Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion
This work offers rigorous theoretical foundations for a widely used practical method, benefiting researchers and practitioners in inverse problems and derivative-free optimization.
The paper provides a complete analysis of the ensemble Kalman inversion with perturbed observations for fixed ensemble size applied to linear inverse problems, establishing well-posedness and convergence results based on continuous time scaling limits. The analysis yields estimates on long-time behavior and insights into convergence properties.
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.