{"id":"0bbda374-b896-4647-aef5-b4569de118d4","arxiv_id":"2506.20777","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A time-reduction and quasi-reversibility scheme is shown to reconstruct a selected minimum-norm initial electric field from noisy boundary data, with a convergence theorem and 3D tests.","lead":"This paper develops a numerical method for recovering the initial electric field in Maxwell's equations from boundary measurements of the field and its derivative over a finite time interval. The method projects time dependence onto a spectral basis and uses regularized least squares, with numerical tests in 3D under 10% noise.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1's key estimate (5.4) omits the n=0 Fourier mode; since Ψ₀″ = Ψ₀, the missing term contributes ∥ε e*₀∥², so the claimed δ² bound is false and the convergence proof collapses.","rationale":"The reader's REJECT verdict is supported, but the most load-bearing defect is more direct than the formal weakest-assumption statement. Theorem 5.1's proof hinges on (5.4), and that estimate is false as written because it drops the n=0 term. Lemma 4.1, the only cited justification, applies to the full sum starting at n=0; switching to n=1 is not harmless because Ψ₀″ = Ψ₀ and s_m0 = δ_{m0}. This is an internal inconsistency in the central convergence claim, not merely a gap between the theorem's target and the physical initial field. The physical-relevance concern remains valid: even if the indexing error were repaired, convergence would be to the minimum-norm solution E* of (3.1), not necessarily to the true E₀, and the piecewise-constant numerical true fields are not in H³. I therefore agree with rejecting the paper in its current form, while noting that the proof may be repairable by reinserting the n=0 term and replacing (5.4) with the actual Lemma 4.1 convergence plus a vanishing remainder.","tokens_in":20089,"tokens_out":9370,"duration_ms":103126,"concrete_test":"Set E*(x,t) = φ(x)Ψ₀(t) with Ψ₀(t) = e^t/√T and any nonzero smooth φ satisfying ∇×(μ^{-1}∇×φ) + εφ = 0, with boundary data F = φΨ₀ and G = (∂_νφ)Ψ₀. Compute the left side of (5.4) for this E*: because e*_0 = φ and e*_n = 0 for n ≥ 1, the expression reduces to ∥εφ∥²_{L²(Ω)^3} for every N. If this is nonzero, (5.4) is false as stated. Then re-derive (5.10)–(5.13) with Σ_{n=0}^N instead of Σ_{n=1}^N; the repaired estimate follows from Lemma 4.1, confirming that the n=0 omission is the specific defect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is estimate (5.4) in Theorem 5.1, which is used to bound J(F_N[E*]) in (5.10)–(5.13). As stated, (5.4) uses the truncation Σ_{n=1}^N s_mn e*_n, but Lemma 4.1 only justifies the corresponding approximation with Σ_{n=0}^N. The missing n=0 term is not negligible. Indeed, Ψ₀(t) = e^t / √T, so Ψ₀″ = Ψ₀, and by the weighted orthonormality of {Ψ_n}, s_m0 = ⟨Ψ₀″, Ψ_m⟩_{L²_{e^{-2t}}} = δ_{m0}. Hence, as N → ∞, the left side of (5.4) tends to ∥ε e*_0∥²_{L²(Ω)^3}, which need not vanish and generally does not. For example, take E*(x,t) = φ(x)Ψ₀(t) with any nonzero smooth φ satisfying ∇×(μ^{-1}∇×φ) + εφ = 0 in Ω and compatible boundary data; then e*_0 = φ, e*_n = 0 for n ≥ 1, and the left side of (5.4) equals ∥εφ∥²_{L²(Ω)^3} for every N, contradicting the claimed δ² bound as δ → 0. Since the bound (5.13) and all subsequent compactness and minimum-norm steps rely on (5.4), Theorem 5.1 does not establish convergence even to E*. This is an internal inconsistency, independent of the separate question of whether E* equals the physical E₀.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the recovery of the initial electric field E0 in a bounded domain from boundary measurements of the electric field and its normal derivative over (0,T), for the time-domain Maxwell system in an inhomogeneous anisotropic medium. The authors formulate an underdetermined lateral boundary value problem (3.1), select a minimum-H^3-norm solution E*, project the field in time onto a Legendre polynomial-exponential basis, reduce the problem to a sequence of coupled spatial systems (4.9), and solve these systems by minimizing a Tikhonov-type functional (5.3). Theorem 5.1 claims convergence of the reconstructed space-time field S_N[V_min] to E* as N tends to infinity, the noise level delta tends to 0, and the regularization parameter epsilon tends to 0 with delta^2 = o(epsilon). Numerical experiments with 10% multiplicative noise on three-dimensional discontinuous test fields show reconstructions of the initial field.","tokens_in":20449,"tokens_out":11047,"duration_ms":124666,"significance":"If Theorem 5.1 were valid as stated, the contribution would be valuable: the time-dimensional reduction converts a (3+1)-dimensional inverse Maxwell problem into a sequence of 3D elliptic systems, with an explicit quasi-reversibility regularization and a convergence theorem. The manuscript also contains a detailed Algorithm 1, three fully three-dimensional numerical tests, and reconstructions that remain accurate at 10% noise. However, as written the central convergence proof contains a concrete missing-mode error and a regularity gap, and the theorem's target is the minimum-norm solution of (3.1), not the physical initial field. The numerical tests use discontinuous true fields that do not satisfy the hypotheses of the theorem. These issues must be resolved before the convergence claim can be accepted. With repairs and a corrected framing, the method itself remains plausible and potentially useful.","major_comments":[{"comment":"Equation (5.4) uses the truncation sum from n=1 to N, while Lemma 4.1, which is invoked to prove it, provides the approximation with the sum from n=0 to N. The omitted n=0 mode is not negligible. Since Psi_0(t)=e^t/sqrt(T), one has Psi_0'' = Psi_0 and s_m0 = delta_m0, so the missing term is epsilon(x) e*_0(x) in the m=0 component. Consequently, as N tends to infinity, the left-hand side of (5.4) tends to ||epsilon e*_0||^2_{L^2(Omega)^3}, which need not vanish. A concrete counterexample is E*(x,t)=phi(x)Psi_0(t) with nonzero smooth phi satisfying the elliptic equation in (3.1); then e*_0=phi and e*_n=0 for n>=1, so the left-hand side of (5.4) equals ||epsilon phi||^2_{L^2(Omega)^3} for every N. The same omission appears in (5.10) when evaluating J_{N,delta,epsilon}(F_N[E*]). Because (5.13) and all subsequent compactness and minimum-norm steps rely on the bound (5.4), Theorem 5.1 is not proved as stated. Replacing n=1 by n=0 in (5.4) and (5.10) appears to repair this specific point, provided the regularity issue in the next comment is also resolved.","section":"§5, Theorem 5.1, Eq. (5.4)"},{"comment":"Lemma 4.1 is stated for u in H^2((0,T);L^2(Omega)), but its proof invokes Proposition 2.1, which requires u in H^k((0,T);L^2(Omega)) for some k>=5. The convergence of the differentiated projection sum_{n=0}^N <u,Psi_n>Psi_n'' to u_tt requires additional temporal regularity and compatibility conditions; for a general H^2 function it is not established by the cited result. The field E* in Theorem 5.1 belongs to S with only H^2((0,T);L^2(Omega)^3), so the hypothesis of Lemma 4.1 as used in the proof of (5.4) is not satisfied. This is a second load-bearing gap in the convergence proof. The authors should either strengthen the assumptions on E* (for example, H^5 temporal regularity) or provide a direct proof of Lemma 4.1 under the stated H^2 assumption.","section":"§4, Lemma 4.1 and §5, Theorem 5.1"},{"comment":"The convergence statements (5.6)-(5.8) are only to E*, the unique minimizer of the H^3 norm over the admissible set S of solutions to (3.1). Nothing in Theorem 3.1 or Section 5 shows that the physical initial electric field E0 satisfies this minimum-norm selection; indeed any solution of (3.1) is admissible, and the true field need not be the norm minimizer. Thus the abstract and Section 1 claims that the method 'recovers the initial electric field' or 'converges to the true solution' are stronger than what is proved. Moreover, in all three numerical tests the true initial fields are discontinuous indicator functions, which are not in H^3(Omega)^3, and the data are generated with partial_t E(x,0)=0; these settings lie outside the hypotheses of Theorem 5.1. The numerical validation should be presented as heuristic, and the theorem should be stated as convergence to the selected minimum-norm solution, or an argument relating E* to the physical E0 should be supplied.","section":"§3, Theorem 3.1 and §5, Theorem 5.1"}],"minor_comments":[{"comment":"The boundary data G* are generated as (curl E*) x nu, whereas the inverse problem in (1.2) uses G = partial_nu E; this discrepancy should be clarified or the two settings should be aligned.","section":"§6, data generation"},{"comment":"Theorem 3.1 states that E* belongs to L^2((0,T);H^3(Omega)^3), but the minimization is performed in the weighted norm L^2_{e^{-2t}}((0,T);H^3(Omega)^3); the norms are equivalent on the finite interval, but the notation should be made consistent.","section":"§3, Theorem 3.1"},{"comment":"The relative errors reported in the numerical section appear to be computed from maximum values in target regions rather than from a global L^2 error; the precise quantity being reported should be defined.","section":"§6, error reporting"},{"comment":"Propositions 2.1, 2.2, and Lemma 2.1 are imported from the authors' preprint [37]; since that reference is not yet published, the statements and proofs should be included or a published source should be cited to make the paper self-contained.","section":"§2 and §4"}],"recommendation":"major_revision","confidential_remarks":"The main theorem currently contains a false estimate caused by the omitted n=0 mode in (5.4), and Lemma 4.1 has an unproved regularity step. Both issues appear repairable within the manuscript's scope by changing the summation index and adding temporal regularity assumptions. However, the mismatch between the claimed 'true solution' and the proved convergence to the minimum-norm solution is a more substantial framing problem that requires revision of the abstract, introduction, and conclusions. I recommend major revision rather than rejection because the proposed method, after repair and reframing, could be a useful contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"J., here's my read on 2506.20777.\n\nThe genuinely new thing is the adaptation of the time-dimensional reduction method, previously used for elastic systems in the authors' companion preprint [37], to Maxwell's equations, with a quasi-reversibility/minimum-norm treatment. That is a reasonable extension, and the numerics are the strongest part: in 3D, with 10% noise, the method recovers piecewise-constant initial fields with errors in the low teens, which is a meaningful computational achievement. The writing is clear and the reduction from a (3+1)-D problem to a sequence of 3D solves is attractive.\n\nThe problems are load-bearing. First, the convergence proof of Theorem 5.1 has a concrete gap. Estimate (5.4) sums n=1 to N, but Lemma 4.1 justifies n=0 to N. The missing term is not negligible: since Ψ₀″=Ψ₀, s_m0=δ_m0, and for m=0 the omitted contribution is εe*₀. The left side of (5.4) therefore tends to ‖εe*₀‖², which is generally nonzero, so (5.4) is false. The counterexample with E*=φ(x)Ψ₀(t) makes this precise. Because (5.13) and the compactness argument rely on (5.4), the theorem as stated is unproven. This looks repairable by including the n=0 mode and absorbing the extra ε² term, but it is a real flaw.\n\nSecond, the theorem proves convergence to E*, the minimum-norm solution of the underdetermined system (3.1), not to the physical initial field E₀. The paper is upfront about the non-uniqueness, but the abstract and title overstate what is recovered. The numerical tests also outrun the theory: the true fields are characteristic functions, not in H³, so the theorem's assumptions don't cover them, and the boundary data in the experiments are (∇×E)×ν, not the ∂νE used in the analysis.\n\nThird, the foundational approximation lemmas come from the authors' own unpublished preprint [37]. That is a heavy dependency for a journal paper.\n\nWho should read it: people working on computational inverse problems for Maxwell, especially time-reduction and quasi-reversibility methods. It deserves a serious referee, because the idea is sound and the numerics are promising, but it is not acceptable in its current form. I would send it to review with a clear request for major revision: fix the n=0 gap, align the numerical and theoretical data, and address the gap between E* and E₀.","headline":"The convergence theorem has a genuine n=0 gap, and even if repaired, it proves convergence to a minimum-norm solution rather than to the physical initial field; the 3D numerics look good but outrun the theory.","tokens_in":1033,"tokens_out":757,"would_cite":false,"duration_ms":94684,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35L50","35Q61","65M32","78A25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that projection onto a Legendre polynomial-exponential basis converts Maxwell initial-field recovery into stable spatial solves converging to a minimum-norm solution.","keywords":["inverse problem","Maxwell equations","initial electric field recovery","time-dimensional reduction","Legendre polynomial-exponential basis","minimum-norm solution","quasi-reversibility method","boundary measurements"],"falsifier":"Take a smooth initial field that satisfies given lateral boundary data but is not the $H^3$-norm minimizer among all compatible fields, generate exact boundary data from it, and run the algorithm with vanishing noise and regularization. If the reconstruction converges to that true field instead of to the independently computed minimum-norm field, the theorem's identification of the convergence target is wrong; if it converges to the minimizer, the method is provably recovering only the selected solution, not the physical field.","tokens_in":1773,"feed_emoji":"⚡","tokens_out":5951,"duration_ms":127592,"temperature":0.7,"pith_summary":"The paper claims that the initial electric field of a time-dependent Maxwell system can be recovered from boundary measurements of the field and its normal derivative over a finite time interval, even when the initial magnetic field and charge density are unknown. The central move is to expand the electric field in a Legendre polynomial-exponential basis in time, turning the space-time inverse problem into a sequence of three-dimensional elliptic systems for time-Fourier coefficients. A quasi-reversibility method with $H^3$-regularization and a minimum-norm selection rule solves these systems. The authors prove that as the truncation order grows and the noise and regularization parameters vanish, with $\\delta^2 = o(\\epsilon)$, the regularized reconstruction converges to the unique minimum-norm solution of the lateral boundary value problem. Numerical experiments in 3D with 10% noise show that the selected solution matches prescribed test fields within roughly 8-16% relative error in peak values.","feed_headline":"Time projection makes Maxwell initial-field recovery stable","feed_subtitle":"Without initial magnetic field or charge data, the method provably converges to the minimum-norm solution.","key_machinery":"The central object is the Legendre polynomial-exponential basis $\\Psi_n(t) = e^t Q_n(t)$, where $\\{Q_n\\}$ is the rescaled orthonormal Legendre basis of $L^2(0,T)$; the basis is orthonormal in the weighted space $L^2_{e^{-2t}}(0,T)$ and, crucially, no $\\Psi_n''$ vanishes identically, so the projection does not annihilate the second time derivative. The machinery applies the time-projection operator $P_N$ to the Maxwell equation, producing the coupled spatial system (4.9) for the Fourier coefficient modes $e_m(x)$, with coupling matrix $s_{mn} = \\int_0^T e^{-2t}\\Psi_n''(t)\\Psi_m(t)\\,dt$. The numerical solver minimizes the Tikhonov-type functional $J_{N,\\delta,\\epsilon}$ that penalizes the equation residual, mismatch of both Cauchy boundary data, and $\\epsilon\\|\\cdot\\|^2_{H^3(\\Omega)^3}$; the minimizer is expanded back in time to give the reconstructed field and its value at $t=0$.","core_discovery":"The paper's central discovery is a stable reconstruction route for an underdetermined electromagnetic inverse problem: by projecting the space-time Maxwell equation onto a Legendre polynomial-exponential basis, the time variable is removed, and the resulting spatial systems can be solved by minimizing a regularized least-squares functional. The rigorously stated payoff is Theorem 5.1: when the noisy boundary data satisfy the stated noise bound and the regularization parameter obeys $\\delta^2 = o(\\epsilon)$, the reconstruction $S_N[V^{\\min}_{N,\\delta,\\epsilon}]$ converges as $(N,\\delta,\\epsilon) \\to (\\infty,0^+,0^+)$ in $L^2((0,T);H^2(\\Omega)^3)$ to $E^*$, the unique minimum-$H^3$-norm solution of the lateral boundary value problem (3.1). The paper frames this convergence as showing that time-dimensional reduction, quasi-reversibility, and minimum-norm selection together turn the underdetermined inverse initial-value problem into a well-posed reconstruction. The numerical experiments demonstrate that the method preserves support and amplitude of discontinuous test fields under 10% multiplicative noise.","pith_inferences":["Editorial inference: because the convergence target is the minimum-norm solution of the boundary value problem, the algorithm's output is best interpreted as the smoothest field consistent with the measurements; the physical initial field is recovered only if it happens to be that selected field.","Editorial inference: the numerical tests use discontinuous true fields that do not satisfy the $H^3$ regularity assumed in Theorem 5.1, so the observed accuracy is evidence of numerical behavior beyond the theorem's stated hypotheses.","Editorial inference: the same time-projection plus quasi-reversibility construction may transfer to other hyperbolic systems with unknown initial velocity, such as elastic or acoustic wave equations, whenever lateral Cauchy data are available.","Editorial inference: a quantitative version of Theorem 5.1, giving explicit rates in $N$, $\\delta$, and $\\epsilon$, would let practitioners set the truncation order and regularization parameter directly from the measured noise level; the paper leaves that open."],"forward_implications":["If Theorem 5.1 holds, the algorithm provably recovers the $H^3$-minimum-norm electric field compatible with the lateral boundary data, without needing the initial magnetic field, the initial time derivative, or the charge density.","The time-dimensional reduction replaces the $(3+1)$-dimensional inverse problem by a sequence of spatial elliptic solves, so the computational cost scales with the number $N$ of temporal modes rather than with a full space-time inversion.","The convergence result gives a parameter recipe: choose $N$ large enough to make the time-projection residual small, and choose $\\epsilon$ so that $\\delta^2/\\epsilon \\to 0$.","The convergence of $S_N[V^{\\min}_{N,\\delta,\\epsilon}](\\cdot,0)$ to $E^*(\\cdot,0)$ in $L^2(\\Omega)^3$ is the precise sense in which the method solves the initial-data recovery problem.","The method also controls the first time derivative in (5.7), so it gives a stable approximation to the full space-time field, not only its initial value."],"supporting_citations":[{"why":"Supplies the Legendre polynomial-exponential basis, its orthonormality and nonvanishing second-derivative properties, and Lemma 2.1 used in the convergence proof.","marker":"[37]"},{"why":"Provides the quasi-reversibility method for hyperbolic inverse source problems that the paper adapts to the time-reduced Maxwell systems.","marker":"[26]"},{"why":"Supplies the quasi-reversibility method and Carleman-based uniqueness context for hyperbolic initial-value problems that motivates the minimum-norm setting.","marker":"[10]"}],"fun_headline_variants":["Time projection stabilizes Maxwell field recovery without B-data","Underdetermined Maxwell inversion solved via time projection","Maxwell initial field reconstruction robust to 10% noise","Minimum-norm Maxwell recovery converges from boundary data","Time reduction yields stable Maxwell inversion with missing data"],"cache_read_input_tokens":23040,"weakest_assumption_plain":"The load-bearing premise is that the true physical initial field coincides with the minimum-norm solution selected by the boundary value problem; the paper proves convergence to that selected field, and nothing in the argument links the minimizer to the physical $E_0$.","fun_headline_variants_meta":{"raw":{"variants":["Time projection stabilizes Maxwell field recovery without B-data","Underdetermined Maxwell inversion solved via time projection","Maxwell initial field reconstruction robust to 10% noise","Minimum-norm Maxwell recovery converges from boundary data","Time reduction yields stable Maxwell inversion with missing data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1549,"prompt_tokens":997,"completion_tokens":552,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":478}},"tokens_in":613,"tokens_out":552,"duration_ms":6563,"temperature":1.0,"reasoning_tokens":478,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:43:46.416891+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a smooth initial field that satisfies given lateral boundary data but is not the $H^3$-norm minimizer among all compatible fields, generate exact boundary data from it, and run the algorithm with vanishing noise and regularization. If the reconstruction converges to that true field instead of to the independently computed minimum-norm field, the theorem's identification of the convergence target is wrong; if it converges to the minimizer, the method is provably recovering only the selected solution, not the physical field.","supporting_citations":[{"cited_title":"Recovery of initial displacement and velocity in anisotropic elastic systems by the time dimensional reduction method","cited_arxiv_id":"2506.13000","evidence_quote":"Supplies the Legendre polynomial-exponential basis, its orthonormality and nonvanishing second-derivative properties, and Lemma 2.1 used in the convergence proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the quasi-reversibility method for hyperbolic inverse source problems that the paper adapts to the time-reduced Maxwell systems."},{"cited_title":"Clason and M","cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-reversibility method and Carleman-based uniqueness context for hyperbolic initial-value problems that motivates the minimum-norm setting."}],"review_version":1}