Optimal reduced model algorithms for data-based state estimation

Reduced model spaces, such as reduced bases and polynomial chaos, are linear spaces Vn of finite dimension n which are designed for the efficient approximation of certain families of parametrized PDEs in a Hilbert space V . The manifold M that gathers the solutions of the PDE for all admissible parameter values is globally approximated by the space Vn with some controlled accuracy ϵn, which is typically much smaller than when using standard approximation spaces of the same dimension such as finite elements. Reduced model spaces have also been proposed in [Y. Maday et al., Internat. J. Numer. Methods Ergrg., 102 (2015), pp. 933-965] as a vehicle to design a simple linear recovery algorithm of the state u ∈ M corresponding to a particular solution instance when the values of parameters are unknown but a set of data is given by m linear measurements of the state. The measurements are of the form lj (u), j = 1, . . .,m, where the lj are linear functionals on V . The analysis of this approach in [P. Binev et al., SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 1-29] shows that the recovery error is bounded by μnϵn, where μn = μ (Vn,W) is the inverse of an inf-sup constant that describe the angle between Vn and the space W spanned by the Riesz representers of (l1, . . ., l m). A reduced model space which is efficient for approximation might thus be ineffective for recovery if μ n is large or infinite. In this paper, we discuss the existence and effective construction of an optimal reduced model space for this recovery method. We extend our search to affine spaces which are better adapted than linear spaces for various purposes. Our basic observation is that this problem is equivalent to the search of an optimal affine algorithm for the recovery of M in the worst case error sense. This allows us to peform our search by a convex optimization procedure. Numerical tests illustrate that the reduced model spaces constructed from our approach perform better than the classical reduced basis spaces.