Convergence of Random Batch Method with replacement for interacting particle systems (2407.19315v1)

Abstract: The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems. In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is the same as the kinetic Monte Carlo (KMC) method for the pairwise interacting particle system of size $N$. In the RBM-r algorithm, one randomly picks a small batch of size $p \ll N$, and only the particles in the picked batch interact among each other within the batch for a short time, where the weak interaction (of strength $\frac{1}{N-1}$) in the original system is replaced by a strong interaction (of strength $\frac{1}{p-1}$). Then one repeats this pick-interact process. This KMC algorithm dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step, and provides an unbiased approximation of the original force/velocity field of the interacting particle system. In this paper, We give a rigorous proof of this approximation with an explicit convergence rate. An improved rate is also obtained when there is no diffusion in the system. Notably, the techniques in our analysis can potentially be applied to study KMC for other systems, including the stochastic Ising spin system.

• The paper rigorously proves the convergence of RBM-r in simulating interacting particle systems using a robust mathematical framework.
• It demonstrates that processing small, randomly selected batches reduces simulation complexity from O(N²) to significantly lower computational cost.
• Explicit convergence rates of O(κ^(1/4)) with diffusion and O(κ^(1/2)) without diffusion validate RBM-r's efficiency and broad applicability.

Convergence of Random Batch Method with Replacement for Interacting Particle Systems

The paper titled "Convergence of Random Batch Method with Replacement for Interacting Particle Systems" explores the efficacy of the Random Batch Method with replacement (RBM-r) in simulating interacting particle systems. Originally proposed by Jin et al., RBM-r is a computational technique rooted in the kinetic Monte Carlo (KMC) method. The authors offer a rigorous mathematical proof of convergence for RBM-r, thus validating its application in various systems that involve interactions between a large number of particles.

The interacting particle systems (IPS) studied here can be described using stochastic differential equations (SDEs) and are typical in numerous fields, including molecular dynamics and plasma physics. The complexity of simulating these systems is significant, traditionally in the order of O(N²) per time step, where N is the number of particles. The RBM dramatically reduces this computational burden by processing small batches of particles, a technique similar in spirit to mini-batch stochastic gradient descent used in machine learning.

Methodology and Results

In RBM-r, particles are repeatedly grouped into small randomly drawn batches, and interactions are calculated only within these batches for short time periods, while in-between reshuffling introduces randomness akin to sampling with replacement. A significant analytical contribution of the paper is demonstrating that under certain assumptions, RBM-r converges to the original IPS in the Wasserstein-2 space. The convergence analysis accounts for strong convexity and polynomial growth of the interaction potential and the Lipschitz continuity of interaction kernels.

One of the paper's key findings is the explicit convergence rate, which is O(κ^1/4) in scenarios with stochasticity and an improved rate of O(κ^1/2) when diffusion is absent. This discovery is nontrivial as it entrenches RBM-r as an efficient algorithm for approximating the dynamics of IPS with substantial computational savings while maintaining a satisfactory level of accuracy.

Implications and Future Directions

The implications of this research are substantial, particularly in simulations of large-scale particle systems where computational resources are constrained. By extending the applications to other complex systems such as the stochastic Ising model, RBM-r could facilitate more intricate exploration of phase transitions or dynamics in condensed matter physics and beyond.

Moreover, this paper's methodological advancements lay groundwork for further explorations into other instances of the KMC method. The intersection between RBM-r and statistical physical systems could yield more sophisticated algorithms that improve the accuracy and efficiency of simulations in computational physics and chemistry.

The future trajectory could also involve refining analytical methods to mitigate the trade-off between computational efficiency and accuracy in models characterized by high dimensional and multi-species parameters. Continued experimental validation and adaptation of RBM-r to non-binary interactions and systems driven by complexities such as cross-diffusion signals another promising direction.

Conclusion

This work fills a critical gap in the literature by providing a rigorous convergence analysis for RBM-r, reinforcing it as an effective method for simulating large-scale interacting particle systems. The robust proof offered in this paper not only supports existing applications but also broadens the horizon for future computational strategies in various scientific fields requiring exhaustive particle interactions modeling.