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Derivative-Free Bayesian Inversion Using Multiscale Dynamics

Pavliotis, G. A. and Stuart, A. M. and Vaes, U. (2022) Derivative-Free Bayesian Inversion Using Multiscale Dynamics. SIAM Journal on Applied Dynamical Systems, 21 (1). pp. 284-326. ISSN 1536-0040. doi:10.1137/21M1397416.

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Inverse problems are ubiquitous because they formalize the integration of data with mathematical models. In many scientific applications the forward model is expensive to evaluate, and adjoint computations are difficult to employ; in this setting derivative-free methods which involve a small number of forward model evaluations are an attractive proposition. Ensemble Kalman-based interacting particle systems (and variants such as consensus-based and unscented Kalman approaches) have proven empirically successful in this context, but suffer from the fact that they cannot be systematically refined to return the true solution, except in the setting of linear forward models [A. Garbuno-Inigo et al., SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 412-441]. In this paper, we propose a new derivative-free approach to Bayesian inversion, which may be employed for posterior sampling or for maximum a posteriori estimation, and may be systematically refined. The method relies on a fast/slow system of stochastic differential equations for the local approximation of the gradient of the log-likelihood appearing in a Langevin diffusion. Furthermore the method may be preconditioned by use of information from ensemble Kalman--based methods (and variants), providing a methodology which leverages the documented advantages of those methods, while also being provably refinable. We define the methodology, highlighting its flexibility and many variants, provide a theoretical analysis of the proposed approach, and demonstrate its efficacy by means of numerical experiments.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Pavliotis, G. A.0000-0002-3468-9227
Stuart, A. M.0000-0001-9091-7266
Vaes, U.0000-0002-7629-7184
Additional Information:© 2022 Society for Industrial and Applied Mathematics. Received by the editors February 8, 2021; accepted for publication (in revised form) by G. Gottwald October 1, 2021; published electronically January 24, 2022. The work of the first author was partially supported by the EPSRC through grant EP/P031587/1 and by J.P. Morgan Chase & Co. under a J.P. Morgan A.I. Research Award 2019. The work of the second author was supported by NSF award DMS-1818977 and by Office of Naval Research award N00014-17-1-2079. The work of the third author was partially funded by the Fondation Sciences Mathematiques de Paris (FSMP) through a postdoctoral fellowship in the "Mathematical Interactions" program.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/P031587/1
J. P. Morgan Chase & Co.UNSPECIFIED
Office of Naval Research (ONR)N00014-17-1-2079
Fondation Sciences Mathématiques de ParisUNSPECIFIED
Subject Keywords:Inverse problems, Multiscale methods, Derivative-free methods
Issue or Number:1
Classification Code:AMS subject classifications. 62F15, 65C35, 65C30, 65N21
Record Number:CaltechAUTHORS:20210719-210152979
Persistent URL:
Official Citation:Derivative-Free Bayesian Inversion Using Multiscale Dynamics. G. A. Pavliotis, A. M. Stuart, and U. Vaes. SIAM Journal on Applied Dynamical Systems 2022 21:1, 284-326; DOI: 10.1137/21M1397416
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109923
Deposited By: George Porter
Deposited On:19 Jul 2021 22:31
Last Modified:25 Mar 2022 18:26

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