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

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 $H^s_{\rm mix}(\mathbb{T}^d)\hookrightarrow L_2(\mathbb{T}^d)$, 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.