{"id":"226830f4-0786-4d0b-817d-0201f63b0e68","arxiv_id":"2507.23613","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"MFWH computes several Helmholtz solutions simultaneously from one wave-equation solve using multiple time filters, with convergence theory and O(N) scaling shown on square domains.","lead":"This paper presents a new algorithm, Multi-Frequency WaveHoltz (MFWH), that solves many Helmholtz equations at different frequencies at the same time by running one wave equation and applying multiple time filters to its solution. It matters because multi-frequency Helmholtz solves appear in acoustics, electromagnetics, and seismology, and the method aims for near-optimal O(N) cost.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Spurious-resonance failure mode is load-bearing: the paper's central claim is unqualified even though Section 6 admits a singular A_h case, and the proposed remedy is only a posteriori.","rationale":"I read the paper in good faith and found the fixed-point analysis (Theorems 1-3), the continuous filter construction, and the O(N) scaling argument internally consistent. The numerical results match the predicted convergence rates, and the multigrid-backed scaling table supports the O(N) claim for the tested geometries. The single most load-bearing weakness is the acknowledged but unresolved spurious-resonance failure mode: the central claim requires A_h nonsingular, yet the paper gives no way to guarantee this and even displays a case where μ crosses 1 away from all ω_m. The unit square, the paper's own test geometry, has a Dirichlet eigenvalue near one of those crossings, so the failure is concretely testable rather than merely pathological. Since the paper already flags this caveat honestly, the reader's CONDITIONAL verdict seems exactly right; I recommend no change. I did not identify a more fundamental internal inconsistency, and I do not treat the reliance on [11] as the primary issue because the relevant formulas are stated in the text and the numerics corroborate them.","tokens_in":15638,"tokens_out":9836,"duration_ms":115726,"concrete_test":"Run MFWH-GMRES on the unit square with Dirichlet boundary conditions for the N_f=2 configuration of Figure 1 (ω_1=5, ω_2=9, N_p=1), using smooth Gaussian forcings. First compute the discrete Dirichlet eigenvalue λ_{h,ν}=π√5≈7.025 and evaluate μ_d at all discrete eigenvalues, marking any λ_{h,ν}≠ω_m with |μ_d(λ_{h,ν})-1|<ε. Then apply GMRES to A_h V_h=b_h and check whether the residual in the original discretized Helmholtz equations reaches the tolerance used in Section 7. If the solver stalls or returns a solution with large Helmholtz residual, the spurious-resonance failure mode is realized on a simple geometry, and the central claim must be explicitly qualified. If it converges, repeat with the domain side length or the frequency pair tuned so a discrete eigenvalue coincides with the μ=1 crossing, which settles whether the singularity is reachable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that MFWH computes N_f Helmholtz solutions from one wave equation requires the linear system A_h V_h = b_h to be nonsingular, i.e., no discrete eigenvalue λ_{h,ν} distinct from all ω_m should have μ_d(λ_{h,ν}) = 1. Section 6 explicitly acknowledges that such a spurious resonance makes A_h singular (or nearly singular) and that the Krylov solver will likely fail. No sufficient condition, parameter choice, or automatic detection is provided; the suggested remedies are a posteriori residual checks and manually changing N_p. This means the advertised algorithm is not guaranteed to work for valid inputs, and the paper's abstract and conclusions do not state this qualification. The failure is not hypothetical: for the N_f=2, ω=(5,9), N_p=1 case of Figure 1, the unit-square Dirichlet Laplacian has an eigenvalue λ≈π√5≈7.025, which lies very close to the spurious μ=1 crossing visible in that figure. Thus a simple standard geometry may already trigger the singular case. A secondary concern is that the discrete time-error correction is delegated to the unpublished manuscript [11], but the spurious-resonance issue is the more direct threat to the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces the Multi-Frequency WaveHoltz (MFWH) algorithm for solving N_f Helmholtz boundary value problems with distinct frequencies ω_m and forcings f^(m) by solving a single forced wave equation whose composite forcing is a sum of f^(m) cos(ω_m t). The iteration applies N_f time filters to the wave-equation solution, updates the component iterates through a small N_f × N_f linear system, and can be accelerated by GMRES. The paper derives a rank-one error-propagation matrix whose nonzero eigenvalue μ(λ) is a weighted sum of filter functions, proves μ(ω_m)=1, and defines a discrete version μ_d with modified frequencies to remove time-discretization error. Numerical experiments on the unit square with Dirichlet conditions confirm the predicted fixed-point rates and show GMRES convergence, and scaling tests with implicit time stepping and multigrid indicate O(N) complexity.","tokens_in":15882,"tokens_out":6504,"duration_ms":69267,"significance":"The core idea is attractive and the continuous-level convergence analysis is clean, self-contained, and consistent with the reported residuals. If the discrete time-error correction can be made fully self-contained, the method offers a genuinely different route to multi-frequency Helmholtz solves: one time-domain wave-equation solve with multiple filters, a definite implicit time-stepping matrix amenable to multigrid, and demonstrated O(N) scaling for fixed frequencies. The paper also provides transparent numerical validation and reports timings up to 4.2×10^6 unknowns. However, two caveats limit the strength of the central claim: the acknowledged spurious-resonance failure mode in Section 6, and the deferral of the discrete time-error analysis to the unpublished companion [11]. Neither appears fatal, but both need to be addressed before the advertised performance can be considered robust.","major_comments":[{"comment":"The statement that the Krylov-accelerated MFWH solves A_h V_h = b_h is load-bearing, but the paper explicitly acknowledges that a discrete eigenvalue λ_{h,ν} distinct from all ω_m with μ_d(λ_{h,ν})=1 (or nearly 1) makes A_h singular (or nearly singular) and will likely cause the Krylov solver to fail. No sufficient condition, parameter choice, or automatic detection is provided; the only proposed remedies are a posteriori residual checks and manually changing N_p. This is not a hypothetical edge case: the N_f=2, ω=(5,9), N_p=1 case plotted in the left panel of Figure 1 shows spurious crossings near λ≈7, and the unit-square Dirichlet Laplacian has an eigenvalue λ=π√5≈7.025, so a standard geometry can fall at or near such a crossing. Since the abstract and conclusions present MFWH as a general simultaneous solver without this caveat, the central claim needs to be either restricted or supported by a condition that rules out spurious resonances, or by a robust automatic remedy.","section":"Section 6, Eq. (48)"},{"comment":"The proof that the discrete iterates converge to the discrete Helmholtz solutions with pristine frequencies ω_m (Eq. (41)) is not self-contained. The modified-frequency definition (39), the discrete filter (42), the choice of α_m in (45), and the time-step recipe are stated, but the derivation that these choices exactly remove the time-discretization error is deferred to the unpublished manuscript [11]. The sentence \"following the recipe in [11]\" in Section 5.3 delegates a central step on which the numerical claims and the O(N) scaling argument rest. The authors should supply a self-contained derivation or a rigorous error estimate for the discrete filter, or at minimum make the argument reproducible within this paper.","section":"Section 5, Eqs. (39)-(45); Section 5.3"}],"minor_comments":[{"comment":"The phrase \"we restrict out attention\" should be \"we restrict our attention.\"","section":"Section 1"},{"comment":"The notation V^{m,k}_j should be V^{(m,k)}_j to match Table 1, and the residual defined in Eq. (52) is an iteration residual (difference of successive iterates) rather than a Helmholtz residual; this should be stated explicitly to avoid confusion.","section":"Section 7, Eq. (52)"},{"comment":"The ACR reported in Figures 3-6 is computed from the same discrete filter function μ_d that defines the iteration, so the match between predicted and observed convergence is partly a consistency check. An independent validation would be a quantitative comparison of the converged MFWH solutions against direct Helmholtz solves for several discretizations; the paper mentions a relative error of about 1e-12 but does not provide a table.","section":"Section 7, Eq. (49)"},{"comment":"The formula for μ in Eq. (37) is given only for N_f=2; a general expression for the weights w_m in Eq. (28) would help readers implement the ACR prediction for larger N_f.","section":"Section 4.2, Eq. (37)"},{"comment":"References [8] and [11] are marked as \"submitted\" or \"unpublished\"; the manuscript should indicate how these can be accessed or should cite published versions if available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main uncertainty for the editor is the dependence on [11]: the single-frequency O(N) result and the discrete time-error correction are claimed there. If the present paper is to stand alone, the authors should add the missing discrete analysis. The spurious-resonance issue is real and should be addressed by an honest qualification and, preferably, a robust detection or avoidance strategy; it does not undermine the method's value when the algorithm does operate away from such resonances."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real contribution. The authors extend WaveHoltz to multiple frequencies by coupling N_f time filters, and the analysis is unusually clean: the error iteration has a rank-one structure, so convergence is governed by one scalar function mu(lambda). Theorems 1-3 are self-contained, and the numerical residuals match the predicted ACR in both explicit and implicit settings. The O(N) scaling demonstration with implicit time-stepping and multigrid is solid evidence for the cost claim.\n\nWhat is new: the coupled multi-frequency filter, the matrix A of cross-couplings, the rank-one B(lambda), and the observation that mu(omega_m)=1 at each target frequency. This is not a trivial superposition; the coupling is necessary and the analysis is nontrivial.\n\nSoft spots, in proportion. The spurious resonance discussion in Section 6 is honest but incomplete, and it is load-bearing. The paper concedes that a discrete eigenvalue away from all omega_m can make mu_d equal to 1, making A_h singular and breaking the Krylov solver. The remedies offered are a posteriori residual checks and changing N_p. There is no sufficient condition, no automatic detection, and no proof that changing N_p removes the problem. This is not a pathological corner: for N_f=2, omega=(5,9), N_p=1, the unit-square Dirichlet Laplacian has an eigenvalue pi*sqrt(5) ≈ 7.025, which sits in the spurious-resonance region of Figure 1. So the advertised 'simultaneously computing multiple solutions' needs a caveat, and the abstract and conclusions currently overstate robustness. Promoting Section 6 and adding a systematic or at least heuristic treatment would materially improve the paper.\n\nTwo smaller issues. The discrete time-error correction relies on the unpublished [11]; the modified frequencies and quadrature weights are central, so that dependency is a real gap. And the ACR comparison is in part a consistency check since the theory and the numerics share the same mu_d; the authors do not fit parameters, which is good, but independent geometries would strengthen the confirmation.\n\nWho this is for: researchers building multi-frequency Helmholtz solvers, especially from time-domain codes. It deserves serious peer review and likely publication after a revision that addresses the spurious-resonance gap. I would accept it for review.","headline":"A genuine multi-frequency extension of WaveHoltz with clean analysis and good scaling, but the spurious-resonance failure mode is a real caveat that should be flagged more prominently.","tokens_in":16428,"tokens_out":3686,"would_cite":true,"duration_ms":37287,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65N22","65N55","65F10","35J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Multi-Frequency WaveHoltz algorithm computes several Helmholtz solutions from one wave equation solve, with O(N) total cost when frequencies are fixed.","keywords":["Helmholtz equation","WaveHoltz","multi-frequency","time-filtering","fixed-point iteration","GMRES","implicit time-stepping","multigrid"],"falsifier":"Compute the discrete spectrum of the operator on a specific grid (for example, the Dirichlet Laplacian on a unit square), plot mu_d(lambda) from Eq. (46), and check whether any eigenvalue lambda_{h,nu} distinct from all omega_m has |mu_d| >= 1; the paper's Figure 5 already exhibits a case where the fixed point diverges because the asymptotic convergence rate exceeds one, and Figure 1 shows spurious resonances near lambda = 7 and 11 for omega_1 = 5, omega_2 = 9 with N_p = 1. A direct test is to run GMRES on (I - S_h)V_h = b_h for a configuration with a spurious resonance and observe the residual in the original discretized Helmholtz equations failing to reach machine precision, confirming that A_h is singular or nearly singular.","tokens_in":15445,"feed_emoji":"🌊","tokens_out":5503,"duration_ms":52096,"temperature":0.7,"pith_summary":"The paper presents Multi-Frequency WaveHoltz (MFWH), an algorithm that solves several Helmholtz boundary-value problems at different frequencies and forcings simultaneously by solving a single forced wave equation and applying multiple time filters. It proves that the fixed-point iteration converges exactly when a scalar filter function mu(lambda) satisfies |mu(lambda_nu)| < 1 for every discrete eigenvalue, and that Krylov acceleration handles cases where the plain iteration diverges. With implicit time stepping and an O(N) multigrid solver for the implicit equations, the scheme achieves O(N) total cost for a fixed set of frequencies. Time-discretization errors are removed by using modified frequencies and adjusted quadrature, so the converged iterates match the discrete Helmholtz solutions with the original frequencies. Numerical results in 2D confirm the convergence theory and the O(N) scaling.","feed_headline":"One wave-equation solve yields many Helmholtz solutions","feed_subtitle":"Simultaneous time filters pull out every frequency response from a single wave solve, with near-linear scaling.","key_machinery":"The machinery is the multi-frequency filter function mu($\\lambda$) = sum_{m=1}^{N_f} w_m beta_m($\\lambda$), where w_m are the column sums of $A^{{-1}}$ and beta_m($\\lambda$) is the single-frequency WaveHoltz filter, a combination of three sinc functions centered at $\\lambda$ = omega_m, $\\lambda$ = -omega_m, and $\\lambda$ = 0. The matrix A has entries beta_i(omega_j) and the rank-one matrix B($\\lambda$) has entries beta_i($\\lambda$); because B is rank one, the iteration matrix M_nu = $A^{{-1}}$B has exactly one nonzero eigenvalue, namely mu($\\lambda$). This reduction turns the multi-frequency convergence question into a one-dimensional check: plot mu against the discrete spectrum and see whether any eigenvalue lands where |mu| >= 1. The discrete version uses modified frequencies (Eq. 39) and quadrature weights (Eq. 45) so the fixed point of the filtered iteration is the discrete Helmholtz solution with the original frequencies.","core_discovery":"The central claim is that multiple Helmholtz solutions can be obtained from one wave-equation solve by forming a composite forcing sum f^(m)(x) cos(omega_m t) and filtering the wave solution with N_f time integrals, one per frequency. The error in each eigenmode contracts by the scalar mu($\\lambda$) = sum w_m beta_m($\\lambda$), the unique nonzero eigenvalue of $A^{{-1}}$B($\\lambda$); the fixed-point iteration converges iff |mu(lambda_nu)| < 1 for all discrete eigenvalues lambda_nu. When the plain iteration stalls, solving the linear system (I - S_h)V_h = b_h with GMRES converges rapidly, and with implicit time stepping at about ten steps per period plus an O(N) multigrid solve per step, the total work scales as O(N) for fixed frequencies.","pith_inferences":["The rank-one structure of B(lambda) means the entire multi-frequency convergence analysis reduces to a one-dimensional filter function; the same reduction is likely available for filters built from other quadrature rules or time-stepping schemes.","The spurious-resonance failure mode suggests a practical safeguard: before committing to a set of frequencies and N_p, scan mu_d(lambda) over the discrete spectrum at negligible cost and adjust N_p or slightly shift frequencies to keep |mu_d| < 1 away from the target frequencies.","The method's current restriction to closed domains with Dirichlet, Neumann, or Robin conditions limits its use in scattering and inverse problems; extending the composite-forcing idea to absorbing boundary conditions or perfectly matched layers would make simultaneous multi-frequency solves viable for those settings.","Because the algorithm only needs a time-domain wave solver, existing mature explicit and implicit time-stepping codes could be wrapped to produce frequency-domain solutions without building a new Helmholtz-specific solver."],"forward_implications":["A single wave-equation solve over one common time interval produces N_f Helmholtz solutions at distinct frequencies, so the per-frequency cost is amortized.","Because the implicit time-stepping matrix is definite, the indefinite Helmholtz system is never formed; multigrid applied to the wave-equation steps keeps total cost O(N) for fixed frequencies.","The fixed-point iteration's convergence rate is exactly predicted by the scalar function mu(lambda), so the number of periods N_p can be tuned by inspecting mu_d against the discrete spectrum.","Krylov acceleration converges even in cases where the plain fixed point diverges (for example, N_p = 1), with similar effective cost per grid point.","Time-discretization errors are eliminated by solving with modified frequencies and filtered quadrature, so the converged iterates match the discrete Helmholtz solutions with the original frequencies."],"supporting_citations":[{"why":"Introduces the WaveHoltz algorithm whose single-frequency time filtering is the building block extended here to multiple frequencies.","marker":"[7]"},{"why":"Analyzes convergence of the semi-discrete WaveHoltz iteration, the single-frequency counterpart of the error analysis used for MFWH.","marker":"[10]"},{"why":"Establishes the O(N) scaling and the modified-frequency and quadrature recipe for removing time-discretization errors that MFWH inherits.","marker":"[11]"},{"why":"Supplies the multigrid solver used for the implicit time-stepping equations in the O(N) scaling experiments.","marker":"[15]"},{"why":"Describes the multigrid method for overlapping grids that the MFWH implementation can rely on for the implicit solves.","marker":"[16]"}],"fun_headline_variants":["One wave solve, many Helmholtz frequencies","Multi-frequency Helmholtz from a single wave equation","WaveHoltz extension solves many frequencies at once","Time filters extract every frequency from one wave solve"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that no discrete eigenvalue of the spatial discretization, other than the target frequencies themselves, makes the multi-frequency filter function equal (or nearly equal) to one; if such a spurious resonance exists, the linear system becomes singular and the Krylov solver will likely fail, with only a posteriori residual checks and changing the number of periods as remedies.","fun_headline_variants_meta":{"raw":{"variants":["One wave solve, many Helmholtz frequencies","Multi-frequency Helmholtz from a single wave equation","WaveHoltz extension solves many frequencies at once","Time filters extract every frequency from one wave solve"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001083,"raw_usage":{"total_tokens":4523,"prompt_tokens":932,"completion_tokens":3591,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":3533}},"tokens_in":548,"tokens_out":3591,"duration_ms":25544,"temperature":1.0,"reasoning_tokens":3533,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:31:58.511090+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the discrete spectrum of the operator on a specific grid (for example, the Dirichlet Laplacian on a unit square), plot mu_d(lambda) from Eq. (46), and check whether any eigenvalue lambda_{h,nu} distinct from all omega_m has |mu_d| >= 1; the paper's Figure 5 already exhibits a case where the fixed point diverges because the asymptotic convergence rate exceeds one, and Figure 1 shows spurious resonances near lambda = 7 and 11 for omega_1 = 5, omega_2 = 9 with N_p = 1. A direct test is to run GMRES on (I - S_h)V_h = b_h for a configuration with a spurious resonance and observe the residual in the original discretized Helmholtz equations failing to reach machine precision, confirming that A_h is singular or nearly singular.","supporting_citations":[{"cited_title":"Appel¨ o, F","cited_arxiv_id":null,"evidence_quote":"Introduces the WaveHoltz algorithm whose single-frequency time filtering is the building block extended here to multiple frequencies."},{"cited_title":"Appel¨ o, J","cited_arxiv_id":null,"evidence_quote":"Establishes the O(N) scaling and the modified-frequency and quadrature recipe for removing time-discretization errors that MFWH inherits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the multigrid solver used for the implicit time-stepping equations in the O(N) scaling experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the multigrid method for overlapping grids that the MFWH implementation can rely on for the implicit solves."}],"review_version":1}