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Optimal approximation order of piecewise constants on convex partitions
Journal of Complexity ( IF 1.8 ) Pub Date : 2019-11-06 , DOI: 10.1016/j.jco.2019.101444
Oleg Davydov , Oleksandr Kozynenko , Dmytro Skorokhodov

We prove that the error of the best nonlinear Lp-approximation by piecewise constants on convex partitions is O(N2d+1), where N is the number of cells, for all functions in the Sobolev space Wq2(Ω) on a cube ΩRd, d2, as soon as 2d+1+1p1q0. The approximation order O(N2d+1) is achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates of the approximation order from the above and below are given for various Sobolev and Sobolev–Slobodeckij spaces Wqr(Ω) embedded in Lp(Ω), some of which also improve the standard estimate O(N1d) known to be optimal on isotropic partitions.



中文翻译:

凸分区上分段常数的最佳逼近阶。

我们证明了最佳非线性误差 大号p凸分区上的分段常数的近似值是 Øñ-2d+1个,在哪里 ñ 是Sobolev空间中所有函数的像元数 w ^q2Ω 在一个立方体上 Ω[Rdd2, 立刻 2d+1个+1个p-1个q0。近似阶数Øñ-2d+1个在通过对自适应二元分区进行各向异性细化而获得的多面体分区上可以实现。对于各种Sobolev和Sobolev–Slobodeckij空间,可以从上下获得进一步的近似阶数估计w ^q[RΩ 嵌入 大号pΩ,其中一些还可以改善标准估算 Øñ-1个d 已知在各向同性分区上是最佳的。

更新日期:2019-11-06
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