Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models

The ensemble Kalman inversion (EKI) is a particle based method which has been introduced as the application of the ensemble Kalman filter to inverse problems. In practice it has been widely used as derivative-free optimization method in order to estimate unknown parameters from noisy measurement data. For linear forward models the EKI can be viewed as gradient flow preconditioned by a certain sample covariance matrix. Through the preconditioning the resulting scheme remains in a finite dimensional subspace of the original high-dimensional (or even infinite dimensional) parameter space and can be viewed as optimizer restricted to this subspace. For general nonlinear forward models the resulting EKI flow can only be viewed as gradient flow in approximation. In this paper we discuss the effect of applying a sample covariance as preconditioning matrix and quantify the gradient flow structure of the EKI by controlling the approximation error through the spread in the particle system. The ensemble collapse on the one side leads to an accurate gradient approximation, but on the other side to degeneration in the preconditioning sample covariance matrix. In order to ensure convergence as optimization method we derive lower as well as upper bounds on the ensemble collapse. Furthermore, we introduce covariance inflation without breaking the subspace property intending to reduce the collapse rate of the ensemble such that the convergence rate improves. In a numerical experiment we apply EKI to a nonlinear elliptic boundary-value problem and illustrate the dependence of EKI as derivative-free optimizer on the choice of the initial ensemble.