Data Assimilation in Reduced Modeling
Abstract
We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from $m$ measurements obtained through known linear functionals on $\mathcal{H}$. Problems of this type are well studied \cite{MRW} under an assumption that $u$ belongs to a prescribed model class, e.g. a known compact subset of $\mathcal{H}$. Motivated by reduced modeling for 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 $\mathcal{H}$ of dimension $n$, or more generally, how well $u$ can be approximated by each $k$dimensional subspace $V_k$ of a sequence of nested subspaces $V_0\subset V_1\cdots\subset V_n$. A recovery algorithm for the onespace formulation, proposed in \cite{MPPY}, is proven here to be optimal and to have 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 for which it is shown that the set of all $u$ satisfying the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multispace problem is to identify any point in this intersection which can provide a much better accuracy than in the onespace problem. Two iterative algorithms based on alternating projections are proposed for recovery in the multispace problem. A detailed analysis of one of them provides a posteriori performance estimates for the iterates, stopping criteria, and 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 $u$.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 arXiv:
 arXiv:1506.04770
 Bibcode:
 2015arXiv150604770B
 Keywords:

 Mathematics  Numerical Analysis;
 62M45;
 65D05;
 68Q32;
 97N50
 EPrint:
 27 pages