Flow Annealed Kalman Inversion for Gradient-Free Inference in Bayesian Inverse Problems

TL;DR

Flow Annealed Kalman Inversion (FAKI) is a novel gradient-free Bayesian inference method that combines Kalman filtering, temperature annealing, and normalizing flows to efficiently approximate complex posterior distributions in inverse problems.

Contribution

FAKI generalizes Ensemble Kalman Inversion by integrating normalizing flows and annealing, enabling better approximation of non-Gaussian targets without gradient information.

Findings

FAKI outperforms standard EKI in accuracy on benchmark problems.

FAKI converges rapidly, typically in around 10 steps.

Demonstrated significant improvements in numerical experiments.

Abstract

For many scientific inverse problems we are required to evaluate an expensive forward model. Moreover, the model is often given in such a form that it is unrealistic to access its gradients. In such a scenario, standard Markov Chain Monte Carlo algorithms quickly become impractical, requiring a large number of serial model evaluations to converge on the target distribution. In this paper we introduce Flow Annealed Kalman Inversion (FAKI). This is a generalization of Ensemble Kalman Inversion (EKI), where we embed the Kalman filter updates in a temperature annealing scheme, and use normalizing flows (NF) to map the intermediate measures corresponding to each temperature level to the standard Gaussian. In doing so, we relax the Gaussian ansatz for the intermediate measures used in standard EKI, allowing us to achieve higher fidelity approximations to non-Gaussian targets. We demonstrate the performance of FAKI on two numerical benchmarks, showing dramatic improvements over standard EKI in terms of accuracy whilst accelerating its already rapid convergence properties (typically in $\mathcal{O}(10)$ steps).