Derivative-free Bayesian Inversion Using Multiscale Dynamics

TL;DR

This paper introduces a novel derivative-free Bayesian inversion method using multiscale stochastic dynamics, enabling systematic refinement and leveraging ensemble Kalman techniques for efficient posterior sampling or MAP estimation.

Contribution

A new systematic, derivative-free Bayesian inversion approach based on multiscale stochastic differential equations that can be refined and preconditioned using ensemble Kalman methods.

Findings

Method is systematically refinable.

Leverages ensemble Kalman advantages.

Demonstrated effectiveness through numerical experiments.

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.