We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold \(\mathcal {M}\) of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on \(\mathcal {M}\) coming from model order reduction. Variational approaches based on linear approximation of \(\mathcal {M}\), such as PBDW, yield a recovery error limited by the Kolmogorov width of \(\mathcal {M}\). To overcome this issue, piecewise-affine approximations of \(\mathcal {M}\) have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to \(\mathcal {M}\). In this paper, we propose a state estimation method relying on dictionary-based model reduction, where space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of \(\ell _1\)-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

我们从一些线性测量中考虑状态估计问题，其中要恢复的状态是参数相关方程解的流形\(\mathcal {M}\)的元素。使用来自模型降阶的\(\mathcal {M}\)的先验知识来估计状态。基于\(\mathcal {M}\)线性近似的变分方法（例如 PBDW）会产生受\(\mathcal {M}\)柯尔莫哥洛夫宽度限制的恢复误差。为了克服这个问题，还考虑了\(\mathcal {M}\)的分段仿射近似，即使用线性空间库，通过最小化到\(\mathcal {M}}的距离来选择其中一个线性空间。 \)。在本文中，我们提出了一种依赖于基于字典的模型简化的状态估计方法，其中使用到流形的距离从快照字典生成的库中选择空间。该选择是在从一组\(\ell _1\)正则化最小二乘问题获得的一组候选空间中执行的。然后，在具有仿射参数化的参数相关算子方程（或 PDE）的框架中，我们提供了基于随机线性代数的高效离线在线分解，确保计算高效稳定，同时保留理论保证。