{"id":"fc59996d-1190-4a73-8b17-8daa1a0be5cf","arxiv_id":"2506.13000","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A Legendre polynomial-exponential basis reduces the inverse elastic wave initial-data problem to elliptic systems solved by quasi-reversibility, with a convergence theorem and 2D reconstructions for isotropic, inhomogeneous, and anisotropic media.","lead":"This paper develops a time-dimensional reduction method that reconstructs the initial displacement and velocity of elastic waves from boundary measurements, using a Legendre-exponential basis and a quasi-reversibility regularization scheme. The method is aimed at seismic, nondestructive testing, and elastography applications where internal initial conditions must be inferred from surface sensors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence theorem needs unverified high time-regularity (series Σ⟨u*,Ψ_n⟩Ψ''_n) not met by piecewise-constant tests; experiments also use δ²≫η, outside theorem's δ²=o(η) regime.","rationale":"The reader's weakest-assumption identifies the core gap: Theorem 4.1's convergence is conditional on a spectral convergence assumption on the second-derivative expansion that is not implied by the minimal-norm solution's natural regularity and is almost certainly violated by the piecewise-constant inclusions used in the numerics. I agree. The proof also contains an invalid inequality in (4.10) — the residual's norm is bounded by √J, not J — but this appears repairable and is not the primary threat. The experimental parameter choice (η=10^{-6}, δ=0.1) gives δ²=10^{-2}, which fails δ²=o(η), and N=30 is fixed rather than N≥N(δ). Thus the numerics operate outside the theorem's hypotheses and cannot be cited as confirmation. A conditional-accept verdict remains appropriate: the method is interesting, but the paper must either verify the regularity assumption for its examples, restrict the claims to smooth cases, or revise the theorem. No change to the reader's verdict is needed.","tokens_in":23025,"tokens_out":7285,"duration_ms":78522,"concrete_test":"Compute the decay of ∥⟨u*, Ψ_n⟩∥_{H^p(Ω)} for the exact solutions of Tests 1–3; if ∥u_n∥_{H^p} does not decay at least as fast as n^{-5}, then ∑ n^{7/2}∥u_n∥ diverges, so the series (2.17) fails to converge and Theorem 4.1's key assumption is violated. This directly settles whether the regularity gap is real for the reported experiments.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is Theorem 4.1: as (N,η,δ)→(∞,0,0) with N≥N(δ) and δ²=o(η), the regularized reconstruction SN[U_min^{N,η,δ}] converges to the minimal-norm solution u* of (3.1) in L²_{e^{-2t}}((0,T);H^p(Ω)^d), together with its first time derivative. The theorem assumes, in addition to the natural regularity u*∈L²(0,T;H^{2+p})∩H²(0,T;H^p), that the series Σ⟨u*,Ψ_n⟩Ψ''_n converges in L²((0,T);H^p). By Theorem 2.1(b), this convergence is guaranteed only when u* has H^k time regularity for k≥5, because ∥Ψ''_n∥_{L²_{e^{-2t}}}=O(n^{7/2}) and the coefficients decay as O(n^{-k}). The minimal-norm solution is only known to have H² time regularity, so the extra condition is a genuine, unverified hypothesis. The numerical tests use piecewise-constant inclusions, for which the elastic wave solution has at most limited Sobolev regularity in time and certainly not H^5; hence the convergence theorem does not cover the experiments as run. In addition, the experimental parameters are η=10^{-6} and δ=10%, so δ²=10^{-2} is not o(η), and N=30 is fixed rather than chosen as N≥N(δ). The numerical results therefore cannot be said to confirm the theorem. A separate proof issue is inequality (4.10): the functional J contains squared norms, so the first residual norm should be bounded by √J, not by J; this is likely repairable but, as written, the boundedness step is not justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a time-dimensional reduction method for recovering initial displacement and velocity in an anisotropic elastic wave equation from lateral Cauchy data. The time variable is expanded in a Legendre-polynomial-exponential basis, reducing the space-time inverse problem to a sequence of coupled elliptic systems in space, which are then solved by a Tikhonov-type quasi-reversibility functional. The paper proves existence and uniqueness of a minimal-norm solution to the lateral Cauchy problem, states a convergence theorem for the regularized reduced solution as the truncation level and regularization parameter are tuned relative to the noise level, and reports two-dimensional numerical reconstructions for isotropic, inhomogeneous, and anisotropic media with 10% noise.","tokens_in":23520,"tokens_out":7525,"duration_ms":85672,"significance":"If the convergence result in Theorem 4.1 is correct, the paper makes a useful methodological contribution: the Legendre-exponential basis is a genuinely new device in this context, and the reduction of a time-dependent ill-posed inverse problem to a sequence of elliptic systems is computationally attractive. The numerical experiments cover an anisotropic elasticity tensor, which goes beyond much of the existing scalar or isotropic literature, and the reconstructed inclusions are qualitatively reasonable. However, the theoretical statement as written rests on an extra regularity assumption that is not verified for the tested discontinuous data, the proof of the key boundedness step uses an invalid inequality, and the numerical parameters do not satisfy the scaling required by the theorem. The central idea is plausible and likely repairable, but the current manuscript oversells the agreement between theory and computation.","major_comments":[{"comment":"The convergence theorem assumes that the series Σ⟨u*,Ψ_n⟩Ψ''_n converges in L²((0,T);H^p(Ω)^d). By Lemma 2.2, ∥Ψ''_n∥ = O(n^{7/2}), so Theorem 2.1(b) guarantees this convergence only when u* has at least H^5 time regularity. The natural regularity supplied by Theorem 3.1 is only u* ∈ L²(0,T;H^{2+p}) ∩ H²(0,T;H^p), and the extra assumption is not verified anywhere. The numerical tests use piecewise-constant initial data, whose elastic wave solutions cannot be expected to possess H^5 time regularity; consequently the experiments are not covered by the convergence theorem as stated. The authors should either prove the series convergence under weaker assumptions, restrict the claim to high-regularity data and run numerical tests with such data, or clearly state that the theorem is not applicable to the experiments.","section":"Theorem 4.1 and Theorem 2.1(b), Eq. (4.4)"},{"comment":"The display after (4.10) bounds ∥div(C:∇S) − ∂ttS∥_{L²H^p} by J_{N,η,δ}(U^{N,η,δ}_min), without a square root. Since J is defined as a sum of squared norms, the residual norm is bounded by √J, not by J. This invalidates the chain (4.10)–(4.12) as written; the boundedness of ∂ttS does not follow. The step appears repairable, because √J remains bounded under δ²=o(η), but the current proof is not correct at this point.","section":"Proof of Theorem 4.1, inequality (4.10)"},{"comment":"The theorem requires η→0, δ→0, and δ²=o(η), with N≥N(δ) chosen accordingly. The numerical experiments set η=10^{-6} and δ=10%, so δ²=10^{-2} is not o(η); moreover N=30 is fixed by trial-and-error for Test 1 rather than chosen as N≥N(δ). Thus the reported computations lie outside Θ, and the abstract's claim that the numerical experiments 'confirm the theory' is not supported. The authors should either run experiments satisfying the theorem's scaling or present the numerics as a heuristic demonstration independent of Theorem 4.1.","section":"Section 5.1, Theorem 4.1 parameter regime"},{"comment":"The remark asserts that the convergence result remains valid when boundary data are available only on a proper subset Γ⊂∂Ω. The proof of Theorem 4.1 identifies the limit z by passing to the limit in boundary integrals over the full ∂Ω and concludes that z satisfies (3.1) with f* and g* on all Γ_T. For a proper subset, this identification is not available, and no alternative argument is supplied. The claim should be removed or substantiated with a separate proof.","section":"Remark 4.1(2)"}],"minor_comments":[{"comment":"The norm subscript in the first term of J appears as [H^p(Ω)^d]^{N+1} inside the sum over m; it should presumably be H^p(Ω)^d, otherwise the expression is not dimensionally consistent.","section":"Equation (4.1)"},{"comment":"The second sum inside the first term runs over n=1 while earlier definitions, such as (3.7), sum over n=0; this inconsistency should be fixed.","section":"Equation (4.1)"},{"comment":"In all three figure captions, the bottom-row labels repeat pcomp_2 for panels (g) and (h); based on the text these should be qcomp_1 and qcomp_2.","section":"Figures 1–3 captions"},{"comment":"The line stating ∥w'_{n_k}∥ → ∥w' L²_{e^{-2t}}(0,T) is missing the norm symbol and argument; it should read ∥w'_{n_k}∥_{L²_{e^{-2t}}(0,T)} → ∥w'∥_{L²_{e^{-2t}}(0,T)}.","section":"Proof of Lemma 4.2"},{"comment":"The abstract and introduction state that the method recovers 'the initial displacement and velocity fields' without always qualifying that, when uniqueness fails, the target is the minimal-norm solution rather than the true physical data. The qualification appears later in the paper but should be prominent from the outset.","section":"Introduction and Abstract"},{"comment":"The display following (4.11) has a missing closing parenthesis in '≤ 2δ²/η + ∥u*∥_{L²...} + ∥u*∥_{L²...', making the inequality hard to parse; it should be rewritten.","section":"Equation (4.11)"}],"recommendation":"major_revision","confidential_remarks":"The paper cites the authors' own time-reduction framework extensively, but the self-citation pattern is not, by itself, inappropriate for a research line. The more important concern is that the advertised convergence theorem is currently a conditional statement whose hypotheses exclude the experiments as run; the inequality error in (4.10) is local and likely repairable. I would recommend revision rather than rejection because the methodological idea is sound enough to warrant a corrected version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine extension of the time-dimensional reduction idea to anisotropic elastic systems with both initial displacement and velocity unknown, and the numerics are honestly reported. The central convergence theorem is not yet airtight: (4.10) bounds an L2 norm by J instead of by sqrt(J), and the high time-regularity hypothesis (the series with Ψ''_n converges, i.e. H^k with k≥5) is assumed without evidence for the minimal-norm solution. The numerical experiments do not cover the theorem as stated: η=1e-6 and δ=10% give δ²=1e-2, which is not o(η), and N=30 is fixed. So the paper is a solid algorithmic package with a proof that needs repair and experiments that are demonstrations, not confirmations.\n\nWhat is actually new: the Legendre-exponential basis is a modest variant of [25], but applying it to the anisotropic elastic inverse problem, together with the spectral estimates in Lemma 2.2 and the minimal-norm selection in Theorem 3.1, is new. Theorem 2.1's characterization of when the second-derivative series converges is a useful technical tool. The numerical results, especially Test 3 with genuinely anisotropic C, look plausible, and the amplitude errors are reported; that is more than many papers in this area do.\n\nWhere it is soft: the stress-test note is right about (4.10). Because J contains squared norms, a bound on J does not give a bound on the L2 norm of the residual; it gives a bound on its square root. The compactness argument needs the L2 bound, so the proof of part 3 of Theorem 4.1 has a real gap. It may be repairable by using the sqrt bound together with δ²/η→0 and the regularization term, but it is not a typo. Also, the extra regularity condition on u* is an unverified assumption; for the piecewise-constant inclusions in the tests it very likely fails. And the experiments run outside the asymptotic regime of the theorem. That does not destroy the numerical claim, but it means the theory and the experiments are not matched.\n\nBottom line: the method is worth knowing about, the application is new, and the numerical evidence is useful. The convergence theorem needs genuine revision: either weaken the claim, add the missing regularity analysis, or show experiments inside the theoretical regime. I would send it to a competent inverse-problems referee; the flaws are repairable and the paper is not incoherent. I would not cite it as providing rigorous convergence until fixed.","headline":"A genuinely new application of time-dimensional reduction to anisotropic elastodynamics, with honest numerics, but the main convergence theorem has a gap in (4.10) and the experiments do not sit inside the theorem's asymptotic regime.","tokens_in":23942,"tokens_out":1718,"would_cite":false,"duration_ms":19340,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","74J05","65M32","35B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Boundary observations of elastic waves can determine both the initial displacement and the initial velocity, provided one selects the minimal-norm solution and approximates it with a Legendre-exponential time reduction.","keywords":["inverse initial data problem","elastic wave equation","anisotropic elasticity","Legendre-exponential basis","time-dimensional reduction","quasi-reversibility method","minimal-norm solution","Cauchy boundary data"],"falsifier":"Take the piecewise-constant initial data used in Test 1, regard the corresponding elastic wave field as $u^*$, and compute the weighted sum $\\sum_{n=0}^\\infty n^{7/2}\\|\\langle u^*, \\Psi_n\\rangle\\|_{H^p(\\Omega)^d}$; if it diverges, then the convergence hypothesis of Theorem 4.1 is violated, and the numerical reconstructions in the paper cannot be explained by the theorem as stated. Alternatively, run the algorithm on a smooth Gaussian initial condition with known truth and check whether the error $\\|S_N[U^{\\,N,\\eta,\\delta}_{\\min}] - u^*\\|$ actually decreases to zero along a sequence satisfying $\\delta^2 = o(\\eta)$; failure to decrease would contradict the theorem's conclusion.","tokens_in":22849,"feed_emoji":"📡","tokens_out":8220,"duration_ms":82687,"temperature":0.7,"pith_summary":"This paper tries to establish that the inverse problem of recovering the initial displacement and velocity of an anisotropic elastic wave from boundary data becomes computationally tractable and provably convergent when the time variable is expanded in a specially chosen orthonormal basis. The basis, made of Legendre polynomials multiplied by an exponential weight, turns the space-time elastic system into a finite chain of elliptic systems for spatial Fourier coefficients. The paper proves that, as the truncation index grows and the noise and regularization parameters tend to zero in the relation $\\delta^2 = o(\\eta)$, the reconstructed field and its first time derivative converge in a weighted $L^2$\\textendash Sobolev norm to the unique minimal-norm solution of the lateral Cauchy problem. A sympathetic reader would care because this offers a practical algorithm for seismology, nondestructive testing, and elastography, where only surface measurements are available and the medium may be inhomogeneous and anisotropic. The numerical experiments with 10% noise indicate that both the geometry and the amplitude of inclusion-like initial data can be recovered.","feed_headline":"Anisotropic elastic sources reconstructed from boundary data alone","feed_subtitle":"A Legendre-exponential time basis reduces the inverse problem to elliptic systems with provable noise convergence.","key_machinery":"The engine of the method is the orthonormal Legendre-exponential basis $\\Psi_n(t) = e^t Q_n(t)$, where $Q_n$ are the normalized Legendre polynomials on $(0,T)$; this basis is orthonormal in the weighted space $L^2_{e^{-2t}}(0,T)$. Its second derivatives $\\Psi_n''$ are never identically zero, so every Fourier mode contributes to the approximation of $\\partial_{tt}u$, and the Sturm\\textendash Liouville eigenvalue structure of the Legendre polynomials gives the decay estimate $\\|\\langle u,\\Psi_n\\rangle\\|_{H^p} \\le C n^{-k}$ together with $\\|\\Psi_n''\\| \\le C n^{7/2}$. The reduction replaces the wave equation by the coupled elliptic system (3.10) with coefficient matrix $s_{mn} = \\int_0^T e^{-2t}\\Psi_n''(t)\\Psi_m(t)\\,dt$, and the quasi-reversibility functional $J_{N,\\eta,\\delta}$ makes that system solvable with noisy data.","core_discovery":"On its own terms, the paper's central claim is Theorem 4.1: for a maximal truncation level $N \\ge N(\\delta)$ and a regularization parameter $\\eta(\\delta)$ tending to zero with $\\delta^2 = o(\\eta)$, the regularized time-reduced reconstruction $S_N[U^{\\,N,\\eta,\\delta}_{\\min}]$ converges to the minimal-norm solution $u^*$ of the lateral Cauchy problem (3.1) in $L^2_{e^{-2t}}((0,T);H^p(\\Omega)^d)$, together with its first time derivative. That means the approximate initial displacement $S_N[\\cdot](\\cdot,0)$ converges strongly in $H^p(\\Omega)^d$ and the approximate initial velocity $\\partial_t S_N[\\cdot](\\cdot,0)$ converges weakly in $H^p(\\Omega)^d$. The proof combines the spectral properties of the Legendre-exponential basis, which justify replacing $\\partial_{tt}u$ by the series $\\sum_n u_n\\Psi_n''$, with the quasi-reversibility functional that enforces the reduced elliptic system and the boundary data while penalizing the $H^{2+p}$ norm.","pith_inferences":["A fair numerical test of the theorem would use smooth initial data, since the piecewise-constant inclusions in the paper's experiments fall outside the $H^k$, $k \\ge 5$ time-regularity hypothesis; comparing convergence on Gaussian versus step profiles would separate the method's performance from the theorem's scope.","The same time-reduction construction should transfer to scalar wave, acoustic, or viscoelastic equations, because the argument only requires a second-order time derivative and the spectral basis; running the scheme on the acoustic wave equation with smooth sources is a direct testable extension.","The condition $\\delta^2 = o(\\eta)$ suggests a concrete parameter-selection rule for practice: choose $\\eta$ to decay slower than the squared noise level, say $\\eta \\sim \\delta^{3/2}$, and then increase $N$ until the projection error in (4.4) drops below $\\delta^2$; this could be automated as a stopping criterion.","The minimal-norm selection implicitly biases the reconstruction toward smoother initial fields, a consequence relevant to seismic imaging where the true source may be rough; the paper does not quantify this bias."],"forward_implications":["If Theorem 4.1 is correct, both the initial displacement and the initial velocity can be stably reconstructed from noisy boundary data without knowing either field in advance.","The method applies to a fully anisotropic and inhomogeneous elasticity tensor, with no isotropy or constant-coefficient restriction, which covers layered geological media and biological tissue.","Because the target is the unique minimal-norm solution, the algorithm returns a well-defined answer even when the lateral Cauchy problem has multiple solutions; if the problem is actually uniquely solvable, the reconstructed field converges to the true physical solution.","The convergence argument carries over when the measurements are restricted to a proper subset $\\Gamma$ of the boundary, so partial sensor coverage does not destroy the guarantee.","At $t=0$, the reconstruction yields the initial displacement strongly and the initial velocity weakly in $H^p(\\Omega)^d$, giving quantitative approximations of both fields."],"supporting_citations":[{"why":"Supplies the quasi-reversibility method used to regularize the reduced elliptic system and prove convergence with noisy data.","marker":"[28]"},{"why":"Provides the Moore-Penrose pseudoinverse/minimal-norm principle that defines the unique target solution of the inverse problem.","marker":"[21]"},{"why":"Introduces the earlier polynomial-exponential basis whose structural limitations the Legendre-exponential basis improves.","marker":"[25]"},{"why":"Earlier time-dimensional reduction method for initial conditions that lacked the spectral convergence framework used here.","marker":"[30]"},{"why":"Establishes inverse source results in elastodynamics via Green's functions, the restrictive setting the paper contrasts with its anisotropic inhomogeneous method.","marker":"[10]"}],"fun_headline_variants":["Time reduction reconstructs elastic source from boundary data","Anisotropic elastic source recovered from boundary measurements","Legendre-exponential basis time-reduces elastic inverse problem","Boundary wave data reveal initial displacement and velocity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence proof rests on the assumption that the true minimal-norm solution is smooth enough in the time variable that a certain infinite series of its second derivatives converges; the numerical tests use sharp step-like inclusions that do not meet this smoothness condition.","fun_headline_variants_meta":{"raw":{"variants":["Time reduction reconstructs elastic source from boundary data","Anisotropic elastic source recovered from boundary measurements","Legendre-exponential basis time-reduces elastic inverse problem","Boundary wave data reveal initial displacement and velocity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000938,"raw_usage":{"total_tokens":4029,"prompt_tokens":982,"completion_tokens":3047,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":2985}},"tokens_in":598,"tokens_out":3047,"duration_ms":25995,"temperature":1.0,"reasoning_tokens":2985,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:38:19.685598+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the piecewise-constant initial data used in Test 1, regard the corresponding elastic wave field as $u^*$, and compute the weighted sum $\\sum_{n=0}^\\infty n^{7/2}\\|\\langle u^*, \\Psi_n\\rangle\\|_{H^p(\\Omega)^d}$; if it diverges, then the convergence hypothesis of Theorem 4.1 is violated, and the numerical reconstructions in the paper cannot be explained by the theorem as stated. Alternatively, run the algorithm on a smooth Gaussian initial condition with known truth and check whether the error $\\|S_N[U^{\\,N,\\eta,\\delta}_{\\min}] - u^*\\|$ actually decreases to zero along a sequence satisfying $\\delta^2 = o(\\eta)$; failure to decrease would contradict the theorem's conclusion.","supporting_citations":[{"cited_title":"Latt` es and J","cited_arxiv_id":null,"evidence_quote":"Supplies the quasi-reversibility method used to regularize the reduced elliptic system and prove convergence with noisy data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier time-dimensional reduction method for initial conditions that lacked the spectral convergence framework used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes inverse source results in elastodynamics via Green's functions, the restrictive setting the paper contrasts with its anisotropic inhomogeneous method."}],"review_version":1}