{"id":"c98b6eea-4395-4cc4-963e-6fe5465b89fc","arxiv_id":"2508.15406","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditional Lipschitz and Holder stability, plus finite element error bounds, are established for a parabolic inverse source problem using Carleman estimates.","lead":"This paper proves that a hidden heat-source term in a parabolic equation can be stably recovered from partial interior measurements, and gives a finite element method with error guarantees. The result matters because inverse source problems appear in many engineering and medical applications where reliability guarantees are rare.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; central claims unverifiable from abstract alone","rationale":"The reader's verdict of UNVERDICTED is appropriate because the abstract alone provides insufficient information to assess the correctness of the Carleman-based stability estimates or the finite element error bounds. My independent read identifies the same weakest assumption: the unspecified hypotheses of the Carleman estimates. Since no full text is available, I cannot identify a specific flaw beyond this absence of verification. The honest finding is a non-finding regarding any concrete mathematical error; the concern is that the central claim is conditional on unstated assumptions. Therefore the verdict should remain UNCHANGED, and the concrete test is to inspect the full Carleman estimates and their applicability.","tokens_in":569,"tokens_out":1660,"duration_ms":18572,"concrete_test":"Obtain the full text and verify that the Carleman estimate (e.g., a named theorem) applies to the parabolic operator with the stated weight and partial interior observation set. Specifically, check that the weight satisfies the standard pseudoconvexity condition, that the observation set covers the Carleman weight's support, and that the source's spatial support is compatible with the estimate's cutoff functions. If these conditions hold, the stability estimates and FE error bounds likely follow; if not, the central claims are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's central claims—novel conditional Lipschitz/Hölder stability and rigorous finite element error bounds—rest entirely on 'suitable Carleman estimates' whose precise hypotheses (weight functions, domain geometry, observation sets, regularity) are not stated. Without these details, one cannot check whether the estimates hold for the stated partial interior measurement geometry. This is not an internal inconsistency; it is an absence of evidence. The most load-bearing concern is that the stability estimates and error bounds may depend on nontrivial assumptions that are not visible at the abstract level. If the Carleman estimates require more restrictive observation or weight conditions than the problem setup allows, the central claims would not follow. However, there is no concrete mathematical error to attack without the full text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The abstract announces a numerical method for recovering a spatially dependent source term in a parabolic equation from partial interior measurements. Conditional Lipschitz stability (with boundary conditions) and Hölder stability (without boundary conditions) are claimed, obtained via Carleman estimates. A conforming finite element discretization in space and time is proposed, with rigorous error bounds derived from the conditional stability estimates. Numerical experiments are reported. The full text is not available; this assessment is based solely on the abstract.","tokens_in":737,"tokens_out":2836,"duration_ms":30520,"significance":"If all claimed results are proved, this would constitute a substantial contribution to the inverse source problem literature: the combination of conditional stability estimates with conforming FEM error bounds for partial interior data is nontrivial and practically relevant. The paper promises a complete numerical analysis package, which is valuable. However, because the abstract contains no equations or proofs, the significance cannot be confirmed at this stage.","major_comments":[{"comment":"The central stability results rest on 'suitable Carleman estimates' without any statement of the weight functions, domain geometry, observation sets, or regularity assumptions. For partial interior measurements, standard global Carleman estimates require specific conditions on the observation subset and weight; if these are not met, both the conditional Lipschitz/Hölder stability and the subsequent FEM error bounds would not follow. The manuscript must state these hypotheses and verify them for the stated partial-measurement geometry.","section":"Abstract"},{"comment":"The claim of 'rigorous error bounds' depends on conditional stability estimates and finite element approximation properties, but the abstract gives no statement of the discrete problem, required regularity, mesh/time-step coupling, or convergence order. Without the actual theorem, the reader cannot judge whether the bounds are meaningful (for example, whether they depend on unknown constants that defeat the reconstruction purpose). The full text must supply these details.","section":"Abstract"},{"comment":"The complete manuscript is absent from the submission; only the abstract is present. All technical content—proofs, theorem statements, numerical implementation details—is omitted. This is a missing-support issue that prevents verification of any of the core claims. I cannot determine whether the results are correct or whether the novelty is as claimed.","section":"Full text (not available)"}],"minor_comments":[{"comment":"The phrase 'partial interior measurements' should specify whether data are taken on a subdomain, at discrete points, or along time-space curves; this affects the Carleman estimate conditions.","section":"Abstract"},{"comment":"The 'new numerical approach' is not identified; the abstract would be more informative if it named the method (e.g., output least squares, quasi-reversibility, Tikhonov regularization) even at a high level.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"The submission as received contains only the abstract. This is unusual; the editor may wish to request the full manuscript before further review. My assessment is based solely on the abstract and should be treated as provisional. I am unable to render a substantive verdict on the mathematical content without the proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is an abstract, not a paper, so my verdict is provisional. The authors claim conditional Lipschitz and Hölder stability for a parabolic inverse source problem with partial interior data, and conforming finite element error bounds that use those stability estimates. That's a worthwhile combination: conditional stability gives the well-posedness backbone, and carrying it through to discrete error bounds is exactly what makes the numerics credible. If the proofs are right, this is a solid contribution to the inverse problems literature.\n\nThe strongest part is the claim itself. Many numerical papers on inverse source problems just propose a scheme and test it; here they tie the reconstruction error to a Carleman-based conditional stability estimate. That's the right way to argue. The FEM setup in both time and space is standard but appropriate. No obvious red flag in the abstract.\n\nThe soft spot is everything that isn't in the abstract. 'Suitable Carleman estimates' is doing a lot of work. For partial interior observations, the weight function and the observation set have to align with the geometry; if they don't, the Hölder/Lipschitz distinction is not just a detail. The abstract also doesn't say what regularity is assumed on the source or the exact data availability. The numerical experiments are mentioned but not described, so I can't tell whether they validate the predicted rates or just show a few pictures. So the central claims are unverifiable at this stage.\n\nOn the literature: conditional stability for parabolic inverse problems using Carleman estimates is a mature area, and I can't tell from the abstract whether 'novel' means a genuinely new distinction or a re-run of known estimates for a slightly different geometry. The authors are credible, so I'd guess there is real work here, but I wouldn't cite it until the proof is available.\n\nBottom line: if this lands in front of a referee, it should get a real review. The topic is well within scope, and the claims are precise enough to be checkable. I'd send it to someone who knows Carleman estimates and FEM—not desk reject it. For me, I'd want to read the full text before deciding whether it's a must-cite.","headline":"A plausible Carleman-based stability + FEM error analysis for a parabolic inverse source problem, but the abstract is too thin to verify the claims.","tokens_in":1156,"tokens_out":3556,"would_cite":false,"duration_ms":37681,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M32","35R30","65M60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Small data errors cause controlled source errors in a parabolic inverse source problem, with formal finite element error bounds.","keywords":["inverse source problem","parabolic equation","Carleman estimates","conditional stability","Hölder stability","finite element method","error bounds","partial interior measurements"],"falsifier":"A direct check: write out the Carleman estimate with explicit constants and the exact conditions on the weight, domain, and observation set, then run the proposed finite element reconstruction with added noise of size δ for both the with-boundary and without-boundary cases. If the observed reconstruction error does not scale as δ or δ^α, the stated stability rates are wrong; if the numerical method remains stable on a configuration where the Carleman conditions are violated, those conditions are not necessary for the conclusion.","tokens_in":514,"feed_emoji":"🔥","tokens_out":4656,"duration_ms":50084,"temperature":0.7,"pith_summary":"This paper aims to show that recovering a spatially varying source term in a parabolic (heat-like) equation from partial interior measurements is not just possible but stable: small errors in the measured data can only lead to controlled errors in the reconstructed source. The authors prove conditional Lipschitz stability when boundary data are available, and conditional Hölder stability when they are not, using Carleman estimates as the main tool. Building on these stability estimates, they analyze a conforming finite element method in both time and space and prove explicit error bounds for the discrete approximation. A sympathetic reader would care because this gives a classical ill-posed inverse problem a rigorous quantitative guarantee, showing that a concrete numerical scheme cannot silently amplify noise beyond a predictable rate.","feed_headline":"Finite elements recover a parabolic source with proven error bounds","feed_subtitle":"Carleman-based conditional stability gives explicit noise-to-error rates and discrete error bounds.","key_machinery":"The load-bearing object is a Carleman estimate for the parabolic operator: a weighted energy inequality with an exponential weight that concentrates control on the part of the solution visible in the measurement region. This estimate does double work: it yields the conditional Lipschitz and Hölder stability inequalities that turn the ill-posed inverse problem into a conditionally stable one, and it provides the quantitative control needed to derive explicit error bounds for the finite element approximation.","core_discovery":"The paper's central claim is that a spatially dependent source component in a parabolic equation can be reconstructed from measurements on only part of the interior, and that the reconstruction inherits a quantitative stability property. If data are perturbed by noise of size δ, the reconstructed source changes by at most a constant times δ when boundary information is available; without boundary information, the change is at most a constant times δ^α for some 0<α<1. The proof of both rates uses Carleman estimates, weighted energy inequalities adapted to the parabolic operator and the partial observation set. The same estimates are then applied to a conforming finite element discretization i","pith_inferences":["Implicit consequence: the same Carleman template should transfer to other parabolic inverse problems, such as coefficient or initial-state recovery, whenever a suitable weight can be constructed for the available measurement set.","Testable extension: one could check whether the Hölder exponent in the no-boundary case is sharp by constructing noise patterns whose amplification matches the bound; confirming that rate would show the stability estimate cannot be strengthened.","Practical extension: the explicit stability constant suggests an adaptive stopping rule for iterative reconstruction—stop when data misfit reaches the noise level, with the Carleman constant determining the safety factor."],"forward_implications":["Any reconstruction algorithm for this inverse source problem inherits a quantitative error guarantee: data noise of size δ leads to reconstruction error at most C δ or C δ^α, with the exponent fixed by whether boundary data are used.","The conforming finite element scheme is certified: its total error splits into a discretization error and a data-noise term, both controlled explicitly rather than by heuristic regularization.","The with-boundary versus without-boundary distinction is meaningful: full boundary data give the stronger Lipschitz rate, while interior-only measurements degrade the rate to Hölder, quantifying the information contributed by the boundary.","Carleman-based stability can serve as a backbone for numerical error analysis, not just for uniqueness, and this paper demonstrates that chain for parabolic inverse source problems."],"supporting_citations":[],"fun_headline_variants":["Finite elements recover parabolic source from partial data","Carleman stability plus FEM delivers error-bounded source recovery","Parabolic source inversion: partial data, full recovery guarantee","Source reconstruction in heat equation with proven accuracy bounds","Fewer measurements? Carleman estimates still ensure stable source recovery"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The entire argument rests on the existence of a Carleman estimate whose weight function and conditions fit the chosen domain and the partial interior measurement geometry; if no such estimate holds there, neither the stability rates nor the finite element error bounds follow.","fun_headline_variants_meta":{"raw":{"variants":["Finite elements recover parabolic source from partial data","Carleman stability plus FEM delivers error-bounded source recovery","Parabolic source inversion: partial data, full recovery guarantee","Source reconstruction in heat equation with proven accuracy bounds","Fewer measurements? Carleman estimates still ensure stable source recovery"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000218,"raw_usage":{"total_tokens":1205,"prompt_tokens":604,"completion_tokens":601,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":348,"completion_tokens_details":{"reasoning_tokens":521}},"tokens_in":348,"tokens_out":601,"duration_ms":6988,"temperature":1.0,"reasoning_tokens":521,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:53:32.875865+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check: write out the Carleman estimate with explicit constants and the exact conditions on the weight, domain, and observation set, then run the proposed finite element reconstruction with added noise of size δ for both the with-boundary and without-boundary cases. If the observed reconstruction error does not scale as δ or δ^α, the stated stability rates are wrong; if the numerical method remains stable on a configuration where the Carleman conditions are violated, those conditions are not necessary for the conclusion.","supporting_citations":[],"review_version":1}