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.
翻译:国家估算是,在参数矢量为$y美元、唯一信息为$1美元线性测量为$1美元的情况下,对参数矢量部分差分方程的解决方案进行大约重建($1美元)的任务。在[Cohen 等人,2021]中,作者建议采用一个方法,将线性缩小空间系列作为通用的非线性缩小模型,用于国家估算。使用可计算替代距离来评估哪些线性估计最接近PDE问题的真正解决方案。在本文中,我们提出了在保持精细网格缩小模型的同时对代孕距离进行粗略计算的战略,因为代孕距离的计算成本与减少的模型任务相比是很大的。我们从数字上表明,粗距离引起的错误是由其他近似差所决定的。