David Kelly

Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time

The ensemble Kalman filter (EnKF) is a method for combining a dynamical model with data in a sequential fashion. Despite its widespread use, there has been little analysis of its theoretical properties. Many of the algorithmic innovations associated with the filter, which are required to make a useable algorithm in practice, are derived in an ad hoc fashion. The aim of this paper is to initiate the development of a systematic analysis of the EnKF, in particular to do so for small ensemble size. The perspective is to view the method as a state estimator, and not as an algorithm which approximates the true filtering distribution. The perturbed observation version of the algorithm is studied, without and with variance inflation. Without variance inflation well-posedness of the filter is established; with variance inflation accuracy of the filter, with respect to the true signal underlying the data, is established. The algorithm is considered in discrete time, and also for a continuous time limit arising when observations are frequent and subject to large noise. The underlying dynamical model, and assumptions about it, is sufficiently general to include the Lorenz 63 and 96 models, together with the incompressible Navier–Stokes equation on a two-dimensional torus. The analysis is limited to the case of complete observation of the signal with additive white noise. Numerical results are presented for the Navier–Stokes equation on a two-dimensional torus for both complete and partial observations of the signal with additive white noise.

Geometric versus non-geometric rough paths

In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in Gubinelli (2004). We first show that branched rough paths can equivalently be defined as

Smooth approximation of stochastic differential equations

gamma

Nonlinear stability and ergodicity of ensemble based Kalman filters

-Holder continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path

Concrete ensemble Kalman filters with rigorous catastrophic filter divergence

mathbf{X}

Rough path recursions and diffusion approximations

lying above a path

Fluctuations in the heterogeneous multiscale methods for fast–slow systems

X

Deterministic homogenization for fast–slow systems with chaotic noise

, there exists a geometric rough path

Nonlinear stability of the ensemble kalman filter with adaptive covariance inflation

bar{mathbf{X}}

Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

lying above an extended path