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EigenWave: An Optimal O(N) Method for Computing Eigenvalues and Eigenvectors by Time-Filtering the Wave Equation

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Pith's one-line read EigenWave computes eigenvalues of elliptic problems by time-filtering the wave equation at a target frequency, and the paper argues the resulting scheme is an optimal O(N) algorithm when implicit time-stepping is solved by multigrid.

desk verdict A genuine new spectral-transform eigensolver with clean analysis and strong experiments; the O(N) headline is empirically supported but not proven, and needs qualification before publication. read the letter →

arxiv 2507.18282 v1 pith:LFKTAFMV submitted 2025-07-24 math.NA cs.NA

Daniel Appelo , Jeffrey W. Banks , William D. Henshaw , Ngan Le , Donald W. Schwendeman This is my paper · ORCID
classification math.NAcs.NA MSC 65F1565N2565N55
keywords EigenWaveHoltzeigenvalueproblemsArnoldialgorithmtime-filteringoversetgridsmultigridO(N)
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes EigenWave, an iterative method for computing selected eigenvalues and eigenvectors of elliptic boundary value problems. Each iteration solves the wave equation with the current eigenvector guess as initial data, then time-filters the solution at a chosen target frequency; this defines a new linear operator with the same eigenvectors and a spectrum concentrated near the target. The method can target eigenvalues anywhere in the spectrum and never has to invert an indefinite shifted matrix. When the implicit time-stepping equations are solved by multigrid, the authors claim the total cost scales linearly with the number of grid points, an optimal O(N) algorithm. That claim rests on the observation, supported by scaling tables, that the number of wave solves per eigenpair stays roughly constant as the mesh is refined.

What carries the argument

The load-bearing object is the WaveHoltz filter function $\beta(\lambda;\omega)=\frac{2}{T_f}\int_0^{T_f}(\cos\omega t-\frac14)\cos(\lambda t)\,dt$, realized discretely as a weighted time-integral accumulated during one wave solve. It acts as a spectral transform that preserves the eigenvectors of $L$ while compressing the spectrum into $[-\frac12,1]$ and amplifying modes near the target frequency. Because the operator $S$ is never formed, each application is just one wave-solve, and with implicit time-stepping the matrix that must be inverted is the definite matrix $M_h=I-\Delta t^2 L_h/2$, which is well suited to multigrid.

What would settle it

Fix a target frequency and compute the same few eigenpairs on a sequence of grids with spacing halved, using implicit time-stepping with 10 steps per period and a multigrid solver. If the total number of wave-solves grows with the number of grid points, or if CPU time per grid point does not stay flat, then the optimal O(N) claim is false; the paper's scaling tables are exactly this experiment and currently show roughly constant wave-solve counts with CPU ratios near 4.

Watch

Extended reading notes

Core claim

The central claim is that eigenvalue problems like $L\phi=-\lambda^2\phi$ can be solved by power or Krylov iteration on the operator $S$ defined by one wave solve plus the filter $v\mapsto \frac{2}{T_f}\int_0^{T_f}(\cos\omega t-\frac14)w(x,t;v)\,dt$. The paper proves that $S$ has the same eigenfunctions as $L$ and eigenvalues $\beta(\lambda_j;\omega)\in[-\frac12,1]$, with the largest values attained near $\lambda_j=\omega$. Consequently an Arnoldi-type solver applied matrix-free to $S$ returns eigenvectors whose original eigenvalues are recovered by a Rayleigh quotient; with implicit time-stepping and multigrid the per-solve work is O(N), so the whole computation is claimed to be O(N).

Load-bearing premise

The O(N) claim stands on the assumption that the number of wave-solves required by the Arnoldi-Krylov iteration to converge stays bounded as the mesh is refined, so total work is dominated by the O(N) multigrid time-step solves.

Editorial extensions

If this is right

  • Eigenpairs anywhere in the spectrum can be computed by choosing the target frequency $\omega$, without forming or inverting an indefinite matrix $L-\sigma I$.
  • A few wave-solves per eigenpair (about 3–5) suffice when EigenWave is embedded in a matrix-free Krylov-Schur or IRAM iteration, with roughly 10 implicit time-steps per period.
  • With a multigrid solver for the implicit equations, the reported CPU times grow about linearly with the number of grid points on refined overset grids, making large two- and three-dimensional eigenvalue problems feasible.
  • Highly multiple eigenvalues are handled accurately because the method computes the invariant eigenspace, and the original eigenvalues are recovered by a simple Rayleigh quotient in a post-processing step.
  • The computed eigenvectors can also be used for deflation in related Helmholtz solvers, so EigenWave provides a practical bridge between eigenvalue computation and wave-based iterative solvers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to use the time filter as a rational or polynomial preconditioner for interior eigenvalue problems beyond Arnoldi methods, since the filter damps high frequencies while enhancing a selected band.
  • If the near-constant wave-solve counts persist at larger scales, the practical bottleneck shifts to multigrid cycle counts; the small cycle increases observed on overset grids are the main threat to strict O(N) scaling and deserve direct measurement.
  • The memory profile suggests the method is especially attractive for extreme-scale three-dimensional problems, where direct sparse factorizations are infeasible; this is implied by the paper's results but not demonstrated at very large N.
  • The filter's tunable peak width offers a concrete memory-versus-cost trade: increasing the number of filter periods narrows the selection window and reduces the number of stored eigenvectors, at the price of proportionally longer wave solves.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper introduces EigenWave, an algorithm for computing eigenvalues and eigenvectors of elliptic boundary value problems. The method defines a linear operator S by solving a related wave equation over one or more periods and applying a time filter that enhances Fourier modes with frequency near a tunable target ω. The operator S shares eigenvectors with the original elliptic operator but has eigenvalues β(λ;ω) that are largest for λ near ω, so that a matrix-free Krylov-Schur or IRAM iteration on S yields the desired eigenvectors; eigenvalues of the original problem are recovered from a Rayleigh quotient. The paper derives the continuous and discrete filter operators, embeds S in standard Arnoldi-based eigensolvers, and demonstrates the method on a range of two- and three-dimensional overset-grid geometries. It further claims that using implicit time-stepping with a fixed number of time-steps per period and a multigrid solver yields an optimal O(N) eigenvalue solver.

Significance. If the O(N) claim is substantiated, the paper describes a practically important method: interior eigenpairs of elliptic problems in complex geometry can be computed with total cost linear in N and without inverting an indefinite shifted matrix. The paper's strengths include a clean self-consistent derivation of the spectral transform, a careful treatment of the discrete time-filtering correction, and an extensive set of numerical experiments over many geometries, including high-multiplicity and three-dimensional cases. The matrix-free integration with high-quality Arnoldi packages is a practical and reproducible contribution. The main weakness is that the optimal-complexity claim is not proven; the paper itself flags both Arnoldi wave-solve counts and multigrid cycle counts as expected or empirically observed rather than theoretically established.

major comments (2)
  1. [Section 7, Tables 15–18] The central claim of an optimal O(N) algorithm rests on two boundedness assumptions: (i) that the number of wave-solves (Arnoldi matvecs) needed to converge the requested eigenpairs remains bounded independent of mesh spacing, and (ii) that the average number of multigrid cycles per wave-solve is bounded as the mesh is refined. The paper explicitly labels (i) as an expectation rather than a proven fact, and the disk data in Tables 17 and 18 show the average multigrid cycles increasing from 6.0 on the coarsest grid to 9.4 (order 2) and 9.8 (order 4) on the finest grid, with CPU ratios of 3.16 and 4.61 in the final refinement steps. The text's statement that the reason for this increase 'needs further investigation' confirms that (ii) is unverified. Since the title and abstract assert optimality, the manuscript should either provide a proof or a rigorous complexity bound for both components, or temper the claim to 'numerically observed near-linear scaling' in the abstract, introduction, and conclusions.
  2. [Section 7, paragraph after Tables 17–18] The CPU-ratio evidence is presented as being 'roughly a factor 4', but the disk results contain ratios such as 3.16 and 4.61, which deviate from 4 by roughly 20% and 15% respectively, and the normalized CPU/N plots in Figure 26 are normalized by the coarsest-grid time, which can mask non-asymptotic slowdowns. The linear-scaling conclusion is therefore substantially better supported for the square geometry than for the disk. The paper should explicitly qualify the disk results and either report more extensive scaling data (for example, an additional refinement level) or explain why the observed growth in multigrid cycles is expected to saturate.
minor comments (6)
  1. [Section 6.5, text after Table 8] The cross-reference 'Table ??' is undefined; the table should be numbered and cited properly.
  2. [Section 6.6, last sentence] The sentence 'about 1.8 (or 1.7) times faster then the explicit scheme at second (or fourth) order accuuracy' contains typos: 'then' should be 'than' and 'accuuracy' should be 'accuracy'.
  3. [Section 4, Eqs. (20a)–(20b)] The choice of αd in (20b) is stated without derivation; a short justification that this value normalizes the discrete filter to one at λ = ω would help the reader.
  4. [Section 7, tables] The tolerance written as '10-10' should be typeset as 10^{-10} to avoid ambiguity.
  5. [Section 6.9, Table 13] The second-order sphere row reports a maximum eigenvector error of 7.13e-5 and a residual of 2.11e-5, which is not 'near full machine precision' as claimed in the introductory paragraph of Section 6; the discrepancy should be explained (for example, by reference eigenvector inaccuracy or near-multiple eigenvalues).
  6. [Section 3, Eq. (15)] The claim that β(λ;ω) lies in [−1/2, 1] is stated but not proved; adding a short argument or a reference would make the spectral-bounds section self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: EigenWave is a designed spectral transform whose eigenpairs are obtained by solving for the eigenvectors of S and then recovering the Laplacian eigenvalues by a Rayleigh quotient, with the O(N) claim presented as an evidenced conjecture rather than a fitted prediction.

full rationale

The paper does not exhibit a circular derivation. Section 3 (Theorem 1) constructs the EigenWave operator S from a wave-equation IBVP and a time filter, then proves by Fourier expansion that S has the same eigenfunctions as L with eigenvalues β(λ_j;ω). This is a genuine derivation, not an assumption of the conclusion: the output eigenvalues are recovered from the computed eigenvectors of S via the Rayleigh quotient (B.1)-(B.2), and the target frequency ω is a user-selected input, not a fitted parameter. No parameter is tuned to the reference eigenvalues computed by SLEPc; the comparison tables validate the method externally. The optimal O(N) claim in Section 7 is an extrapolation supported by the scaling Tables 15-18, with wave-solve counts roughly constant; the paper itself flags the two assumptions behind the claim ('it is expected that the convergence of the Arnoldi algorithm should also be roughly independent of the mesh spacing') and notes 'the average number of multigrid cycles per wave-solve increases somewhat in some cases; the reason for this needs further investigation.' These are unproven complexity assumptions and limitations, not circularity. Self-citations to the authors' WaveHoltz papers [1,4-6] provide context and prior technology, but the load-bearing properties are either re-derived in this paper (the β filter and eigenfunction identity) or evidenced by this paper's own numerical tables; no uniqueness theorem or ansatz is imported solely through self-citation. The algorithm is a legitimate spectral-transform method with independent numerical content.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on standard spectral theory for elliptic operators, a discrete eigenfunction expansion for the finite-difference operator, and the assumed O(N) performance of the external multigrid solver. The O(N) complexity claim additionally assumes that Arnoldi iteration counts are mesh-independent, which is empirically supported but not proven.

free parameters (2)
  • NITS (implicit time-steps per period) = 10 (typical); minimum 5 for stability; studied from 5 to 15 in Appendix C.5
    Number of implicit time-steps per period. Chosen by numerical experiments to minimize time-steps per eigenvalue; not derived from theory. The abstract's claim that about 10 time-steps per period suffice depends on this choice.
  • Np (number of filter periods) = 1 in most runs; values 2, 4, 6, 8, 12 used; optimized per requested count in Appendix C.6
    Controls filter-peak width. Chosen empirically as a trade-off between cost per wave-solve and number of eigenpairs returned. The optimal value depends on the number of requested eigenpairs.
assumptions (4)
  • standard math The elliptic operator L with boundary conditions has a complete orthonormal set of eigenfunctions with real non-positive eigenvalues, used in the modal expansion of the wave equation solution.
    Invoked in Theorem 1 in Section 3 and is standard spectral theory for self-adjoint elliptic operators on bounded domains.
  • domain assumption The discrete operator L_h has a complete set of eigenvectors that diagonalize the implicit time-stepping scheme in Appendix A, Eq. (A.1).
    Needed for the discrete filter analysis; not proven for overset-grid finite differences, though it is expected for consistent discretizations of self-adjoint operators.
  • domain assumption The multigrid solver Ogmg solves the definite implicit systems in O(N) work with iteration counts independent of N.
    The O(N) claim relies on the external solver's complexity, cited from [54,55,63], not proven in this paper.
  • ad hoc to paper The number of Arnoldi or wave-solve iterations needed to converge requested eigenpairs is roughly independent of mesh spacing.
    Stated as an expectation in Section 7 and supported by scaling tables, but not proven; if false, the O(N) total complexity fails.

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Pith. "Pith review of EigenWave: An Optimal O(N) Method for Computing Eigenvalues and Eigenvectors by Time-Filtering the Wave Equation." pith.science (2026). https://pith.science/paper/LFKTAFMV

@misc{pith2026250718282,
  author       = {Pith},
  title        = {Pith review of: EigenWave: An Optimal O(N) Method for Computing Eigenvalues and Eigenvectors by Time-Filtering the Wave Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LFKTAFMV}},
  note         = {Machine review of arXiv:2507.18282}
}
abstract

An algorithm named EigenWave is described to compute eigenvalues and eigenvectors of elliptic boundary value problems. The algorithm, based on the recently developed WaveHoltz scheme, solves a related time-dependent wave equation as part of an iteration. At each iteration, the solution to the wave equation is filtered in time. As the iteration progresses, the filtered solution generally contains relatively larger and larger proportions of eigenmodes whose eigenvalues are near a chosen target frequency (target eigenvalue). The ability to choose an arbitrary target frequency enables the computation of eigenvalues anywhere in the spectrum, without the need to invert an indefinite matrix, as is common with other approaches. Furthermore, the iteration can be embedded within a matrix-free Arnoldi algorithm, which enables the efficient computation of multiple eigenpairs near the target frequency. For efficiency, the time-dependent wave equation can be solved with implicit time-stepping and only about $10$ time-steps per-period are needed, independent of the mesh spacing. When the (definite) implicit time-stepping equations are solved with a multigrid algorithm, the cost of the resulting EigenWave scheme scales linearly with the number of grid points $N$ as the mesh is refined, giving an optimal $O(N)$ algorithm. The approach is demonstrated by finding eigenpairs of the Laplacian in complex geometry using overset grids. Results in two and three space dimensions are presented using second-order and fourth-order accurate approximations.

Figures

Figures reproduced from arXiv: 2507.18282 by the authors.

Figure 1
Figure 1. Absolute value of selected eigenvectors of the Laplacian with Dirichlet boundary conditions computed with the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Overset grid (and magnification) for the letters in RPI. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. and discussed further below, βpλ; ωq P r´1 2 , 1s. Thus, all eigenvalues βj of S are real and lie in the interval r´1 2 , 1s. This completes the proof. l 0 1 2 3 4 5 / -0.5 0 0.5 1 WaveHoltz beta function : N p =1 : N p =2 : N p =3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Eigenpairs of a square with target frequency [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: At left is overset grid G p2q disk for a disk. The middle and right show graphs of the filter function β with the computed eigenvalues marked with black circles for 2nd and 4th-order accurate discretizations respectively, both using grid G p4q disk. The target frequenc…
Figure 6
Figure 6. Figure 6: Some computed eigenfunctions of a disk where [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Left: overset grid G p2q cic for a circle in a channel. Right: graph of the filter function β with the computed eigenvalues marked with black circles for a fourth-order accurate computation on grid G p4q cic . The target frequency ω “ 4 is marked as a vertical black li…
Figure 8
Figure 8. Figure 8: Circle in a square: some computed eigenvectors. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Left: three-shapes overset grid G p4q ts . Middle: magnified view of the grid. Right: graph of the filter function β with the computed eigenvalues marked with black circles. ϕ p4q -1.0 1.0 ϕ p20q -1.0 1.0 ϕ p44q -1.0 1.0 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Three shapes. Selected computed eigenvectors. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Left: overset grid for the letters R, P, and I, and a magnified portion. Right: graph of the filter function [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: RPI grid: computed eigenvectors. ϕ p284q -1.0 1.0 ϕ p308q -1.0 1.0 ϕ p334q -1.0 1.0 [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: RPI grid: Selected eigenvectors computed with the EigenWave algoritm. [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Computing eigenpairs of the RPI domain with more periods (larger [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: At left is double ellipse geometry and overset grid [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Selected eigenvectors of the double ellipse geometry on [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: Box: computing multiple eigenpairs, order=2. The computed eigenvalues are marked with black circles. [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]
Figure 18
Figure 18. Figure 18: Left: overset grid G p2q disk for a cylindrical pipe. Middle and right: graphs of the filter function β with the computed eigenvalues marked with black circles for second-order accurate computations on grid G p2q pipe (middle) and on the finer grid G p4q pipe [PITH_F…
Figure 19
Figure 19. Figure 19: Pipe: some computed eigenvectors. EigenWave: grid=pipeze2, ts=implicit, order=2, Np “ 1, KrylovSchur num wave time-steps wave-solves time-steps max max max eigs solves per period per eig per-eig eig-err evect-err eig-res 35 124 10 3.5 35 1.85e-13 7.67e-09 6.39e-13 [P…
Figure 20
Figure 20. Figure 20: At left is exploded view of the surface patches of the overset grid for a solid sphere. At right is a plot of the filter [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
Figure 21
Figure 21. Figure 21: Three computed eigenvectors on a sphere using [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: Double ellipsoid overset grid G p1q des (grid lines coarsened by a factor of 2). Let G pjq des denote the overset grid for the double-ellipsoid geometry with typical grid spacing ∆s pjq “ 1{p10jq. As shown in [PITH_FULL_IMAGE:figures/full_fig_p024_22.png]
Figure 23
Figure 23. Figure 23: Filter function and computed eigenvalues for the double ellipsoid. [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]
Figure 24
Figure 24. Figure 24: Selected eigenvectors for the double ellipsoid using [PITH_FULL_IMAGE:figures/full_fig_p025_24.png]
Figure 25
Figure 25. Figure 25: The asymptotic convergence rate of the WaveHoltz filter is normally determined by the discrete eigenvalue closest [PITH_FULL_IMAGE:figures/full_fig_p026_25.png]
Figure 26
Figure 26. Figure 26: Grid scaling with implicit time-stepping and multigrid. Normalized values of CPU [PITH_FULL_IMAGE:figures/full_fig_p028_26.png]

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