论文标题
在索波列夫空间中连续功能的低排名近似，混合平滑度
Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
论文作者
论文摘要
令$ω_i\ subset \ mathbb {r}^{n_i} $，$ i = 1，\ ldots，m $，以命令为域。在本文中，我们研究了$ l^2（ω_1\ times \ dots \ times \timesΩ_m）$的低级别近似值，来自Sobolev空间具有主导的混合平滑度的功能。为此，我们首先估计了双变量近似的等级，即连续奇异值分解的等级。与具有各向同性平滑度的Sobolev空间的功能相比，比较\ cite {gh14，gh19}，由于额外的混合平滑度，我们获得了改进的结果。然后，将这种收敛结果用于研究张量火车分解，以构建来自Sobolev空间的多元低级别近似值的方法，具有主导的混合光滑度。我们证明这种方法能够击败维度的诅咒。
Let $Ω_i\subset\mathbb{R}^{n_i}$, $i=1,\ldots,m$, be given domains. In this article, we study the low-rank approximation with respect to $L^2(Ω_1\times\dots\timesΩ_m)$ of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare \cite{GH14,GH19}, we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.
