Second Order Ensemble Langevin Method for Sampling and Inverse Problems
This work addresses sampling challenges in Bayesian inverse problems, presenting an incremental improvement with a gradient-free and affine-invariant approach.
The authors tackled the problem of sampling from complex distributions in Bayesian inverse problems by proposing a second-order ensemble Langevin method, which demonstrated potential as a numerical sampler through numerical results.
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.