Data Assimilation in Banach spaces

The study of how to fuse measurement data with parametric models for PDEs has led to a new spectrum of problems in optimal recovery. A classical setting of optimal recovery is that one has a bounded set K in a Banach spaceX and a nite collection of linear functionals lj, j = 1;:::;m, from X . Given a function which is known to be in K and to have known measurements lj(f) = wj, j = 1;:::;m, the optimal recovery problem is to construct the best approximation to f from this information. Since there are generally innitely many functions in K which share these same measurements, the best approximation is the center of the smallest ballB, called the Chebyshev ball, which contains the set K of allf inK with these measurements. Most results in optimal recovery study this problem for classical Banach spacesX such as the Lp spaces, 1 p1 , and for K the unit ball of a smoothness space inX . The aforementioned parametric PDE model assumes instead, that K is the solution manifold of a parametric PDE or, as it will be the case in this paper, the assumption that K is the solution manifold is replaced by an assumption that all elements in K can be approximated by a known linear space V = Vn of dimension n to a known accuracy " = "n. This model arises because the solution manifold is complicated and usually only understood through how well it can be approximated by some known nite dimensional spaces with known a priori error estimates. Optimal recovery in this new setting was formulated and analyzed in [19] whenX is a Hilbert space, and further studied in [6]. In particular, it was shown in the latter paper that a certain numerical algorithm proposed in [19], based on least squares approximation, is optimal. The purpose of the present paper is to study this new setting for optimal recovery in a general Banach spaceX in place of a Hilbert space. While the optimal recovery has a simple theoretical description as the center of the Chebyshev ball and the optimal performance, i.e., the best error, is given by the radius of the Chebyshev ball, this is far from a satisfactory solution to the problem since it is not clear how to nd the center and the radius of the Chebyshev ball. This leads to the two fundamental problems studied in the paper. The rst centers on building numerically executable algorithms which are optimal or perhaps only near optimal. The second problem is to give sharp a priori bounds for the best error in terms of easily computed quantities. We show how these two problems are connected with well studied concepts in Banach space theory. Firstly, a priori bounds that are within twice the best error are given using the angle between the space V and the null spaceN X , consisting of all f2X whose measurements lj(f) = 0, j = 1;:::;m. Secondly, it is shown that the problem of constructing optimal or near optimal algorithms is connected to the construction of Banach space liftings. Examples are given of how these theoretical results can be implemented in concrete algorithms whenX is an Lp(D) space, with 1 p <1, or the space C(D), corresponding to p =1.