Coarse reduced model selection for nonlinear state estimation

State estimation is the task of approximately reconstructing a solution $u$ of a parametric partial differential equation when the parameter vector $y$ is unknown and the only information is $m$ linear measurements of $u$. In [Cohen et. al., 2021] the authors proposed a method to use a family of linear reduced spaces as a generalised nonlinear reduced model for state estimation. A computable surrogate distance is used to evaluate which linear estimate lies closest to a true solution of the PDE problem. In this paper we propose a strategy of coarse computation of the surrogate distance while maintaining a fine mesh reduced model, as the computational cost of the surrogate distance is large relative to the reduced modelling task. We demonstrate numerically that the error induced by the coarse distance is dominated by other approximation errors.