In this paper we will present an adaptive wavelet scheme to solvethe generalized Stokes problem. Using divergence free wavelets, theproblem is transformed into an equivalent matrix vector system, thatleads to a positive definite system of reduced size for thevelocity. This system is solved iteratively, where the applicationof the infinite stiffness matrix, that is sufficiently compressible,is r...

Several almost optimal results are obtained about N-term nonlinear approximation by dilated integer translates of a refinable function associated with a finite mask and a rather general matrix dilation A ∈ GLn(Z). §

We study nonlinear approximation in Lp(R) (0 < p < ∞, d > 1) from (a) n-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of Rd. To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces (B-spaces) which can be viewed as an anisotropic variation of Besov spaces. We use the B-spac...

We study nonlinear n-term approximation in Lp(R) (0 < p <∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel nested triangulations. We call them B-spaces because they...

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 consist...

In [GOS99], adaptive sparse grid spaces spanned by a finite number of tensor-product L2-orthogonal Haar functions have been applied to capacitance calculations on a unit screen. In this note, we state asymptotically optimal approximation rates for this problem when choosing the best possible adaptive sparse grid space of a given dimension N . We also compare the results with other recent approa...

We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focus on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions F(2) near electron-electron and electro...