Approximation of mixed order Sobolev functions on the d-torus --\n Asymptotics, preasymptotics and d-dependence

2013/12/22 by Thomas Kuehn, Kuehn, Thomas, Winfried Sickel +3 · 1 citation
Decision Sciences · Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.1312.6386

openalex publication_date 2013/12/22 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We investigate the approximation of d-variate periodic functions in Sobolev\nspaces of dominating mixed (fractional) smoothness s>0 on the d-dimensional\ntorus, where the approximation error is measured in the L₂-norm. In other\nwords, we study the approximation numbers of the Sobolev embeddings H^s _rm\nmix( mathbbT^d) hookrightarrow L₂( mathbbT^d), with particular emphasis\non the dependence on the dimension d. For any fixed smoothness s>0, we find\nthe exact asymptotic behavior of the constants as d\→\∞. We observe\nsuper-exponential decay of the constants in d, if n, the number of linear\nsamples of f, is large. In addition, motivated by numerical implementation\nissues, we also focus on the error decay that can be achieved by low rank\napproximations. We present some surprising results for the so-called\n``preasymptotic'' decay and point out connections to the recently introduced\nnotion of quasi-polynomial tractability of approximation problems.\n

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