The velocity jump Langevin process and its splitting scheme: long time convergence and numerical accuracy

Abstract: The Langevin dynamics is a diffusion process extensively used, in particular in molecular dynamics simulations, to sample Gibbs measures. Some alternatives based on (piecewise deterministic) kinetic velocity jump processes have gained interest over the last decade. One interest of the latter is the possibility to split forces (at the continuous-time level), reducing the numerical cost for sampling the trajectory. Motivated by this, a numerical scheme based on hybrid dynamics combining velocity jumps and Langevin diffusion, numerically more efficient than their classical Langevin counterparts, has been introduced for computational chemistry in [42]. The present work is devoted to the numerical analysis of this scheme. Our main results are, first, the exponential ergodicity of the continuous-time velocity jump Langevin process, second, a Talay-Tubaro expansion of the invariant measure of the numerical scheme on the torus, showing in particular that the scheme is of weak order 2 in the step-size and, third, a bound on the quadratic risk of the corresponding practical MCMC estimator (possibly with Richardson extrapolation). With respect to previous works on the Langevin diffusion, new difficulties arise from the jump operator, which is non-local.

• The paper establishes exponential ergodicity of the velocity jump Langevin process using advanced Sobolev space techniques.
• It demonstrates that the splitting scheme achieves a weak order of 2, significantly reducing computational costs in molecular dynamics.
• Introducing a suitable Lyapunov function, the study confirms long-time stability and precise convergence of the discretization bias.

Analysis of the Velocity Jump Langevin Process and Its Splitting Scheme

The paper "The velocity jump Langevin process and its splitting scheme: long time convergence and numerical accuracy" presents a detailed numerical analysis of a novel sampling scheme that integrates velocity jump processes with Langevin dynamics. This hybrid approach aims to improve sampling efficiency and computational cost-effectiveness in molecular dynamics (MD) simulations.

Context and Motivation

The Langevin dynamics are widely utilized in MD to simulate stochastic processes that sample Gibbs measures. However, alternative methods, such as kinetic velocity jump processes, have emerged due to their potential for reducing the numerical cost by splitting forces at the continuous-time level. The paper builds on this concept by focusing on a specific numerical scheme that combines velocity jumps with Langevin diffusion for enhanced efficiency.

Key Contributions

1. Ergodicity and Numerical Analysis:
   • The authors establish the exponential ergodicity of the continuous velocity jump Langevin process. They employ sophisticated Sobolev space methods and utilize a modified norm to illustrate the convergence to equilibrium. This is crucial for demonstrating the long-time stability of the scheme.
   • A Talay-Tubaro expansion of the invariant measure indicates that the proposed scheme achieves a weak order of 2 in terms of step-size accuracy. Such a result underscores the potential for achieving high numerical precision with reduced computational demands.
2. Lyapunov Function and Stability:
   • By introducing a suitable Lyapunov function, the authors confirm the stability of the scheme, ensuring that the process remains well-behaved over extended periods. This insight is critical for validating the practical utility of the method in realistic MD scenarios.
3. Weak Error Analysis:
   • The research demonstrates that the discretization bias of the BJAOAJB chain converges quadratically with the step-size. This finding aligns with the expected behavior of a second-order accurate scheme, thereby providing a robust foundation for its application in large-scale MD simulations.

Implications and Future Directions

The integration of velocity jump processes with Langevin dynamics has the potential to significantly impact various fields beyond MD, such as Bayesian statistics, where sampling efficiency is paramount. The demonstrated improvements in computational cost without compromising accuracy make this approach highly attractive for systems with complex interactions, such as those found in pharmaceuticals and other high-dimensional spaces.

Furthermore, the theoretical framework laid out in this study could be extended to other domains where stochastic differential equations play a crucial role. Future research might explore hybrid dynamics more deeply, potentially incorporating machine learning techniques for adaptive tuning of simulation parameters based on system feedback.

Conclusion

The authors provide a comprehensive mathematical and numerical framework for a novel velocity jump Langevin process, successfully addressing the limitations of conventional Langevin dynamics in specific applications. While rooted in classic sampling theory, this work opens new avenues for efficient computational simulations, holding promise for impactful future advancements in statistical physics and related fields.