A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters

Despite the widespread usage of discrete generation ensemble Kalman particle filtering methodology to solve nonlinear and high-dimensional inverse problems, little is known about their mathematical foundations. As genetic-type filters (a.k.a. sequential Monte Carlo), this ensemble-type can also be interpreted as mean-field approximations Kalman–Bucy equation. In contrast with conventional type interacting methods equipped a globally Lipschitz drift-type function, Ensemble depend on quadratic-type interaction function defined in terms sample covariance particles. Most literature applied mathematics computer science these sophisticated amounts designing different classes useable observer-type methods. These are based variety inconsistent but judicious auxiliary transformations or include additional inflation/localisation-type algorithmic innovations, order avoid inherent time-degeneracy an insufficient size when solving problem unstable signal. To best our knowledge, first only rigorous analysis developed pioneering articles by Le Gland–Monbet–Tran Mandel–Cobb–Beezley, which were published early 2010s. Nevertheless, besides fact that studies prove asymptotic consistency filter, they provide exceedingly pessimistic mean-error estimates grow exponentially fast respect time horizon, even for linear Gaussian problems stable one-dimensional signals. present article we develop novel self-contained complete stochastic perturbation fluctuations, stability, long-time performance filters, including time-uniform nonasymptotic apply possibly results class filters. The Riccati difference equations considered work interest own right, prototype new rational