Analysis of Hierarchical Ensemble Kalman Inversion
For researchers in Bayesian inverse problems, this work provides theoretical insights into hierarchical EnKF, though it is incremental as it focuses on linear cases.
The paper analyzes hierarchical Bayesian inversion via the ensemble Kalman filter (EnKF), showing that hierarchical approaches can break the subspace property of the initial ensemble. Continuous-time limits are derived for hierarchical inversion and variants like covariance inflation and localization, with numerical verification on a linear elliptic PDE.
We discuss properties of hierarchical Bayesian inversion through the ensemble Kalman filter (EnKF). Our focus will be primarily on deriving continuous-time limits for hierarchical inversion in the linear case. An important characteristic of the EnKF for inverse problems is that the updated particles are preserved by the linear span of the initial ensemble. By incorporating certain hierarchical approaches we show that we can break away from the induced subspace property. We further consider a number of variants of the EnKF such as covariance inflation and localization, where we derive their continuous-time limits. We verify these results with various numerical experiments through a linear elliptic partial differential equation.