Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

Abstract

This paper is concerned with the filtering problem in continuous-time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter which provides an exact solution for the linear Gaussian problem, (ii) the ensemble Kalman-Bucy filter (EnKBF) which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems, and (iii) the feedback particle filter (FPF) which represents an extension of the EnKBF and furthermore provides for an consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of non-uniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

Data assimilation: The Schrödinger perspective

“…Here we instead follow the presentation of Taghvaei and Mehta (2016) and Taghvaei et al (2017) and assume that we have M samples z i from a PDF π. The method is based on Since T reproduces constant functions, (155) determines φ i up to a constant contribution, which we fix by requiring for tackling this problem.…”

“…and Taghvaei, de Wiljes, Mehta and Reich (2017). Here we would like to mention in particular an approximation based on diffusion maps which we summarise in Appendix A.…”

The Hitchhiker’s guide to nonlinear filtering

“…For a linear state space model the FBPF with constant-gain approximation becomes exact and is identical 20 to the ensemble Kalman-Bucy filter (EnKBF, Bergemann & Reich, 2012;Taghvaei et al, 2018), which can be shown to be asymptotically exact (Künsch, 2013). More precisely, for a linear model with f (x) = Ax, G(x) = Σ 1/2 x and h(x) = Bx, the gain K(x) can be solved for in closed form, using the knowledge that the posterior is Gaussian at all times, and is given by K =Σ t B.…”

Gain function approximation in the feedback particle filter

“…In the continuous-time setting, two formulations of the EnKF have been developed: stochastic EnKF, and the more recent deterministic EnKF [8,51]. As has already been noted, the deterministic EnKF is in fact identical to the FPF algorithm (1.7) in the linear Gaussian setting [8,59].…”