Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical integrators for confined Langevin dynamics

B. Leimkuhler, A. Sharma, M. V. Tretyakov

TL;DR

This work addresses the challenge of simulating underdamped Langevin dynamics within bounded domains by developing weak-sense numerical integrators that enforce elastic boundary reflections. It introduces first-order Euler-type schemes and second-order splitting schemes based on a natural A_c,B,O generator decomposition, with theoretical finite-time and ergodic error analyses. A striking finding is that certain stochastic symmetric splittings achieve second-order weak convergence with a single gradient evaluation, explained by the average mid-step collision timing, and corroborated by numerical experiments. The methods enable efficient sampling from constrained distributions and accurate long-time behavior in molecular dynamics and related optimization problems, while laying groundwork for extensions to multi-particle interactions and McKean–Vlasov settings.

Abstract

We derive and analyze numerical methods for underdamped (kinetic) Langevin dynamics in a domain with elastic reflection at the boundary. First-order approximations are based on an Euler-type scheme incorporating collision-handling at the boundary. To achieve second order, composition schemes are derived based on decomposition of the generator into collisional drift, impulse, and stochastic momentum evolution. In a deterministic setting, this approach would typically lead to first-order approximation, even in symmetric compositions, but we find that the stochastic method can provide second-order weak approximation with a single gradient evaluation, both at finite times and in the ergodic limit. We provide analysis of this observation, as well as numerical demonstration, and we compare and contrast the performance of different variants of the integration method using model problems.

Numerical integrators for confined Langevin dynamics

TL;DR

This work addresses the challenge of simulating underdamped Langevin dynamics within bounded domains by developing weak-sense numerical integrators that enforce elastic boundary reflections. It introduces first-order Euler-type schemes and second-order splitting schemes based on a natural A_c,B,O generator decomposition, with theoretical finite-time and ergodic error analyses. A striking finding is that certain stochastic symmetric splittings achieve second-order weak convergence with a single gradient evaluation, explained by the average mid-step collision timing, and corroborated by numerical experiments. The methods enable efficient sampling from constrained distributions and accurate long-time behavior in molecular dynamics and related optimization problems, while laying groundwork for extensions to multi-particle interactions and McKean–Vlasov settings.

Abstract

We derive and analyze numerical methods for underdamped (kinetic) Langevin dynamics in a domain with elastic reflection at the boundary. First-order approximations are based on an Euler-type scheme incorporating collision-handling at the boundary. To achieve second order, composition schemes are derived based on decomposition of the generator into collisional drift, impulse, and stochastic momentum evolution. In a deterministic setting, this approach would typically lead to first-order approximation, even in symmetric compositions, but we find that the stochastic method can provide second-order weak approximation with a single gradient evaluation, both at finite times and in the ergodic limit. We provide analysis of this observation, as well as numerical demonstration, and we compare and contrast the performance of different variants of the integration method using model problems.
Paper Structure (26 sections, 11 theorems, 197 equations, 15 figures, 4 tables, 1 algorithm)

This paper contains 26 sections, 11 theorems, 197 equations, 15 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and related work
  3. Sampling
  4. Molecular dynamics
  5. Computational fluid dynamics
  6. Optimization
  7. Specular boundary value problems
  8. Contributions and overview
  9. Preliminaries
  10. Kolmogorov equations for (\ref{['cld']})
  11. Ergodicity
  12. Numerical methods
  13. First-order schemes
  14. [A_c, B, O] splitting schemes
  15. Comparison with other numerical integrators for (\ref{['cld']})
  16. ...and 11 more sections

Key Result

Theorem 3.1

Under Assumptions cld_as:1-cld_as:4a, the order of weak convergence of scheme [PA$_c$] for (cld) is $\mathcal{O}(h)$, i.e. the following inequality holds, for sufficiently small $h>0$: where $(Q(t),P(t))$ is from (cld) and $C>0$ is independent of $h$.

Figures (15)

  • Figure 3.1: Description of [PA$_c$] and [A$_c$P] schemes.
  • Figure 3.2: Description of [OBA$_c$BO], [BA$_c$OA$_c$B], [OA$_c$BA$_c$O] schemes for (\ref{['ecld']}).
  • Figure 3.3: A depiction of multi-collisional step of Algorithm \ref{['A$_c$ step with multiple']}.
  • Figure 4.1: Plots showing the distribution of $\tau_1$.
  • Figure 4.2: Left plot shows second-order in $h$ decrease of the statistics $\hat{\Lambda}_1$ (blue line); red line gives the second order slope. Right plot compares the statistics $\hat{\Lambda}_2$ (green line) vs the constant dashed red line, i.e. it shows that $\hat{\Lambda}_2$ remains almost constant.
  • ...and 10 more figures

Theorems & Definitions (28)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.1
  • proof
  • Remark 3.1
  • ...and 18 more