Dictionary-based model reduction for state estimation

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.