Nonlinear n-term Approximation from Hierarchical Spline Bases

This article is a survey of some recent developments which concern two multilevel approximation schemes: (a) Nonlinear n-term approximation from piecewise polynomials generated by anisotropic dyadic partitions in R, and (b) Nonlinear n-term approximation from sequences of hierarchical spline bases generated by multilevel triangulations in R. A construction is given of sequences of bases consisting of differentiable (in C with r ≥ 1) piecewise polynomials (splines) over multilevel triangulations, which allow triangles with arbitrarily sharp angles. Both schemes are based on multiscale decompositions that are defined through multilevel nested partitions and their common features are best captured by the term “multiresolution”, which also relates them to wavelets. In contrast to the wavelet case, these are highly nonlinear approximation methods from redundant systems with a great deal of flexibility. It is shown that the rates of nonlinear n-term spline approximation, when using the above schemes with an arbitrary but fixed multilevel partition or triangualtion, are governed by certain smoothness spaces, called Bspaces. Unlike the commonly used Besov spaces, the B-spaces allow to characterize all rates of approximation, which gives more complete results in the isotropic case as well. An effective algorithm is described for finding, for a given function, an anisotropic dyadic partition which minimizes the corresponding B-norm of the function and thus provides an optimal rate of highly nonlinear approximation using the first scheme above. Scalable algorithms are given for nonlinear approximation which both capture the rate of the best approximation and provide the basis for numerical implementation.