# Second Order Ensemble Langevin Method for Sampling and Inverse Problems

arXiv: https://arxiv.org/abs/2208.04506 · HTML: https://mlsanity.com/paper/2208.04506/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 58 references that are in the map.
- First posted on arXiv 9 Aug 2022.

## Builds on (most cited first, 9 of 9)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks](https://mlsanity.com/paper/1808.03620/) (2018)
- [Geometric integrators and the Hamiltonian Monte Carlo method](https://mlsanity.com/paper/1711.05337/) (2017)
- [The Geometric Foundations of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1410.5110/) (2014)
- [On the Geometric Ergodicity of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1601.08057/) (2016)
- [Ensemble preconditioning for Markov chain Monte Carlo simulation](https://mlsanity.com/paper/1607.03954/) (2016)
- [Never look back - A modified EnKF method and its application to the training of neural networks without back propagation](https://mlsanity.com/paper/1805.08034/) (2018)
- [Kalman Filter and its Modern Extensions for the Continuous-time Nonlinear Filtering Problem](https://mlsanity.com/paper/1702.07241/) (2017)

## Built on by (most cited first, 6 of 6)

- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Accelerating optimization over the space of probability measures](https://mlsanity.com/paper/2310.04006/) (2023)
- [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024)
- [New affine invariant ensemble samplers and their dimensional scaling](https://mlsanity.com/paper/2505.02987/) (2025)
- [Nesterov Acceleration for Ensemble Kalman Inversion and Variants](https://mlsanity.com/paper/2501.08779/) (2025)

## Abstract (arXiv)

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.