On the Optimality of Reduced-Order Models for Band Structure Computations: A Kolmogorov $n$-Width Perspective

TL;DR

This paper uses Kolmogorov n-widths to establish optimality benchmarks for reduced-order models in band structure calculations, demonstrating exponential decay of approximation error and validating a greedy algorithm's near-optimal performance.

Contribution

It introduces a Kolmogorov n-width perspective to quantify the fundamental limits of reduced-order models for band structure problems, providing sharp error bounds and insights into basis selection.

Findings

Kolmogorov n-widths decay exponentially with spectral gap.

Spectral projectors simplify analysis of band crossings.

Greedy algorithms nearly achieve the theoretical optimal convergence.

Abstract

In this paper, we exploit the concept of Kolmogorov $n$-widths to establish optimality benchmarks for reduced-order methods used in phononic, acoustic, and photonic band structure calculations. The Bloch-transformed operators are entire holomorphic functions of the wave vector~$\kk$, and by Kato's analytic perturbation theory the eigenpairs inherit this holomorphy wherever the spectral gap is positive. The Kolmogorov $n$-width of the solution manifold therefore decays exponentially, at a rate controlled by the minimum spectral gap between the band of interest and its neighbors. For clusters of bands, we show that working with spectral projectors rather than individual eigenvectors renders all internal crossings -- avoided, symmetry-enforced, or conical -- irrelevant: only the gap separating the cluster from the remaining spectrum matters. These results provide a sharp lower bound on the error of any linear reduction method, against which existing approaches can be measured. Numerical experiments on one- and two-dimensional problems confirm the predicted exponential decay and demonstrate that a greedy algorithm achieves near-optimal convergence. It also provides a principled justification for the choice of basis vectors in highly successful reduced-order models like RBME.