Decay of the Kolmogorov $N$-width for wave problems

Abstract

The Kolmogorov $N$-width $d_N(\mathcal{M})$ describes the rate of the worst-case error (w.r.t.\ a subset $\mathcal{M}\subset H$ of a normed space $H$) arising from a projection onto the best-possible linear subspace of $H$ of dimension $N\in\mathbb{N}$. Thus, $d_N(\mathcal{M})$ sets a limit to any projection-based approximation such as determined by the reduced basis method. While it is known that $d_N(\mathcal{M})$ decays exponentially fast for many linear coercive parametrized partial differential equations, i.e., $d_N(\mathcal{M})=\mathcal{O}(e^{-βN})$, we show in this note, that only $d_N(\mathcal{M}) =\mathcal{O}(N^{-1/2})$ for initial-boundary-value problems of the hyperbolic wave equation with discontinuous initial conditions. This is aligned with the known slow decay of $d_N(\mathcal{M})$ for the linear transport problem.