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Null-Space-Free 6D Spectral Embedding with Local Rayleigh Quotient Recovery for 3D Quasiperiodic Maxwell's Eigenproblems

Teng-Chao Sun, Tiexiang Li, Wen-Wei Lin, Xing-Long Lyu

arXiv:2609.06965v2math.NAcs.NA

TL;DR

Quasiperiodic Maxwell eigenproblems lack a finite physical-space unit cell, and their six-dimensional discretization has a large gradient kernel while physical-field recovery can be costly. The paper combines a null-space-free spectral reduction, inverse Lanczos, and physical-space reconstruction and validation. The reported large-scale experiments show an approximately threefold speedup over the original shift-and-invert realization.

  • Problem

    The 6D projected discretization has a large gradient kernel, and direct recovery of physical fields can require costly Fourier-to-grid evaluation.

    Source: §1 · p. 2

  • Method

    The framework combines a null-space-free reduction and inverse Lanczos with multi-center Taylor field reconstruction and local weighted Rayleigh quotient validation.

    Source: §1 · p. 2, §4 · p. 7, §5.3 · p. 14

  • Results

    Approximately threefold speedup over the original shift-and-invert realization on eigenproblems of dimension exceeding 107.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The framework connects positive-spectrum computation with physical-field recovery and physical-space consistency assessment for 3D quasiperiodic Maxwell modes.

    Source: §1 · p. 2, §7 · p. 20

    The analysis concerns a fixed Fourier-pseudospectral discretization without asserting ordered-eigenvalue convergence as the cutoff increases; Yee postprocessing assumes coordinate-axis-diagonal permittivity.

    Source: §2 · p. 3, §5.1 · p. 11

Abstract

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We develop a numerical framework for three-dimensional quasiperiodic Maxwell eigenvalue problems obtained through a six-dimensional periodic embedding. A projected Bloch--Fourier discretization yields a structured generalized eigenvalue problem with a large gradient-type kernel. Explicit orthonormal bases for the longitudinal and transverse subspaces remove this kernel exactly and reduce the original generalized eigenvalue problem to a null-space-free standard eigenvalue problem containing only the positive spectrum. An explicit inverse representation of the reduced operator avoids nested inner--outer linear solves and leads to an inverse Lanczos method whose main inner computation is a Hermitian positive definite conjugate-gradient solve with condition number bounded by that of the mass matrix; a residual estimate quantifies the effect of inner solves on the inverse Ritz pairs. To recover the computed modes in physical space, the six-dimensional Fourier eigenvectors are reconstructed on a three-dimensional Yee grid by a separated multi-center Taylor expansion, which avoids the dense Fourier-to-grid phase matrix and remains practical when direct dense reconstruction becomes prohibitively expensive. Local weighted Rayleigh quotients provide an independent physical-space validation, and their mass-weighted expectation is proved to equal the cropped Yee Rayleigh quotient under a partition-of-unity condition. Numerical experiments confirm the accuracy and computational effectiveness of the proposed framework and its physical-space recovery of three-dimensional quasiperiodic Maxwell modes.

1 Introduction

The introduction motivates a six-dimensional projection framework for computing quasiperiodic Maxwell spectra and interpreting their modes in physical space. Its contributions address positive-spectrum computation, field reconstruction, and physical-space consistency assessment.

1 Introduction

  • Quasiperiodic media lack translational periodicity, while periodic approximants may require rapidly growing domains and introduce band folding and mode mixing.The paper situates its computational focus alongside prior higher-dimensional projection and divergence-free projection methods for quasiperiodic Maxwell problems.Source: §1 · pp. 1–2
  • A modewise longitudinal-transverse decomposition removes the gradient kernel and yields a null-space-free standard eigenvalue problem.An explicit inverse representation eliminates nested solves; the resulting inner system is Hermitian positive definite.Source: §1 · p. 2
  • On eigenproblems exceeding 107 dimensions, inverse Lanczos achieves approximately a threefold speedup over the original shift-and-invert realization.The reduced inner system’s condition number is bounded directly by the permittivity contrast.Source: §1 · p. 3
  • A blockwise low-rank multi-center Taylor representation exploits local phase structure to recover Fourier eigenvectors on large 3D Yee grids without the dense Fourier-to-grid phase matrix.Direct and blockwise evaluations retain arithmetic cost proportional to Fourier modes times physical sampling points; blockwise evaluation reduces auxiliary storage only.Source: §1 · pp. 2–3
  • A 3D local weighted Rayleigh quotient framework estimates the spectrum and assesses reconstructed staggered Maxwell fields against the 3D Yee discretization.The motivation is that a Fourier-truncated eigenpair alone does not quantify how well its reconstructed field satisfies that discretization.Source: §1 · pp. 2–3

2 6D Projection Formulation and Bloch-Fourier Discretization of the 3D Quasiperiodic Maxwell’s Problem

The paper lifts the 3D quasiperiodic Maxwell problem to a periodic problem in six dimensions and discretizes its Bloch-periodic factor in a tensor-product Fourier space. The resulting fixed discrete generalized eigenproblem retains a gradient kernel, while its positive eigenvectors satisfy a mass-weighted discrete divergence constraint.

2 6D Projection Formulation and Bloch-Fourier Discretization of the 3D Quasiperiodic Maxwell’s Problem

  • The quasiperiodic permittivity is represented as the restriction of a periodic tensor field on a 6D torus, and the physical field is the restriction of a lifted field to the physical cut.The projection matrix has rank three; the setting assumes uniformly symmetric positive definite permittivity and nonmagnetic media.Source: §2 · pp. 3–4
  • The lifted field uses a Bloch representation, and its periodic factor is discretized by a tensor-product Fourier-pseudospectral method on the 6D torus.The analysis studies the fixed discretization without asserting convergence of ordered eigenvalues as the Fourier cutoff increases.Source: §2 · p. 4
  • The discrete curl-curl generalized eigenproblem is positive semidefinite in stiffness and has a gradient-type longitudinal null space.For each retained mode with nonzero effective wave vector, the curl-curl operator has one longitudinal zero mode and two transverse positive modes.Source: §2 · p. 5
  • The discrete mass matrix is Hermitian positive definite, and its action is evaluated using componentwise FFTs and pointwise multiplication by sampled permittivity tensors.Each application costs O(NF log NF) operations and requires O(NF) auxiliary storage.Source: §2 · p. 5
  • Every positive eigenvector satisfies the mass-weighted discrete divergence constraint, the discrete counterpart of the physical divergence constraint.This follows from the discrete compatibility relations between the gradient, curl, and stiffness operators.Source: §2 · p. 6

3 null-space Free Reduction to a Standard Eigenvalue Problem

This section constructs orthonormal longitudinal and transverse bases for the projected curl-curl operator. Restricting the generalized eigenproblem to the transverse range removes the gradient kernel while preserving the complete positive spectrum of the fixed discretization.

3 null-space Free Reduction to a Standard Eigenvalue Problem

  • The unitary basis [U0, U1, U2] separates the kernel of K from its range, with Ur spanning range(K).The construction assumes nonzero retained effective wave vectors and a reference direction nonparallel to each one; a zero effective wave vector must instead be assigned to the null space.Source: §3 · pp. 6–7
  • Every positive eigenvector lies in the transverse range, allowing the generalized eigenproblem to be reduced to a standard eigenproblem on that range.The construction uses the discrete mass-weighted divergence relation to place positive eigenvectors in the range of Ur.Source: §3 · pp. 6–7
  • The reduction exactly removes the gradient kernel while preserving the complete positive spectrum of the fixed Fourier-pseudospectral discretization.The resulting null-space-free formulation is the input to the inverse Lanczos method developed next.Source: §3 · p. 7

4 Inverse Lanczos Method with Orthogonalized Inner Solves

The section derives an explicit inverse for the reduced positive-spectrum operator and uses it in a null-space-free inverse Lanczos solver. An orthogonalized inner system bounds the conditioning of its CG solve by that of the mass matrix, while a residual estimate accounts for finite inner solves.

4 Inverse Lanczos Method with Orthogonalized Inner Solves

  • The explicit inverse representation avoids nested linear solves and makes a Hermitian positive definite system the dominant iterative solve.The inverse Lanczos procedure targets the smallest positive eigenpairs of the reduced problem.Source: §4 · p. 7
  • The orthogonalized inner formulation solves with c M, whose condition number is bounded by that of M independently of the singular values of W.The bound follows from λmin(M) ≤ λmin(c M) ≤ λmax(c M) ≤ λmax(M).Source: §4 · pp. 8–9
  • A problem-specific residual estimate separates the effects of finite-tolerance inner solves and full reorthogonalization on inverse Ritz residuals.With independently terminated inner iterations, the additional reorthogonalization coefficients need not vanish.Source: §4 · p. 9
  • The stated stopping indicator monitors the nominal Lanczos recurrence residual but omits the additional terms arising in the computed recurrence.Those terms are associated with full reorthogonalization when inner iterations are independently terminated.Source: §4 · pp. 9–10
  • The inverse Lanczos implementation uses CG to solve c M systems, selects the largest inverse Ritz pairs, and reciprocates them to obtain the smallest positive eigenvalues.The algorithm also reconstructs the corresponding Fourier eigenvectors.Source: §4 · p. 10

5 Physical-Space Reconstruction and Local Rayleigh Quotient Analysis

The section reconstructs Fourier eigenmodes on a three-dimensional staggered Yee grid and evaluates their physical-space consistency. It replaces dense Fourier-to-grid evaluation with a separated Taylor representation and develops local weighted Rayleigh quotients whose mass-weighted expectation matches the cropped Yee quotient under stated conditions.

5 Physical-Space Reconstruction and Local Rayleigh Quotient Analysis

  • The reconstructed field is assessed on a cropped three-dimensional Yee grid using residuals and Rayleigh quotients, independently of its six-dimensional Fourier representation.The cropped operators are used for physical-space Rayleigh-quotient and residual computations.Source: §5 · p. 10, §5.1 · p. 11

5.1 Yee operators and cropped sampling windows

  • The Yee postprocessing uses a curl-curl stencil and componentwise mass operator for permittivity tensors diagonal in physical coordinates.The mass operator's diagonal entries are uniformly positive; sampling uses an enlarged grid region to supply neighboring stencil values.Source: §5.1 · p. 11

5.2 Memory-efficient reconstruction of the 3D Yee field

  • Because the dense phase matrix requires O(NgNF) storage, the field is reconstructed with a separated multi-center Taylor representation instead.Reducing local phase radii tightens the reconstruction error bound; the stated work is O(SLpNF + LpNg) per Fourier field.Source: §5.2 · pp. 11–12

5.3 LWRQs and physical-space consistency

  • The Maxwell LWRQ averages local energy and mass densities separately before taking their ratio, avoiding potentially ill-conditioned componentwise quotients.This construction accounts for coupling of the staggered components by the curl-curl operator.Source: §5.3 · p. 12
  • Under the partition-of-unity weight condition and positive local masses, the mass-weighted expectation of LWRQs equals the cropped Yee Rayleigh quotient.The section also relates local quotient variation to weight distance and bounds Taylor-reconstruction contributions to the residual by the reconstruction error estimate.Source: §5.3 · pp. 13–14

6 Numerical experiments

The experiments validate the solver, reconstruction, and physical-space consistency measures, then apply the full framework to a representative 3D quasiperiodic Maxwell problem.

6.1 Experiment 1: Large-scale comparison of original and reduced inverse realizations

  • NRI and ERI recover the same positive eigenvalues to numerical precision and average 65.83 Lanczos steps, versus 76.92 Arnoldi steps for OSI; ERI remains less costly than OSI as dimension grows.At N = 5, NRI reaches its time limit before completion.Source: Figure 1 · p. 16, §6.1 · p. 16

6.2 Experiment 2: 3D Yee-field reconstruction

  • With p = 10 and 64 centers, reconstruction errors are e2 = 8.69×10−10 and eA = 1.38×10−5.Localization reduces the phase radius and field error at each Taylor order, while increasing the order improves accuracy at a fixed partition.Source: Figure 2 · p. 17, §6.2 · p. 17
  • At the largest window completed by both methods, Taylor reconstruction is approximately 12.9 times faster; direct reconstruction exceeds the time budget at the next size.Field error remains at a similar level as the sampling window grows under the phase-radius criterion.Source: Figure 2 · p. 17, §6.2 · p. 17

6.3 Experiment 3: Physical-space consistency and LWRQ validation

  • Yee refinement lowers η3D, νLWRQ, and δλ; for half-widths L ≳2, η3D and νLWRQ remain in the 10−5 range and δλ near 10−6–10−5.The measures indicate agreement between reconstructed fields, the independently discretized 3D system, and the 6D eigenpairs.Source: Figure 3 · p. 18, §6.3 · p. 18
  • For modes 11 and 114, local LWRQ-deviation maps on z = 0 and x = 0 resolve spatial variation consistent with small global νLWRQ.The maps use a common logarithmic color scale for the two sections of each mode, with values below 10−6 saturated.Source: Figure 3 · p. 18, §6.3 · p. 19

6.4 Experiment 4: A representative 3D quasiperiodic Maxwell problem

  • At four representative point-mode pairs, LWRQ means differ from 6D eigenvalues by at most 1.369 × 10−3; reconstructed slices show directional field patterns.The experiment connects the low-frequency spectrum, reconstructed fields, and physical-space consistency assessment in one N = 5 computation.Source: Figure 4 · p. 20, §6.4 · p. 20

7 Conclusions

The framework combines null-space-free spectral reduction, inverse Lanczos, and physical-space recovery for 3D quasiperiodic Maxwell eigenproblems. Future work will examine how projection geometry affects spectra and eigenvalue-spacing statistics.

7 Conclusions

  • Numerical experiments confirm the effectiveness of the proposed reduction, eigensolver, reconstruction, and physical-space recovery framework for 3D quasiperiodic Maxwell problems.Source: §7 · p. 20
  • A modewise longitudinal-transverse decomposition removes the gradient kernel and reduces the projected Bloch-Fourier generalized eigenproblem to a null-space-free standard problem containing only the positive spectrum.Source: §7 · p. 20
  • An explicit inverse representation and orthogonalized inner formulation enable inverse Lanczos with a Hermitian positive definite conjugate-gradient system.Source: §7 · p. 20
  • Six-dimensional Fourier eigenvectors are reconstructed on a three-dimensional staggered Yee grid, while cropped Yee residuals and local weighted Rayleigh quotients assess physical-space consistency using a separately discretized Yee operator.Source: §7 · p. 20
  • Future work will study projection geometry's influence on spectral structure, including band-structure comparisons across projection matrices, transitions among periodic, quasiperiodic, and aperiodic regimes, and nearest-neighbor eigenvalue-spacing statistics.Source: §7 · pp. 20–21
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