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

Ankit Srivastava

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.

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

Abstract

In this paper, we exploit the concept of Kolmogorov -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~, and by Kato's analytic perturbation theory the eigenpairs inherit this holomorphy wherever the spectral gap is positive. The Kolmogorov -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.

Paper Structure

This paper contains 41 sections, 3 theorems, 52 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Affine structure in $\mathbf{k}$
  4. Solution manifold and reduced order strategy
  5. Kolmogorov $n$-Widths
  6. The $n$-width as an optimality benchmark
  7. Decay rates
  8. Exponential decay.
  9. Algebraic decay.
  10. The regularity connection.
  11. Parametric regularity and the role of holomorphy
  12. The size of the holomorphic extension.
  13. Computable bounds via SVD
  14. Constructive near-optimality: the greedy algorithm
  15. $n$-Width Analysis for the Bloch Eigenvalue Problem
  16. ...and 26 more sections

Key Result

Proposition 3.1

Let $j$ be a band index and $\mathbf{k}_0 \in \mathcal{B}$ a point at which $\omega_j^2(\mathbf{k}_0)$ is a simple eigenvalue with spectral gap $\delta_j(\mathbf{k}_0) > 0$ as defined in eq:spectral-gap. Let $L > 0$ be the Lipschitz constant for the eigenvalue curves, as in eq:eval-lipschitz. Then t where the holomorphy radius satisfies the lower bound $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: One-dimensional parametric curves (solution manifolds) embedded in a three-dimensional ambient space $V = \mathbb{R}^3$. (a) A curve lying entirely within a two-dimensional subspace $V_2$: two basis vectors $\phi_1, \phi_2$ suffice to represent every point on $\mathcal{M}$ exactly. (b) A curve winding through all three dimensions. (c) Two separate one-dimensional curves, each confined to its own two-dimensional subspace which together span all three dimensions.
  • Figure 2: Geometric interpretation of the Kolmogorov $n$-width. The red curve is the solution manifold $\mathcal{M}$ embedded in $V = \mathbb{R}^3$, and the blue shaded plane is a two-dimensional subspace $V_2$. The dashed blue curve is the orthogonal projection of $\mathcal{M}$ onto $V_2$; the black stick marks the largest perpenducular distance, $\sup_{\mu}\|\mathbf{u}(\mu) - \Pi_{V_2}\mathbf{u}(\mu)\|$. This is shown for two different planes.
  • Figure 3: 1D phononic crystal with continuously varying properties. (a) Unit cell property distributions $E(x)$ and $\rho(x)$. (b) Band structure showing the first six dispersion branches. (c) Normalized singular value decay of the snapshot matrices for each band, with fitted exponential rates $\beta$.
  • Figure 4: Oracle greedy algorithm for the first $J = 10$ bands. (a) Worst-case projection error versus basis dimension $n$ for the SVD-optimal subspace and the oracle greedy. (b) Band structure with greedy selections marked. Open circles denote the initialization point; filled circles indicate subsequent greedy selections, with marker size decreasing in the order of selection.
  • Figure 5: Residual-based greedy algorithm for the first $J = 10$ bands, initialized with $J$ eigenvectors at the Brillouin zone center. (a) Worst-case projection error versus basis dimension $n$ for the SVD-optimal subspace and the residual-based greedy. The algorithm converges to machine precision at $n = 30$ using $21$ full eigenvalue solves. (b) Band structure with greedy selections marked. Open circles denote the initialization point; filled circles indicate subsequent greedy selections, with marker size decreasing in the order of selection.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 2.1: Kolmogorov $n$-width
  • Proposition 3.1: Holomorphy radius for Bloch eigenpairs
  • Theorem 3.2: $n$-width bound for an isolated band
  • Theorem 3.3: $n$-width bound for a multi-band manifold