  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 candidate method (the locally weighted Ensemble Kalman method
for inversion) with the motivation of combining some of the strengths of the
particle filter (ability to cope with nonlinear maps and non-Gaussian
distributions) and the Ensemble Kalman filter (no filter degeneracy).
