  Filtering in spatially-extended dynamical systems is a challenging problem
with significant practical applications such as numerical weather prediction.
Particle filters allow asymptotically consistent inference but require
infeasibly large ensemble sizes for accurate estimates in complex spatial
models. Localisation approaches, which perform local state updates by
exploiting low dependence between variables at distant points, have been
suggested as a potential resolution to this issue. Naively applying the
resampling step of the particle filter locally however produces implausible
spatially discontinuous states. The ensemble transform particle filter replaces
resampling with an optimal-transport map and can be localised by computing maps
for every spatial mesh node. The resulting local ensemble transport particle
filter is however computationally intensive for dense meshes. We propose a new
optimal-transport based local particle filter which computes a fixed number of
maps independent of the mesh resolution and interpolates these maps across
space, reducing the computation required and allowing it to be ensured
particles remain spatially smooth. We numerically illustrate that, at a reduced
computational cost, we are able to achieve the same accuracy as the local
ensemble transport particle filter, and retain its improved robustness to
non-Gaussianity and ability to quantify uncertainty when compared to local
ensemble Kalman filters.
