Second-order accurate ensemble transform particle filters

TL;DR

This paper develops second-order accurate extensions of the ensemble transform particle filter (ETPF) to improve computational efficiency and ensemble spread estimation, demonstrating enhanced accuracy in chaotic models and limitations in complex scene-viewing applications.

Contribution

It introduces second-order accurate extensions of the ETPF, including Sinkhorn approximation, and shows how the nonlinear ensemble transform filter (NETF) is a special case.

Findings

Significant accuracy improvements over standard ensemble Kalman filter and ETPF in Lorenz models.

Second-order corrections can cause statistical inconsistencies in complex models.

The Sinkhorn approximation reduces computational costs of the linear transport step.

Abstract

Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) (S.~Reich, {\it A non-parametric ensemble transform method for Bayesian inference}, SIAM J.~Sci.~Comput., 35, (2013), pp. A2013--A2014) replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second-order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter (NETF) arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.