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

Nicolaï Gouraud, Lucas Journel, Pierre Monmarché

TL;DR

The paper develops and analyzes a velocity jump Langevin framework that splits forces to reduce costly gradient evaluations while preserving the Gibbs measure μ. It proves geometric ergodicity of the continuous-time process via hypocoercivity in higher-order Sobolev spaces, and establishes a second-order weak error expansion for the Strang-like splitting scheme, yielding μδ(f)=μ(f)+c_f δ^2+O(δ^3). It also provides ergodicity and explicit invariant-measure expansion for the discrete scheme, along with quadratic-risk bounds for MCMC estimators and potential Richardson extrapolation. Collectively, these results justify the numerical method’s accuracy and stability for long-time sampling in molecular dynamics, offering practical speedups by replacing some force evaluations with velocity jumps while rigorously controlling discretization bias and estimator variance.

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

TL;DR

The paper develops and analyzes a velocity jump Langevin framework that splits forces to reduce costly gradient evaluations while preserving the Gibbs measure μ. It proves geometric ergodicity of the continuous-time process via hypocoercivity in higher-order Sobolev spaces, and establishes a second-order weak error expansion for the Strang-like splitting scheme, yielding μδ(f)=μ(f)+c_f δ^2+O(δ^3). It also provides ergodicity and explicit invariant-measure expansion for the discrete scheme, along with quadratic-risk bounds for MCMC estimators and potential Richardson extrapolation. Collectively, these results justify the numerical method’s accuracy and stability for long-time sampling in molecular dynamics, offering practical speedups by replacing some force evaluations with velocity jumps while rigorously controlling discretization bias and estimator variance.

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.

Paper Structure

This paper contains 27 sections, 30 theorems, 344 equations, 1 figure.

Table of Contents

  1. Motivations and overview
  2. Setting and main results
  3. Notations
  4. Definitions and assumptions
  5. Main results
  6. Discussion and applications
  7. Decomposition of the forces.
  8. Conditions on $U$.
  9. Choice of $\Psi$.
  10. Role of $\rho$.
  11. Simulation of the jumps.
  12. Complexity of the algorithm.
  13. Application to molecular dynamics.
  14. Numerical illustrations.
  15. Geometric ergodicity
  16. ...and 12 more sections

Key Result

Theorem 1

Under Assumption assu, for all $f\in\mathcal{A}$ and all multi-index $\alpha\in\mathbb N^{2d}$, there exist $C,q>0$, $b\in (0,1)$, such that for all $t>0$, $(x,v)\in\mathbb R^{2d}$,

Figures (1)

  • Figure 1: Trajectories of a splitting scheme for the velocity jump Langevin process

Theorems & Definitions (57)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Proposition 6
  • Proposition 7
  • proof : Proof of Theorem \ref{['thm:estimates']}
  • Proposition 8
  • Proposition 9
  • ...and 47 more