Data Assimilation in Reduced Modelling

This paper considers the problem of optimal recovery of an element $u$ of a Hilbert space $\cH$ from measurements of the form $\ell_j(u)$, $j=1,\dots,m$, where the $\ell_j$ are known linear functionals on $\cH$. Problems of this type are well studied \cite{MRW} and usually are carried out under an assumption that $u$ belongs to a prescribed model class, typically a known compact subset of $\cH$. Motivated by reduced modeling for solving parametric partial differential equations, this paper considers another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\cH$ of dimension $n$, or more generally, in the form of how well $u$ can be approximated by each of a sequence of nested subspaces $V_0\subset V_1\cdots\subset V_n$ with each $V_k$ of dimension $k$. A recovery algorithm for the one-space formulation was proposed in \cite{MPPY}. Their algorithm is proven, in the present paper, to be optimal. It is also shown how the recovery problem for the one-space problem, has a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. The major contribution of the present paper is to analyze the multi-space case. It is shown that, in this multi-space case, the set of all $u$ that satisfy the given information can be described as the intersection of a family of known ellipsoids in $\cH$. It follows that a near optimal recovery %recovery recovery algorithm in the multi-space problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of $u$ in the multi-space setting can be much than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$.