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., $d_N(\mathcal{ M})=\mathcal{ O}(e^{ -\beta N})$, 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.
Categories: math.NA