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Nonlinear approximation. (English) Zbl 0931.65007
Acta Numerica 7, 51-150 (1998).
Summary: This is a survey of nonlinear approximation, especially that part of the subject which is important in numerical computation. Nonlinear approximation means that the approximants do not come from linear spaces but rather from nonlinear manifolds. The central question to be studied is what, if any, are the advantages of nonlinear approximation over the simpler, more established, linear methods. This question is answered by studying the rate of approximation which is the decrease in error versus the number of parameters in the approximant. The number of parameters usually correlates well with computational effort. It is shown that in many settings the rate of nonlinear approximation can be characterized by certain smoothness conditions which are significantly weaker than required in the linear theory. Emphasis in the survey will be placed on approximation by piecewise polynomials and wavelets as well as their numerical implementation. Results on highly nonlinear methods such as optimal basis selection and greedy algorithms (adaptive pursuit) are also given. Applications to image processing, statistical estimation, regularity for partial differential equations and adaptive algorithms are discussed.
For the entire collection see [Zbl 0894.00025].

MSC:
65D15 Algorithms for approximation of functions
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
41A25 Rate of convergence, degree of approximation
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
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