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Dynamical reweighting for estimation of fluctuation formulas

Raphaël Gastaldello, Gabriel Stoltz, Urbain Vaes

TL;DR

This work addresses the high-variance challenge of estimating transport coefficients in molecular dynamics via Green–Kubo by introducing a dynamical reweighting scheme based on Girsanov's theorem. It develops a biased-drift SDE with a scalar parameter ${α}$, constructs a GK-like estimator with Girsanov weights, and proves the existence and uniqueness of a variance-minimizing ${α_T^*}$ for fixed horizon ${T}$, along with its asymptotic ${T\to\infty}$ scaling. The analysis shows that, while one can achieve variance reduction, the gains are modest and scale as ${O(1/T)}$ in the long-time limit; numerical experiments in 1D and 2D overdamped Langevin dynamics corroborate this, highlighting limited practical impact in higher dimensions or less favorable regimes. The work provides a rigorous framework for dynamical reweighting and offers guidance for future extensions, such as optimizing the biasing function or applying the approach to more effective variance-reduction strategies.

Abstract

We propose a variance reduction method for calculating transport coefficients in molecular dynamics using an importance sampling method via Girsanov's theorem applied to Green--Kubo's formula. We optimize the magnitude of the perturbation applied to the reference dynamics by means of a scalar parameter~$α$ and propose an asymptotic analysis to fully characterize the long-time behavior in order to evaluate the possible variance reduction. Theoretical results corroborated by numerical results show that this method allows for some reduction in variance, although rather modest in most situations.

Dynamical reweighting for estimation of fluctuation formulas

TL;DR

This work addresses the high-variance challenge of estimating transport coefficients in molecular dynamics via Green–Kubo by introducing a dynamical reweighting scheme based on Girsanov's theorem. It develops a biased-drift SDE with a scalar parameter , constructs a GK-like estimator with Girsanov weights, and proves the existence and uniqueness of a variance-minimizing for fixed horizon , along with its asymptotic scaling. The analysis shows that, while one can achieve variance reduction, the gains are modest and scale as in the long-time limit; numerical experiments in 1D and 2D overdamped Langevin dynamics corroborate this, highlighting limited practical impact in higher dimensions or less favorable regimes. The work provides a rigorous framework for dynamical reweighting and offers guidance for future extensions, such as optimizing the biasing function or applying the approach to more effective variance-reduction strategies.

Abstract

We propose a variance reduction method for calculating transport coefficients in molecular dynamics using an importance sampling method via Girsanov's theorem applied to Green--Kubo's formula. We optimize the magnitude of the perturbation applied to the reference dynamics by means of a scalar parameter~ and propose an asymptotic analysis to fully characterize the long-time behavior in order to evaluate the possible variance reduction. Theoretical results corroborated by numerical results show that this method allows for some reduction in variance, although rather modest in most situations.

Paper Structure

This paper contains 42 sections, 10 theorems, 174 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Objectives of this work
  3. Our contributions
  4. Outline of the paper
  5. Mathematical framework
  6. General setting
  7. Computation of transport coefficients and Green--Kubo formula
  8. Variance reduction using importance sampling
  9. Approximation based on Girsanov formula
  10. Differentiability
  11. Strict convexity
  12. Coercivity
  13. Conclusion of the proof
  14. Estimator of the optimal value
  15. Asymptotic estimations
  16. ...and 27 more sections

Key Result

Proposition 2.2

\newlabelproposition:asymptotic_variance_gk0 Let $R, S \in L^2_0(\mu)$ be two smooth observables. Under assumption:inv_decay, the variance of the estimator $\widehat{\rho}_{T,J}$ given in eq:estimator_continuous_time scales asymptotically as $T$:

Figures (8)

  • Figure 1: Time behavior of $F_T'(0)$ and $F_T"(0)$.
  • Figure 2: Comparison between $F_T(\alpha)$, \ref{['eq:taylor_expansion']} and a quadratic fit around $\widehat{\alpha}_T^*$ at $T=2.0$.
  • Figure 3: Comparison of $\alpha_T^*$ and $\widehat{\alpha}_T^*$ with respect to time.
  • Figure 4: Numerical results for the one-dimensional dynamics considered in \ref{['ssub:num_1d']}.
  • Figure 5: Visualization of the level sets of the potential $V$ and trajectories of the reference dynamics \ref{['eq:ovd_2D']}.
  • ...and 3 more figures

Theorems & Definitions (22)

  • Remark 2.1
  • Proposition 2.2
  • Proof 1
  • Proposition 3.1
  • Proof 2
  • Remark 3.2
  • Proposition 3.3
  • Proof 3
  • Lemma 3.4
  • Proof 4
  • ...and 12 more