To minimise systematic errors in Monte Carlo simulations of charged particles, long range electrostatic interactions have to be calculated accurately and efficiently. Standard approaches, such as Ewald summation or the naive application of the classical Fast Multipole Method, result in a cost per Metropolis-Hastings step which grows in proportion to some positive power of the number of particles N in the system. This prohibitively large cost prevents accurate simulations of systems with a sizeable number of particles. Currently, large systems are often simulated by truncating the Coulomb potential which introduces uncontrollable systematic errors. In this paper we present a new multilevel method which reduces the computational complexity to $\mathcal{O}(\log (N))$ per Metropolis-Hastings step, while maintaining errors which are comparable to direct Ewald summation. We show that compared to related previous work, our approach reduces the overall cost by better balancing time spent in the proposal- and acceptance- stages of each Metropolis-Hastings step. By simulating large systems with up to $N={10}^5$ particles we demonstrate that our implementation is competitive with state-of-the-art MC packages and allows the simulation of very large systems of charged particles with accurate electrostatics.

为了使带电粒子的蒙特卡洛模拟中的系统误差最小，必须准确而有效地计算远距离静电相互作用。标准方法（例如Ewald求和或经典快速多极方法的天真应用）导致每个Metropolis-Hastings步骤的成本与系统中粒子N数量的某些正幂成比例地增长。如此高昂的成本阻碍了具有大量粒子的系统的精确仿真。当前，通常通过截断库仑电势来模拟大型系统，这会引入不可控的系统误差。在本文中，我们提出了一种新的多级方法，该方法将计算复杂度降低到$\mathcal{Ø}（\mathrm{日志}（ñ））$保持Metropolis-Hastings步骤，同时保持与直接Ewald求和可比的误差。我们证明，与以前的相关工作相比，我们的方法通过更好地平衡在Metropolis-Hastings步骤的建议和接受阶段所花费的时间来降低了总成本。通过模拟多达$ñ={10}^5$ 我们证明了我们的粒子与先进的MC封装相比具有竞争力，并且可以模拟具有精确静电的超大型带电粒子系统。