On the Compressibility of Operators in Wavelet Coordinates

preprint No. 1249, Department of Mathematics, University of Utrecht, July 2002. Submitted to SIAM J. Math. Anal. In [CDD00], Cohen, Dahmen and DeVore proposed an adaptive wavelet algorithm for solving operator equations. Assuming that the operator defines a boundedly invertible mapping between a Hilbert space and its dual, and that a Riesz basis of wavelet type for this Hilbert space is available, the operator equation can be transformed into an equivalent well-posed infinite matrix-vector system. This system is solved by an iterative method, where each application of the infinite stiffness matrix is replaced by an adaptive approximation. Assuming that the stiffness matrix is sufficiently compressible, i.e., that it can be sufficiently well approximated by sparse matrices, it was proven that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best N -term approximations for the solution assuming it would be explicitly available. With the available results concerning compressibility however, this optimality was actually restricted to solutions with limited Besov regularity. In this paper we derive improved results concerning compressibility, which imply that with wavelets that have sufficiently many vanishing moments and that are sufficiently smooth, the adaptive wavelet method has optimal computational complexity independent of the regularity of the solution.