Nonlinear approximation describes to some extent an ideal state in assuming for example the knowledge of all infinitly many wavelet coefficients of a function. Therefore one cannot hope to achieve proved possible convergence orders in a strong sense by any realistic adaptive numerical method. Hence one has to weaken this ideal state somehow. This was one of the starting points for a systematic study of restricted nonlinear approximation in a recent paper by Cohen, DeVore and Hochmuth. Their analysis provides complete characterizations in terms of scales of Besov spaces.
In this talk we explain how restricted nonlinear approximation covers in particular numerical discretizations with respect to graded meshes adapted to apriori known singularities in the solutions of boundary integral equations. Moreover we describe how restricted nonlinear approximation takes place within the poles uniform approximation and nonlinear approximation. Finally we discuss (besides others) various kinds of tree-type and tresholding algorithms on this background.
The work of the author has been supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant Ho 1846/1-11.