Approximation numbers of Sobolev embeddings-Sharp constants and tractability

作者：Kuehn Thomas; Sickel Winfried; Ullrich Tino^*

来源：Journal of Complexity, 2014, 30(2): 95-116.

DOI：10.1016/j.jco.2013.07.001

• Approximation numbers
• Sobolev embeddings
• Sharp constants
• Weak tractability
• Curse of dimensionality

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H-s(T-d) of fractional smoothness s %26gt; 0 for various equivalent norms including the classical one. The error is always measured in L-2 (T-d). Particular emphasis is given to the dependence of all constants on the dimension d. We capture the exact decay rate in n and the exact decay order of the constants with respect to d, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.

• 出版日期2014-4