caa a22 4500 38805736X CHVBK 20180307125111.0 cr unu---uuuuu 161130e199801 xx s 000 0 eng 10.1017/S0962492900002816 doi S0962492900002816 pii (NATIONALLICENCE)cambridge-10.1017/S0962492900002816 DeVore Ronald A. Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA E-mail: devore@math.sc.edu Nonlinear approximation [Elektronische Daten] [Ronald A. DeVore] 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 PDEs, and adaptive algorithms are discussed. Copyright © Cambridge University Press 1998 Acta Numerica Cambridge University Press 7(1998-01), 51-150 0962-4929 7<51 1998 7 ANU https://doi.org/10.1017/S0962492900002816 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0962492900002816 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- DeVore Ronald A. Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA E-mail: devore@math.sc.edu NATIONALLICENCE 773 0- Acta Numerica Cambridge University Press 7(1998-01), 51-150 0962-4929 7<51 1998 7 ANU CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge