A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation

In this work, we apply a particle filter with three additional procedures (model reduction, tempering and jittering) to a damped and forced incompressible 2D Euler dynamics defined on a simply connected bounded domain. We show that using the combined algorithm, we are able to successfully assimilate data from a reference system state (the ``truth") modelled by a highly resolved numerical solution of the flow that has roughly $3.1\times10^6$ degrees of freedom for $10$ 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. Roy. Soc. A (2015). This work follows on the numerical implementation for SALT presented by Cotter et al, SIAM Multiscale Model. Sim. (2019) for the flow in consideration. The model reduction is substantial: The reduced SALT model has $4.9\times 10^4$ degrees of freedom. Forecast reliability and estimated asymptotic behaviour of the particle filter are also presented.