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Efficient Langevin sampling with position-dependent diffusion

Eugen Bronasco, Benedict Leimkuhler, Dominic Phillips, Gilles Vilmart

TL;DR

This work tackles efficient sampling of the invariant measure for Brownian dynamics with position-dependent diffusion. It introduces the Second-Order Post-processed Variable-Diffusion (PVD-2) scheme, which achieves invariant-measure order $p=2$ while requiring only one force evaluation per timestep, by pairing a drift evaluation with a post-processor built on a weak-order-2 noise integrator. The authors develop a comprehensive convergence framework based on exotic aromatic B-series, including integration-by-parts and tree-formalisms, to prove the $p=2$ result, and analyze mean-square stability with several modification strategies. Extensive numerical experiments across 1D, 2D, and high-dimensional problems confirm the method’s second-order accuracy for the invariant measure and highlight its efficiency and robustness relative to established approaches, particularly in high dimensions or with nontrivial diffusion tensors.

Abstract

We introduce a numerical method for Brownian dynamics with position dependent diffusion tensor which is second order accurate for sampling the invariant measure while requiring only one force evaluation per timestep. Analysis of the sampling bias is performed using the algebraic framework of exotic aromatic Butcher-series. Numerical experiments confirm the theoretical order of convergence and illustrate the efficiency of the new method.

Efficient Langevin sampling with position-dependent diffusion

TL;DR

This work tackles efficient sampling of the invariant measure for Brownian dynamics with position-dependent diffusion. It introduces the Second-Order Post-processed Variable-Diffusion (PVD-2) scheme, which achieves invariant-measure order while requiring only one force evaluation per timestep, by pairing a drift evaluation with a post-processor built on a weak-order-2 noise integrator. The authors develop a comprehensive convergence framework based on exotic aromatic B-series, including integration-by-parts and tree-formalisms, to prove the result, and analyze mean-square stability with several modification strategies. Extensive numerical experiments across 1D, 2D, and high-dimensional problems confirm the method’s second-order accuracy for the invariant measure and highlight its efficiency and robustness relative to established approaches, particularly in high dimensions or with nontrivial diffusion tensors.

Abstract

We introduce a numerical method for Brownian dynamics with position dependent diffusion tensor which is second order accurate for sampling the invariant measure while requiring only one force evaluation per timestep. Analysis of the sampling bias is performed using the algebraic framework of exotic aromatic Butcher-series. Numerical experiments confirm the theoretical order of convergence and illustrate the efficiency of the new method.
Paper Structure (16 sections, 8 theorems, 65 equations, 8 figures, 1 table)

This paper contains 16 sections, 8 theorems, 65 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Weakly accurate integration methods for constant diffusion
  3. Position-dependent diffusion
  4. New post-processed method for variable diffusion
  5. Convergence analysis
  6. Preliminaries
  7. Integration by parts
  8. Tree formalism
  9. Proof of Theorem \ref{['thm:main_thm']}
  10. Mean-square stability analysis
  11. Stability domain for mean-square stiff problems
  12. Improving the stability
  13. Numerical experiments
  14. One dimension
  15. Two dimensions
  16. ...and 1 more sections

Key Result

Theorem 3.1

Assume that $V,\Sigma$ are of class $C^\infty$ with all partial derivatives having polynomial growth, and assume that $F,\Sigma$ are globally Lipchitz. Assuming that it is ergodic, PVD-2 given by eq:new_scheme and applied to eq:SDE0 has order two with respect to the invariant measure eq:defrho, prec for all smooth test function $\phi$ and all initial condition $X_0=x$ where the constant $c, C,\lam

Figures (8)

  • Figure 1: Mean-square stability domains of the PVD-2 method (\ref{['eq:new_scheme']}) and modifications (\ref{['eq:modification_1']}) and (\ref{['eq:modification_2']}), for which $\mathbb{E}\xspace(X_n^2) \rightarrow 0$ for the scalar test problem \ref{['eq:testproblem']} in the $(p,q^2)$--plane where $p=\lambda h,q=\mu \sqrt h$.
  • Figure 2: The potentials and diffusion coefficients used in one-dimensional experiments.
  • Figure 3: Convergence for sampling the invariant measure in a quadratic potential $V(x) = x^2 / 2$. Left: diffusion $\Sigma(x) = \frac{3}{2} + \frac{1}{2} \cos(x)$. Right: diffusion $\Sigma(x) = \frac{3}{2} + \frac{1}{2} \sin(x)$.
  • Figure 4: Convergence for sampling the invariant measure in a quartic potential $V(x) = x^4 / 4$. Left: diffusion $\Sigma(x) = \frac{3}{2} + \frac{1}{2} \cos(x)$. Right: diffusion $\Sigma(x) = \frac{3}{2} + \frac{1}{2} \sin(x)$.
  • Figure 5: Convergence for sampling the invariant measure in a double-well potential $V(x) = \frac{x^2}{2} + \sin(1+3x)$ with diffusion $\Sigma(x) = \exp{\left(\frac{1}{4}V(x)\right)}$. The left figure shows the error convergence against stepsize and the right figure shows the error convergence against the number of $F$ evaluations, a proxy for computational cost.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Definition 3.8
  • Proposition 3.9
  • Theorem 3.10: IBP
  • ...and 3 more