Dictionary-based model reduction for state estimation

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.