An $O(\log N)$ Monte Carlo method for periodic Coulomb systems

TL;DR

The paper introduces DMK-MC, an accelerated Monte Carlo method that achieves $O( ext{log} N)$ computational complexity for sampling periodic Coulomb systems, significantly improving efficiency over previous methods.

Contribution

It adapts the dual-space multilevel kernel-splitting framework to Monte Carlo sampling, enabling $O( ext{log} N)$ work per move for long-range electrostatics.

Findings

DMK-MC outperforms recent FMM-based methods in speed.

Achieves consistent speedups across various system configurations.

Maintains accuracy comparable to existing approaches.

Abstract

Efficient Monte Carlo (MC) sampling of many-body systems with long-range electrostatics is often limited by the cost of per-move energy-difference evaluation under periodic boundary conditions. We present DMK-MC, an accelerated MC method that adapts the dual-space multilevel kernel-splitting (DMK) framework to single-particle Metropolis updates. DMK-MC computes the energy change and, upon acceptance, updates the stored incoming plane-wave fields with $O(1)$ work per tree level, yielding an overall $O(\log N)$ expected work per trial move for fixed accuracy. The method decomposes the Coulomb kernel into three components: a global, periodized smooth part; a multilevel sequence of smooth difference kernels whose interactions are restricted to same-level colleague boxes; and a singular residual kernel whose short-range interactions are evaluated directly. Benchmarks on uniform, highly nonuniform, and implicit-solvent electrolyte and colloidal configurations show that DMK-MC consistently outperforms a recent FMM-based $O(\log N)$ Monte Carlo method, delivering several-fold speedups at comparable tolerances.