Decay of the Kolmogorov [formula omitted]-width for wave problems

Abstract

The Kolmogorov N-width dN(M) describes the rate of the worst-case error (w.r.t. a subset M⊂H of a normed space H) arising from a projection onto the best-possible linear subspace of H of dimension N∈N. Thus, dN(M) sets a limit to any projection-based approximation such as determined by the reduced basis method. While it is known that dN(M) decays exponentially fast for many linear coercive parameterized partial differential equations, i.e., dN(M)=O(e−βN), we show in this note, that only dN(M)=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 dN(M) for the linear transport problem.