A Particle Filter for Stochastic Advection by Lie Transport: A Case Study for the Damped and Forced Incompressible Two-Dimensional Euler Equation

Abstract

In this work, we combine a stochastic model reduction with a particle filter augmented with tempering and jittering, and apply the combined algorithm to a damped and forced incompressible two-dimensional Euler dynamics defined on a simply connected bounded domain. We show that using the combined algorithm, we are able to assimilate data from a reference system state (the “truth'') modeled by a highly resolved numerical solution of the flow that has roughly $3.1 \times 10^6$ degrees of freedom, into a stochastic system having two orders of magnitude less degrees of freedom, which is able to approximate the true state reasonably accurately for five large-scale eddy turnover times, using modest computational hardware. The model reduction is performed through the introduction of a stochastic advection by Lie transport (SALT) model as the signal on a coarser resolution. The SALT approach was introduced as a general theory using a geometric mechanics framework from Holm [Proc. A, 471 (2015)]. This work follows on the numerical implementation for SALT presented by Cotter et al. [SIAM Multiscale Model. Simul., 17 (2019), pp. 192--232] for the flow in consideration. The model reduction is substantial: the reduced SALT model has $4.9 \times 10^4$ degrees of freedom. Results from reliability tests on the assimilated system are also presented.