Coarse reduced model selection for nonlinear state estimation

TL;DR

This paper introduces a strategy to efficiently select coarse reduced models for nonlinear state estimation in PDEs, reducing computational costs while maintaining accuracy.

Contribution

It proposes a coarse computation approach for surrogate distances in reduced model selection, balancing efficiency and accuracy.

Findings

Coarse surrogate distance computation is effective for model selection.

Error from coarse distance is dominated by other approximation errors.

Method reduces computational cost without sacrificing significant accuracy.

Abstract

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.