Improved Exponential Convergence Rates by Oversampling Near the Boundary
Improved Exponential Convergence Rates by Oversampling Near the Boundary
复制标题
通过边界附近的过采样提高指数收敛率
DOI：
10.1007/s00365-013-9211-5
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发表时间：
2014
影响因子：
2.7
通讯作者：
B. Zwicknagl
中科院分区：
文献类型：
--
作者：
C. Rieger;B. Zwicknagl
光滑函数的采样不等式在离散范数和误差项方面限制了连续范数，当离散数据集变得稠密时，误差项以指数方式趋于零。改进的估计来自离散点集，集群附近的边界，特别是分散的点集，分布在边界层中的二次，张量化切比雪夫网格。如果应用于稳定重建过程的残差，这样的不等式产生指数收敛阶。我们的研究结果同意的观察，指数确定性的近似率往往提高全球范围内，如果数据集分布更密集的边界附近。
Sampling inequalities for smooth functions bound a continuous norm in terms of a discretized norm and an error term that tends to zero exponentially as the discrete data set becomes dense. Improved estimates are derived for discrete point sets that cluster near the boundary, in particular for scattered point sets that are distributed quadratically in a boundary layer, and for tensorized Chebyshev grids. If applied to residuals of stable reconstruction processes, such inequalities yield exponential convergence orders. Our results agree with the observation that exponential deterministic approximation rates are often improved globally if the data sets are distributed more densely near the boundary.