Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion

TL;DR

This paper provides a comprehensive analysis of the ensemble Kalman inversion method, focusing on its well-posedness and convergence properties for linear inverse problems, using continuous time scaling limits and stochastic differential equations.

Contribution

It offers the first complete theoretical analysis of the ensemble Kalman inversion with perturbed observations, including well-posedness and convergence results for fixed ensemble size.

Findings

Establishes well-posedness of the method for linear inverse problems.

Derives convergence estimates based on continuous time limits.

Provides insights into the long-term behavior of the algorithm.

Abstract

The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing in various areas of application. We present a complete analysis of the ensemble Kalman inversion with perturbed observations for a fixed ensemble size when applied to linear inverse problems. The well-posedness and convergence results are based on the continuous time scaling limits of the method. The resulting coupled system of stochastic differential equations allows to derive estimates on the long-time behaviour and provides insights into the convergence properties of the ensemble Kalman inversion. We view the method as a derivative free optimization method for the least-squares misfit functional, which opens up the perspective to use the method in various areas of applications such as imaging, groundwater flow problems, biological problems as well as in the context of the training of neural networks.