We introduce novel 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 the competitiveness of our samplers with numerical experiments on a Bayesian imaging inverse problem and a system of interacting particles.

翻译：我们提出了一系列新颖的马尔可夫链蒙特卡洛（MCMC）算法，这些算法基于通过分裂格式框架获得的分段确定性马尔可夫过程的数值逼近。我们同时介绍了未经调整和经过调整的算法，其中通过应用不可逆的Metropolis-Hastings过滤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的弱误差在步长上具有二阶精度，同时每次迭代通常仅需对负对数目标函数进行一次梯度计算。随后，我们聚焦于基于弹跳粒子和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，并考虑了步长不变测度的展开。我们分析了该展开中主导项对刷新速率和分裂格式结构的依赖性，从而为哪种结构最佳提供了指导。最后，我们通过贝叶斯成像逆问题和相互作用粒子系统的数值实验，展示了我们采样器的竞争力。