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Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Inigo, Alfredo and Hoffmann, Franca and Li, Wuchen and Stuart, Andrew M. (2020) Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler. SIAM Journal on Applied Dynamical Systems, 19 (1). pp. 412-441. ISSN 1536-0040. https://resolver.caltech.edu/CaltechAUTHORS:20190722-103410192

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1137/19M1251655DOIArticle
https://arxiv.org/abs/1903.08866arXivDiscussion Paper
ORCID:
AuthorORCID
Garbuno-Inigo, Alfredo0000-0003-3279-619X
Hoffmann, Franca0000-0002-1182-5521
Li, Wuchen0000-0002-2218-5734
Alternate Title:Gradient Structure Of The Ensemble Kalman Flow With Noise
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.
Funders:
Funding AgencyGrant Number
Schmidt Futures ProgramUNSPECIFIED
Earthrise AllianceUNSPECIFIED
Paul G. Allen Family FoundationUNSPECIFIED
NSFAGS-1835860
NSFDMS-1818977
CaltechUNSPECIFIED
Air Force Office of Scientific Research (AFOSR)FA9550-18-1-0502
Subject Keywords:ensemble Kalman inversion, Kalman-Wasserstein metric, gradient flow, mean-field Fokker-Planck equation
Issue or Number:1
Classification Code:AMS subject classifications: 65N21, 35Q84, 62F15, 65C30, 65C35, 82C80
Record Number:CaltechAUTHORS:20190722-103410192
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190722-103410192
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97319
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:22 Jul 2019 20:59
Last Modified:31 Mar 2020 15:49

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