  The Ensemble Kalman Filter (EnKF) has achieved great successes in data
assimilation in atmospheric and oceanic sciences, but its failure in
convergence to the right filtering distribution precludes its use for
uncertainty quantification. We reformulate the EnKF under the framework of
Langevin dynamics, which leads to a new particle filtering algorithm, the
so-called Langevinized EnKF. The Langevinized EnKF inherits the
forecast-analysis procedure from the EnKF and the use of mini-batch data from
the stochastic gradient Langevin-type algorithms, which make it scalable with
respect to both the dimension and sample size. We prove that the Langevinized
EnKF converges to the right filtering distribution in Wasserstein distance
under the big data scenario that the dynamic system consists of a large number
of stages and has a large number of samples observed at each stage. We
reformulate the Bayesian inverse problem as a dynamic state estimation problem
based on the techniques of subsampling and Langevin diffusion process. We
illustrate the performance of the Langevinized EnKF using a variety of
examples, including the Lorenz-96 model, high-dimensional variable selection,
Bayesian deep learning, and Long Short Term Memory (LSTM) network learning with
dynamic data.
