Second Order Ensemble Langevin Method for Sampling and Inverse Problems
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910987601313792 |
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| author | Liu, Ziming Stuart, Andrew M. Wang, Yixuan |
| author_facet | Liu, Ziming Stuart, Andrew M. Wang, Yixuan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_04506 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Second Order Ensemble Langevin Method for Sampling and Inverse Problems Liu, Ziming Stuart, Andrew M. Wang, Yixuan Dynamical Systems Machine Learning Numerical Analysis Methodology 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. |
| title | Second Order Ensemble Langevin Method for Sampling and Inverse Problems |
| topic | Dynamical Systems Machine Learning Numerical Analysis Methodology |
| url | https://arxiv.org/abs/2208.04506 |