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Derivative-free Bayesian Inversion Using Multiscale Dynamics

Pavliotis, G. A. and Stuart, A. M. and Vaes, U. (2021) Derivative-free Bayesian Inversion Using Multiscale Dynamics. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20210719-210152979

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Abstract

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. 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, whilst also being provably refineable. 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:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2102.00540arXivDiscussion Paper
ORCID:
AuthorORCID
Pavliotis, G. A.0000-0002-3468-9227
Stuart, A. M.0000-0001-9091-7266
Vaes, U.0000-0002-7629-7184
Additional Information:G.A.P. was partially supported by the EPSRC through the grant number EP/P031587/1 and by JPMorgan Chase & Co under a J.P. Morgan A.I. Research Award 2019. (Any views or opinions expressed herein are solely those of the authors listed, and may differ from the views and opinions expressed by JPMorgan Chase & Co. or its affiliates. This material is not a product of the Research Department of J.P. Morgan Securities LLC. This material does not constitute a solicitation or offer in any jurisdiction.) The work of A.M.S. is supported by NSF (award DMS-1818977) and by the Office of Naval Research (award N00014-17-1-2079). The work of U.V. was partially funded by the Fondation Sciences Mathématique de Paris (FSMP), through a postdoctoral fellowship in the "mathematical interactions" program.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/P031587/1
J. P. Morgan Chase & Co.UNSPECIFIED
NSFDMS-1818977
Office of Naval Research (ONR)N00014-17-1-2079
Fondation Sciences Mathématiques de ParisUNSPECIFIED
Subject Keywords:Inverse problems, Multiscale methods, Derivative-free methods
Classification Code:AMS subject classifications. 62F15, 65C35, 65C30, 65N21
Record Number:CaltechAUTHORS:20210719-210152979
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210719-210152979
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109923
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:19 Jul 2021 22:31
Last Modified:19 Jul 2021 22:31

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