We consider the problem of state estimation from $m$ linear measurements, where the state $u$ to recover is an element of the manifold $\mathcal{M}$ of solutions of a parameter-dependent equation. The state is estimated using a prior knowledge on $\mathcal{M}$ coming from model order reduction. Variational approaches based on linear approximation of $\mathcal{M}$, such as PBDW, yields a recovery error limited by the Kolmogorov $m$-width of $\mathcal{M}$. To overcome this issue, piecewise-affine approximations of $\mathcal{M}$ have also be 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 a 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 the path of a $\ell_1$-regularized least-squares problem. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parameterizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

翻译：我们研究从$m$个线性测量中进行状态估计的问题，其中待恢复的状态$u$是参数依赖方程解流形$\mathcal{M}$上的元素。利用来自模型降阶的$\mathcal{M}$先验知识进行状态估计。基于$\mathcal{M}$线性近似的变分方法（如PBDW）所产生的恢复误差受限于$\mathcal{M}$的Kolmogorov $m$-宽度。为克服这一局限，已有研究采用$\mathcal{M}$的分片仿射近似方法，即构建线性空间库并选择其中与$\mathcal{M}$距离最小的空间。本文提出一种基于字典模型降阶的状态估计方法，该方法通过快照字典生成的空间库，利用到流形的距离选择最优空间。该选择过程通过$\ell_1$正则化最小二乘问题的解路径获得一组候选空间。进一步地，在具有仿射参数化的参数依赖算子方程（或偏微分方程）框架下，我们基于随机线性代数提供高效的离线-在线分解策略，在保证理论可证明性的同时实现高效稳定的计算。