Perspectives on locally weighted ensemble Kalman methods

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a possible candidate (the locally weighted Ensemble Kalman method) for combining the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy), and can also be applied to optimisation and sampling tasks.