Dictionary-based model reduction for state estimation
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We consider the problem of state estimation from m linear measurements, where the state u to recover is an element of the manifold ℳ of solutions of a parameter-dependent equation. The state is estimated using a prior knowledge on ℳ coming from model order reduction. Variational approaches based on linear approximation of ℳ, such as PBDW, yields a recovery error limited by the Kolmogorov m-width of ℳ. To overcome this issue, piecewise-affine approximations of ℳ have also be considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to 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 ℓ_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.
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