Randomized approximation of Sobolev embeddings, III

Abstract

We continue the study of randomized approximation of embeddings between Sobolev spaces on the basis of function values. The source space is a Sobolev space with nonnegative smoothness order; the target space has negative smoothness order. The optimal order of approximation (in some cases only up to logarithmic factors) is determined. Extensions to Besov and Bessel potential spaces are given and a problem recently posed by Novak and Woźniakowski is partially solved. The results are applied to the complexity analysis of weak solution of elliptic PDE.