Adaptive and anisotropic piecewise polynomial approximation

Albert Cohen () and Jean-Marie Mirebeau ()
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Albert Cohen: Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions
Jean-Marie Mirebeau: Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions

A chapter in Multiscale, Nonlinear and Adaptive Approximation, 2009, pp 75-135 from Springer

Abstract: Abstract We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given function f which is approximated. We focus our discussion on (i) the properties that describe an optimal partition for f, (ii) the smoothness properties of f that govern the rate of convergence of the approximation in the L p -norms, and (iii) fast refinement algorithms that generate near optimal partitions. While these results constitute a fairly established theory in the univariate case and in the multivariate case when dealing with elements of isotropic shape, the approximation theory for adaptive and anisotropic elements is still building up. We put a particular emphasis on some recent results obtained in this direction.

Keywords: Decision Rule; Besov Space; Optimal Partition; Cartoon Function; Adaptive Approximation (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-03413-8_4

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DOI: 10.1007/978-3-642-03413-8_4

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