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.

翻译：本文基于分裂格式框架对分段确定性马尔可夫过程进行数值逼近，提出了一系列马尔可夫链蒙特卡罗（MCMC）算法。我们介绍了未经调整的算法以及经过调整的算法，后者通过应用不可逆的Metropolis-Hastings滤波器消除了由离散化误差引起的渐近偏差。在一般框架下，我们证明了未经调整的方案在步长上具有二阶弱误差，同时通常每次迭代仅需计算负对数目标函数的一次梯度评估。随后，我们聚焦于基于弹跳粒子采样器和Zig-Zag采样器的未经调整方案，给出了确保几何遍历性的条件，并考虑了不变测度按步长的展开式。我们分析了该展开式中主导项对更新率及分裂格式结构的依赖性，从而为最优结构选择提供了指导原则。最后，我们通过在贝叶斯成像反问题和相互作用粒子系统上的数值实验，展示了所提出采样器的优异性能。