Convergence analysis of the generalized empirical interpolation method (Q2814454)

Using an adaptive greedy algorithm, interpolation schemes are studied to approximate functions from a Banach space \(\mathcal X\) (later-on also for a Hilbert space, where improved error estimates are found), where the approximand is restricted to a compact subset \(F\) and the approximant is from an \(n\)-dimensional subspace of \(\mathcal X\). That subspaces adapts itself to \(F\) and is constructed by a greedy method. The interpolation errors are estimated in particular by relating them to the error of the best approximation, the \(n\)-width. Up to an increased error due to the operator norm of the approximation, these errors in the actual approximation algorithm are shown to be asymptotically the same as the \(n\)-width so long as it has polynomial or exponential decay. The implementations of the methods are discussed too.