Adaptive Solution of Operator Equations Using Wavelet Frames

Preprint No. 1239, Department of Mathematics, University of Utrecht, May 2002. Submitted to SIAM J. Numer. Anal. In “Adaptive Wavelet Methods II Beyond the Elliptic case” of Cohen, Dahmen and DeVore ([CDD00]), an adaptive method has been developed for solving general operator equations. Using a Riesz basis of wavelet type for the energy space, the operator equation is transformed into an equivalent matrix-vector system. This system is solved iteratively, where the 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 appproximated by sparse matrices, it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best N -term approximation for the solution assuming it would be explicitly available. The condition concerning compressibility requires that, dependent on their order, the wavelets have sufficiently many vanishing moments, and that they are sufficiently smooth. Yet, except on tensor product domains, wavelets that satisfy this smoothness requirement are difficult to construct. In this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains, each of them being the image under a smooth parametrization of the hypercube. By lifting wavelets on the hypercube to the the subdomains we obtain a frame for the energy space. With this frame the operator equation is transformed into a matrix-vector system, after which this system is solved iteratively by an adaptive method similar to the one from [CDD00]. With this approach, frame elements that have sufficiently many vanishing moments and are sufficiently smooth, which is needed for the compressibility, are easily constructed. By handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix-vector system, we prove that this adaptive method has optimal computational complexity.