An Optimal Transport Formulation of the Ensemble Kalman Filter

Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and feedback (FPF) are numerical algorithms to approximate solution of nonlinear filtering problem in continuous time. The distinguishing feature these is that Bayesian update step implemented using a control law. It has been noted literature law not unique. This main addressed this article. To obtain unique law, formulated here an optimal transportation problem. An explicit formula for (mean-field type) derived linear Gaussian setting. Comparisons made with laws different types EnKF described literature. Via empirical approximation mean-field finite-N controlled algorithm obtained. For algorithm, equations mean covariance shown be identical filter. allows strong conclusions on convergence error properties based classical stability theory that, under certain technical conditions, squared converges zero even finite number particles. A detailed propagation chaos analysis carried out algorithm. used prove weak distribution N? ?. simplified problem, analytical comparison mse importance sampling-based described. helps explain favorable scaling control-based reported several studies recent