Derivative-free Bayesian inversion using multiscale dynamics
File(s)GAPAMSUV2022.pdf (2.22 MB)
Accepted version
Author(s)
Pavliotis, GA
Stuart, AM
Vaes, U
Type
Journal Article
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 [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.
Date Issued
2022-01
Date Acceptance
2021-10-01
Citation
SIAM Journal on Applied Dynamical Systems, 2022, 21 (1), pp.284-326
ISSN
1536-0040
Publisher
Society for Industrial and Applied Mathematics
Start Page
284
End Page
326
Journal / Book Title
SIAM Journal on Applied Dynamical Systems
Volume
21
Issue
1
Copyright Statement
© by SIAM. Unauthorized reproduction of this article is prohibited.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Identifier
https://epubs.siam.org/doi/10.1137/21M1397416
Grant Number
EP/P031587/1
Subjects
0102 Applied Mathematics
Fluids & Plasmas
Publication Status
Published
Date Publish Online
2022-01-24