The Kolmogorov N-width for linear transport: Exact representation and the influence of the data

Abstract

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 ${c_r N^{-r}}$ for functions {with Sobolev regularity} $g\in H^{r{-\varepsilon}}$ for all $\varepsilon>0$ even if $g\not\in H^{r}$. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight.