NDLI: Adaptive Solution of Operator Equations Using Wavelet Frames

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Stevenson, Rob
Copyright Year	2003
Abstract	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.
Starting Page	1074
Ending Page	1100
Page Count	27
File Format	PDF
ISSN	00361429
DOI	10.1137/S0036142902407988
e-ISSN	10957170
Issue Number	3
Volume Number	41
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-26
Access Restriction	Subscribed
Subject Keyword	Stability and convergence of numerical methods Wavelets optimal computational complexity best \(N\)-term approximation frames Iterative methods for linear systems wavelets Approximation by arbitrary nonlinear expressions; widths and entropy operator equations Multigrid methods; domain decomposition adaptive methods Rate of convergence, degree of approximation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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