The Kolmogorov $N$-width describes the best possible error one can achieve by elements of an $N$-dimensional linear space. Its decay has extensively been studied in Approximation Theory and for the solution of Partial Differential Equations (PDEs). Particular interest has occurred within Model Order Reduction (MOR) of parameterized PDEs e.g.\ by the Reduced Basis Method (RBM). While it is known that the $N$-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to $N^{-1/2}$. On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the $N$-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the $N$-width seems to lack. In this paper, we use techniques from Fourier Analysis to derive exact representations of the $N$-width in terms of initial and boundary conditions of the linear transport equation modeled by some function $g$ for half-wave symmetric data. For arbitrary functions $g$, we derive bounds and prove that these bounds are sharp. In particular, we prove that the $N$-width decays as $c_r N^{-(r+1/2)}$ for functions in the Sobolev space, $g\in H^r$. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight.

翻译：Kolmogorov $N$-宽度描述了由$N$维线性空间中的元素所能实现的最佳逼近误差。其衰减性已在逼近论以及偏微分方程（PDEs）解的框架下得到广泛研究。在参数化偏微分方程的模型降阶（MOR）中，例如通过约化基方法（RBM），该问题尤受关注。尽管已知对于某些问题，$N$-宽度呈指数级快速衰减（从而可实现高效的MOR），但在线性输运和波动方程的例子中，衰减速率却降为$N^{-1/2}$。另一方面，普遍认为光滑的参数依赖关系有助于$N$-宽度的快速衰减。然而，关于数据属性（如正则性或斜率）对$N$-宽度衰减速率影响的详细分析似乎尚属空白。本文利用傅里叶分析技术，针对半波对称数据，给出了线性输运方程（由某函数$g$建模）在初边值条件下的$N$-宽度的精确表示。对于任意函数$g$，我们推导了误差界并证明了这些界的紧致性。具体而言，我们证明：对于Sobolev空间中的函数$g\in H^r$，$N$-宽度按$c_r N^{-(r+1/2)}$衰减。我们的理论分析辅以数值实验，验证了所给误差界的紧致性，并提供了额外的定量洞察。