其收敛速度，类似于加速梯度优化方法。通过一系列数值实验（包括非盲图像反卷积，高光谱分解和层析成像重建，具有总变化和$ \ ell_1 $型先验。与Euler型近端蒙特卡洛方法的比较证实，用我们的方法生成的马尔可夫链表现出明显更快的收敛速度，达到了更大的有效样本量，并且在相等的计算预算下产生了较低的均方估计误差。