Abstract

This talk is concerned with the approximation of embeddings between Besov-type spaces defined on bounded multidimensional domains or (patchwise smooth) manifolds. We compare the quality of approximations of three different strategies based on wavelet expansions. For this purpose, sharp rates of convergence corresponding to classical uniform refinement, best $N$-term, and best $N$-term tree approximation will be presented. In particular, we will see that whenever the embedding of interest is compact, greedy tree approximation schemes are as powerful as abstract best $N$-term approximation and that (for a large range of parameters) they can outperform uniform schemes based on a priori fixed (hence non-adaptively chosen) subspaces. This observation justifies the usage of adaptive non-linear algorithms in computational practice, e.g., for the approximate solution of boundary integral equations arising from physical applications. If time permits, implications for the related concept of approximation spaces associated to the three approximation strategies will be discussed.