{"id":"f953385e-a1f1-4be9-90f3-0ba7796581a8","arxiv_id":"2606.20875","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":2,"one_line_summary":"Uniqueness is established for the constant-coefficient case via Fourier transform and Volterra analyticity; a Legendre spatial reduction plus Tikhonov scheme is proved to converge to the minimum-norm solution in the variable-coefficient case for fixed truncation order.","lead":"The paper develops a method to recover the unknown initial condition of a convection-diffusion equation with memory from final-time data, using Legendre polynomial reduction in space and Tikhonov regularization to handle ill-posedness. A smart generalist might read it to see how spectral methods and regularization can stabilize reconstructions in PDE inverse problems that include history-dependent terms.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Convergence holds only under an a priori suitable choice of regularization parameter vanishing with noise level","rationale":"The reader's weakest assumption directly identifies the conditional nature of the convergence result; no stronger internal inconsistency (e.g., in the Legendre reduction or Volterra uniqueness argument) is visible from the given material, so the reg-parameter choice remains the single load-bearing point.","tokens_in":1739,"tokens_out":326,"duration_ms":20293,"concrete_test":"Extract the precise statement of the convergence theorem (likely in the section on Tikhonov analysis) and check whether it supplies an explicit alpha(delta) rule or only assumes existence of a sequence alpha_n(delta_n) satisfying the standard conditions; if only the latter, recompute the numerical examples with a discrepancy-principle choice of alpha and verify whether the reported error still tends to zero at the claimed rate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim is the convergence of Tikhonov minimizers to the finite-dimensional minimum-norm solution for fixed truncation order N as delta, alpha -> 0. This requires alpha = alpha(delta) satisfying alpha -> 0 and delta^2/alpha -> 0 (standard source condition + qualification for Tikhonov). The abstract states the result holds 'under a suitable choice' but gives no indication that an a posteriori rule (discrepancy, balancing, etc.) is derived or that the proof supplies an explicit rate or constructive selection independent of the unknown solution. In the reduced finite-dimensional least-squares problem this choice remains external to the data.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies an inverse initial-data problem for a convection-diffusion equation with memory term. For spatially constant coefficients it proves uniqueness via Fourier transform reducing the problem to a scalar Volterra equation whose analyticity yields uniqueness. For variable coefficients it introduces a Legendre-Galerkin spatial reduction that converts the PDE inverse problem into a finite-dimensional terminal-value ODE system for the coefficient vector; this reduced problem is solved by Tikhonov regularization with an H² penalty. The central theoretical result states that, for any fixed truncation order N, the regularized minimizers converge to the minimum-norm solution of the finite-dimensional problem as the noise level δ and the regularization parameter α both tend to zero, provided α is chosen suitably (i.e., α→0 and δ²/α→0). Two-dimensional numerical illustrations are presented.","tokens_in":1885,"tokens_out":623,"duration_ms":16516,"significance":"If the convergence statement can be made constructive, the combination of exact spatial reduction with a provably convergent regularization scheme would supply a theoretically supported numerical method for a class of severely ill-posed inverse problems that include memory. The explicit reduction to a finite terminal-value system and the convergence proof for that reduced system are genuine strengths; the numerical examples provide at least preliminary evidence of practical behavior.","major_comments":[{"comment":"Abstract and convergence statement: the claim that the regularized minimizers converge to the finite-dimensional minimum-norm solution 'under a suitable choice of the regularization parameter' is load-bearing for the method, yet the manuscript gives no indication that an a-posteriori rule (discrepancy principle, balancing principle, etc.) is derived or that the proof supplies a constructive selection of α(δ) independent of the unknown solution. The standard source-condition argument requires α→0 and δ²/α→0, both of which remain external to the data.","section":"Abstract / convergence theorem"},{"comment":"Reduction step: the passage from the original memory PDE to the finite tensor-product Legendre system for the time-dependent coefficients must preserve the memory integral exactly; any truncation or quadrature error introduced at this stage would propagate into the subsequent Tikhonov analysis. The manuscript should state explicitly whether the memory term is discretized exactly or approximated and, if the latter, quantify the additional consistency error.","section":"Legendre spatial reduction section"}],"minor_comments":[{"comment":"The abstract states that uniqueness holds 'in a spatially independent coefficient setting' but does not clarify whether the same Fourier-Volterra argument extends, even formally, to the variable-coefficient case before the numerical reduction is applied.","section":"Abstract"},{"comment":"Notation for the memory kernel and the precise form of the H² penalty should be introduced once and used consistently; several symbols appear only in the abstract and are not redefined in the main text.","section":"Notation / preliminaries"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting both the strengths of the spatial reduction and the limitations of the convergence result. We respond to each major comment below.","responses":[{"response":"We agree that the convergence theorem is stated under the external conditions α→0 and δ²/α→0 and that no a-posteriori selection rule is derived or analyzed in the manuscript. The result therefore remains non-constructive with respect to practical parameter choice. In the revision we will modify the abstract and the theorem statement to make this assumption explicit and will add a short remark in the conclusions noting that discrepancy-principle or balancing-principle rules could be considered in applications, while acknowledging that their rigorous justification would require additional source conditions not treated here.","revision_made":"partial","referee_comment":"[Abstract / convergence theorem] Abstract and convergence statement: the claim that the regularized minimizers converge to the finite-dimensional minimum-norm solution 'under a suitable choice of the regularization parameter' is load-bearing for the method, yet the manuscript gives no indication that an a-posteriori rule (discrepancy principle, balancing principle, etc.) is derived or that the proof supplies a constructive selection of α(δ) independent of the unknown solution. The standard source-condition argument requires α→0 and δ²/α→0, both of which remain external to the data."},{"response":"The reduction is obtained by applying the orthogonal projection onto the tensor-product Legendre space directly to the entire PDE, including the memory integral. Because the projection is linear, the memory term is transferred exactly into the finite-dimensional system; no truncation or quadrature approximation is introduced at the reduction stage. The resulting terminal-value ODE system is therefore the exact Galerkin projection of the original problem. We will insert an explicit clarifying paragraph in the Legendre spatial reduction section stating that the memory integral is preserved exactly and that no additional consistency error arises from the reduction itself.","revision_made":"yes","referee_comment":"[Legendre spatial reduction section] Reduction step: the passage from the original memory PDE to the finite tensor-product Legendre system for the time-dependent coefficients must preserve the memory integral exactly; any truncation or quadrature error introduced at this stage would propagate into the subsequent Tikhonov analysis. The manuscript should state explicitly whether the memory term is discretized exactly or approximated and, if the latter, quantify the additional consistency error."}],"tokens_in":1499,"tokens_out":518,"duration_ms":27537,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing here is a workable numerical scheme for recovering initial data in a convection-diffusion equation that includes a memory integral over past Laplacian values. They first prove uniqueness in the constant-coefficient case by turning the problem into a scalar Volterra equation whose analyticity forces the initial datum to be zero when final data vanish. For variable coefficients they project onto a tensor-product Legendre basis in space, reduce the PDE to a system of ODEs in the coefficients, and apply Tikhonov regularization with an H² penalty to the resulting finite-dimensional least-squares problem.\n\nWhat works is the reduction step itself and the convergence statement: for fixed truncation order they show that the regularized minimizers approach the minimum-norm solution of the reduced system as noise level δ and regularization parameter α both go to zero, provided α is chosen suitably (the usual source-condition requirement that δ²/α → 0). The two-dimensional numerical examples are included to show the method in action.\n\nThe soft spot is exactly the one flagged in the stress test. The proof gives no a-posteriori rule for picking α from the data alone; the “suitable choice” remains external and must be supplied by the user. That is standard for Tikhonov but still limits practical use. The paper also stays at fixed truncation order, so there is no analysis of how the spatial error interacts with the regularization error. The examples are only illustrative; no systematic study of mesh size, memory kernel, or noise levels appears.\n\nThis is a narrow but competent extension aimed at people who already work on inverse problems for nonlocal or memory PDEs. It does not open a new direction, but the reduction-plus-regularization combination is cleanly executed and the uniqueness argument is self-contained. A serious editor should send it to referees; the gaps are fixable with modest additional work on parameter selection and error balancing.","headline":"Standard Legendre reduction plus Tikhonov for a memory PDE inverse problem; convergence proof holds only under an external parameter choice.","tokens_in":2380,"tokens_out":445,"would_cite":false,"duration_ms":12065,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Legendre spatial reduction combined with Tikhonov regularization recovers the initial condition from final-time data for a memory convection-diffusion equation.","keywords":["inverse initial data problem","memory convection-diffusion equation","Legendre polynomial expansion","Tikhonov regularization","ill-posed problems","finite-dimensional reduction","numerical reconstruction"],"falsifier":"A concrete sequence of noisy final-time data and corresponding regularization parameters, both tending to zero, for which the regularized Legendre-coefficient vectors fail to approach the minimum-norm solution of the reduced system at a fixed truncation order.","tokens_in":2639,"feed_emoji":"","tokens_out":687,"duration_ms":12268,"temperature":0.7,"pith_summary":"The paper targets the inverse problem of recovering the unknown initial state of a convection-diffusion equation that includes a nonlocal memory term from observations at the final time. The backward problem is severely ill-posed. Uniqueness is established for spatially constant coefficients by Fourier analysis that reduces the question to an analytic Volterra equation. For variable coefficients the authors replace the original PDE by a finite tensor-product Legendre expansion in space, which converts the inverse problem into a finite-dimensional terminal-value system; this reduced system is then solved by Tikhonov regularization with an H-squared penalty. For any fixed truncation order the regularized solutions are shown to converge to the minimum-norm solution of the reduced system once the noise level and the regularization parameter both tend to zero under a suitable linking rule.","feed_headline":"Legendre reduction plus Tikhonov regularization recovers initial data","feed_subtitle":"The approach converts the severely ill-posed backward problem into a convergent finite-dimensional regularized system for any fixed truncati","key_machinery":"Finite tensor-product Legendre expansion in space that reduces the inverse problem to a terminal-value system for time-dependent coefficients, solved by Tikhonov-regularized least squares with an H^2 penalty.","core_discovery":"For a fixed truncation order, the Tikhonov-regularized minimizers converge to the finite-dimensional minimum-norm solution as the noise level and the regularization parameter vanish, under a suitable choice of the regularization parameter.","pith_inferences":["Separate analysis of the truncation error is needed before the method can be applied to data with arbitrarily fine spatial features.","The same reduction-plus-regularization pattern may apply directly to other linear parabolic equations that contain nonlocal time operators.","Practical implementation would still require an a-posteriori rule such as the discrepancy principle to select the regularization parameter from data alone."],"forward_implications":["Uniqueness of the initial datum holds when coefficients are independent of space, proved via Fourier transform and analytic continuation of a scalar Volterra equation.","The original infinite-dimensional ill-posed problem is replaced by a finite-dimensional, computable least-squares problem whose regularized solutions converge under the stated parameter rule.","Two-dimensional numerical tests confirm that the reduced regularized reconstructions remain stable for moderate noise levels.","The memory term is retained exactly inside the reduced system of ordinary differential equations for the Legendre coefficients."],"fun_headline_variants":["Tikhonov regularized Legendre method recovers initial data","Legendre spatial reduction with Tikhonov solves inverse memory problem","Convergent recovery of initial data via reduced Legendre Tikhonov","Tikhonov regularization in Legendre reduced system recovers initial data"],"cache_read_input_tokens":64,"weakest_assumption_plain":"A regularization parameter can be chosen that vanishes with the noise level in a way that does not require knowledge of the exact solution.","fun_headline_variants_meta":{"raw":{"variants":["Tikhonov regularized Legendre method recovers initial data","Legendre spatial reduction with Tikhonov solves inverse memory problem","Convergent recovery of initial data via reduced Legendre Tikhonov","Tikhonov regularization in Legendre reduced system recovers initial data"]},"model":"grok-4.3","cost_usd":0.007785,"raw_usage":{"total_tokens":3534,"prompt_tokens":625,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":77849500,"prompt_tokens_details":{"text_tokens":625,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2842,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":625,"tokens_out":67,"duration_ms":17547,"temperature":1.0,"reasoning_tokens":2842,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T16:07:14.066252+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete sequence of noisy final-time data and corresponding regularization parameters, both tending to zero, for which the regularized Legendre-coefficient vectors fail to approach the minimum-norm solution of the reduced system at a fixed truncation order.","supporting_citations":[],"review_version":1}