Data Assimilation in Reduced Modeling

Source abstract

This paper considers the problem of optimal recovery of an element $u$ of a Hilbert space ${\cal H}$ from measurements of the form $\ell_j(u)$, $j=1,\dots,m$, where the $\ell_j$ are known linear functionals on ${\cal H}$. Problems of this type are well studied [C. A. Micchelli, T. J. Rivlin, and S. Winograd, Numer. Math., 26 (1976), pp. 191--200] and usually are carried out under an assumption that $u$ belongs to a prescribed model class, typically a known compact subset of ${\cal H}$. 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 ${\cal H}$ of dimension $n$, or more generally, in the form of how well $u$ can be approximated by each subspace from 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 [Y. Maday, A. T. Patera, J. D. Penn and M. Yano, Internat. J. Numer. Methods Engrg., 102 (2015), pp. 933--965]. Their algorithm is proved, 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 multispace case by exploiting additional information derived from the whole hierarchy of spaces $V_j$ rather than only from the largest space $V_n$. It is shown that in this multispace case, the set of all $u$ that satisfy the given information can be described as the intersection of a family of known ellipsoids in ${\cal H}$. It follows that a near optimal recovery algorithm in the multispace problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of $u$ in the multispace setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multispace 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 $u$.