Resumen
In "Adaptive wavelet methods II---Beyond the elliptic case" of Cohen, Dahmen, and DeVore [Found. Comput. Math., 2 (2002), pp. 203--245], 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 approximated 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 that the latter would be explicitly available. The condition concerning compressibility requires that, dependent on their order, the wavelets have sufficiently many vanishing moments, and that they be sufficiently smooth. However, except on tensor product domains, wavelets that satisfy this smoothness requirement are not easy 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 subdomains, we obtain a {\em 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 the work of Cohen, Dahmen, and DeVore. With this approach, frame elements that have sufficiently many vanishing moments and are sufficiently smooth, something 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.

Resumen