Ensemble Kalman Inversion as an Inertial Interacting Particle System

Abstract

Ensemble Kalman Inversion (EKI) is a derivative-free, ensemble-based method for inverse and optimization problems. A central limitation of its continuous-time dynamics is premature covariance collapse, which can restrict the directions accessible to the inversion and increase sensitivity to the initial ensemble. We introduce a second-order interacting particle system that combines the Kalman force with inertia, damping, attraction towards the ensemble mean, and short-range repulsion. We establish a generalized subspace property showing that nonzero initial velocities may enlarge the affine space accessible to the dynamics, while a finite-ensemble restriction remains. For linear inverse problems, we derive the mean and fluctuation dynamics, identify a regime in which fully collapsed configurations are linearly unstable, characterize asymptotic optimality on the subspace retained by the limiting covariance, and prove exponential decay for the associated frozen-covariance mean dynamics. Numerical experiments on high-dimensional linear and nonlinear inverse problems investigate subspace enlargement, discrepancy-based regularization, and ensemble collapse.