  In this paper, we present a new ensemble-based filter method by
reconstructing the analysis step of the particle filter through a transport
map, which directly transports prior particles to posterior particles. The
transport map is constructed through an optimization problem described by the
Maximum Mean Discrepancy loss function, which matches the expectation
information of the approximated posterior and reference posterior. The proposed
method inherits the accurate estimation of the posterior distribution from
particle filtering. To improve the robustness of Maximum Mean Discrepancy, a
variance penalty term is used to guide the optimization. It prioritizes
minimizing the discrepancy between the expectations of highly informative
statistics for the approximated and reference posteriors. The penalty term
significantly enhances the robustness of the proposed method and leads to a
better approximation of the posterior. A few numerical examples are presented
to illustrate the advantage of the proposed method over the ensemble Kalman
filter.
