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

Abstract

We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other words, we study the approximation numbers of the Sobolev embeddings $H_{\mathrm{mix}}^s({\mathbb{T}}^d)\hookrightarrow L_2({\mathbb{T}}^d)$, with particular emphasis on the dependence on the dimension $d$. For any fixed smoothness $s>0$, we find the exact asymptotic behavior of the constants as $d\to \infty$. We observe super-exponential decay of the constants in $d$, if $n$, the number of linear samples of $f$, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by low rank approximations. 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.

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