@article{PathirajaReichStannat2021,
  author    = {Pathiraja, Sahani Darschika and Reich, Sebastian and Stannat, Wilhelm},
  title     = {McKean-Vlasov SDEs in nonlinear filtering},
  series = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics},
  volume    = {59},
  journal   = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics},
  number    = {6},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {0363-0129},
  doi       = {10.1137/20M1355197},
  pages     = {4188 -- 4215},
  year      = {2021},
  abstract  = {Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.},
  language  = {en}
}
@article{Reich2012,
  author    = {Reich, Sebastian},
  title     = {A Gaussian-mixture ensemble transform filter},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {138},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {662},
  publisher = {Wiley-Blackwell},
  address   = {Malden},
  issn      = {0035-9009},
  doi       = {10.1002/qj.898},
  pages     = {222 -- 233},
  year      = {2012},
  abstract  = {We generalize the popular ensemble Kalman filter to an ensemble transform filter, in which the prior distribution can take the form of a Gaussian mixture or a Gaussian kernel density estimator. The design of the filter is based on a continuous formulation of the Bayesian filter analysis step. We call the new filter algorithm the ensemble Gaussian-mixture filter (EGMF). The EGMF is implemented for three simple test problems (Brownian dynamics in one dimension, Langevin dynamics in two dimensions and the three-dimensional Lorenz-63 model). It is demonstrated that the EGMF is capable of tracking systems with non-Gaussian uni- and multimodal ensemble distributions.},
  language  = {en}
}