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 be lacking. In this paper, we state that the optimal linear space is a direct sum of shift-isometric eigenspaces corresponding to the largest eigenvalues, yielding an exact representation of the N-width as their sum. For the linear transport problem, which is modeled by half-wave symmetric initial and boundary conditions g, we obtain such an optimal decomposition by sorted trigonometric functions with eigenvalues that match the Fourier coefficients of g. Further, for normalized g in the Sobolev space \(H^r\) of broken order \(r>0\), the sorted eigenfunctions give the sharp upper bound of the N-width, which is a reciprocal of a certain power sum. Yet, for ease, we also provide the decay \((\pi N)^{-r}\), obtained by the non-optimal space of ordering the trigonometric functions by frequency rather than by eigenvalue. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight.