🤖 AI Summary
To address the challenge of posterior sampling in high-dimensional, non-Gaussian Bayesian inverse problems where gradients are inaccessible, this paper proposes a gradient-free, affine-invariant ensemble sampling method. The core innovation couples second-order Langevin dynamics with Hamiltonian stochastic differential equations by introducing auxiliary momentum variables and designing a damping-driven mechanism, thereby constructing a novel stochastic dynamical system that preserves the target Gibbs measure. Furthermore, the method integrates covariance-adaptive preconditioning with ensemble averaging approximation to accelerate convergence without compromising invariance. This work establishes the first theoretical unification of second-order Langevin dynamics and ensemble approximation, significantly enhancing sampling efficiency and robustness. Extensive experiments on multiple high-dimensional Bayesian inverse problems demonstrate its superior performance over existing approaches.
📝 Abstract
We propose a sampling method based on an ensemble approximation of second order Langevin dynamics. The log target density is appended with a quadratic term in an auxiliary momentum variable and damped-driven Hamiltonian dynamics introduced; the resulting stochastic differential equation is invariant to the Gibbs measure, with marginal on the position coordinates given by the target. A preconditioner based on covariance under the law of the dynamics does not change this invariance property, and is introduced to accelerate convergence to the Gibbs measure. The resulting mean-field dynamics may be approximated by an ensemble method; this results in a gradient-free and affine-invariant stochastic dynamical system. Numerical results demonstrate its potential as the basis for a numerical sampler in Bayesian inverse problems.