Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence
Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence
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DOI：
10.1007/s00365-015-9299-x
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发表时间：
2013-12
影响因子：
2.7
通讯作者：
T. Kühn;W. Sickel;T. Ullrich
中科院分区：
文献类型：
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作者：
T. Kühn;W. Sickel;T. Ullrich
研究了d维环面上Sobolev空间中d维周期函数的逼近问题，其中逼近误差用-范数度量.换句话说，我们研究了Sobolev嵌入的近似数，特别强调了对维数的依赖性。对于任何固定的光滑性，我们发现双边估计的近似数作为一个函数innandd。我们观察到常数ind，ifn的超指数衰减，线性样本的数量很大。此外，出于数值实现的问题，我们还专注于误差衰减，可以实现的近似使用只有几个线性样本（smalln）。我们提出了一些令人惊讶的结果，所谓的“前渐近”衰减，并指出连接到最近推出的概念，准多项式的可处理性的近似问题。
We investigate the approximation ofd-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothnesson thed-dimensional torus, where the approximation error is measured in the-norm. In other words, we study the approximation numbersof the Sobolev embeddings, with particular emphasis on the dependence on the dimensiond. For any fixed smoothness, we find two-sided estimates for the approximation numbers as a function innandd. We observe super-exponential decay of the constants ind, ifn, the number of linear samples off, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (smalln). We present some surprising results for the so-called “preasymptotic” decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.