Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler
Abstract
Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of overdamped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker--Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker--Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.
Additional Information
© 2020 Society for Industrial and Applied Mathematics. Received by the editors March 25, 2019; accepted for publication (in revised form) by Y. Bakhtin October 10, 2019; published electronically February 4, 2020. The work of the first and fourth authors was supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program, by Earthrise Alliance, by the Paul G. Allen Family Foundation, and by the National Science Foundation (NSF grant AGS-1835860). The work of the fourth author was also supported by NSF grant DMS-1818977. The work of the second author was partially supported by Caltech's von Karman postdoctoral instructorship. The work of the third author was supported by AFOSR MURI FA9550-18-1-0502. The authors are grateful to José A. Carrillo, Greg Pavliotis, and Sebastian Reich for helpful input which improved this paper.Attached Files
Published - 19m1251655.pdf
Submitted - 1903.08866.pdf
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Additional details
- Alternative title
- Gradient Structure Of The Ensemble Kalman Flow With Noise
- Eprint ID
- 97319
- Resolver ID
- CaltechAUTHORS:20190722-103410192
- Schmidt Futures Program
- Earthrise Alliance
- Paul G. Allen Family Foundation
- AGS-1835860
- NSF
- DMS-1818977
- NSF
- Caltech
- FA9550-18-1-0502
- Air Force Office of Scientific Research (AFOSR)
- Created
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2019-07-22Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field