{"id":"1fbdbd67-79b2-4e74-823d-e2180b5e82ec","arxiv_id":"2506.21699","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A three-stage numerical pipeline, WKB phase retrieval plus Fourier dimension reduction plus Carleman convexification, recovers scatterer location and contrast from phaseless 3D backscattering data.","lead":"This paper combines a travel-time phase guess, a Fourier filter, and a convex-optimization trick to reconstruct a hidden object's dielectric constant from wave-intensity measurements alone. The tests on noisy synthetic data recover simple inclusions, but the 'globally convergent' label applies only to part of the pipeline.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'globally convergent' result is proven for the approximate system (4.8) with the hard-coded zero boundary data on ∂Ω\\Γ, not for the original phaseless inverse problem; Eq. (4.10) contains no term for this model error.","rationale":"The reader's weakest-assumption analysis targets exactly the gap I consider most load-bearing: the Carleman convexification theorem is rigorous but applies to a model that already incorporates the unvalidated zero boundary condition (3.15). The proof of strict convexity is plausible and consistent with the authors' prior work, and the numerical demonstrations are suggestive, but the Section 6 claim of convergence to the true solution conflates convergence to the minimizer of an approximate functional with solution of the original phaseless problem. Eq. (4.10) is the only quantitative link between the minimizer and a solution of a boundary value problem, and that boundary value problem is (4.8), not the physical problem. A forward-solver rerun with the true unmeasured boundary values is a direct, decisive check: it isolates the contribution of (3.15) from all other algorithmic choices. The phase-retrieval step (2.10) is a second unquantified component, but the boundary zero-fill is the more fundamental gap because it is embedded in the theoretical statement itself. I therefore keep the reader's CONDITIONAL verdict: the method may be useful, but the claimed global convergence for the phaseless problem must be qualified to the approximate model or substantiated by controlling this boundary mismatch.","tokens_in":18230,"tokens_out":8370,"duration_ms":103048,"concrete_test":"Recompute Test 1 with the zero-fill assumption (3.15) replaced by the true boundary data: use the forward solver to evaluate v(x,k) from (3.1), compute the Fourier coefficients (3.7) on ∂Ω\\Γ, and rerun Algorithm 1 with those values instead of zero, leaving all other parameters unchanged. Compare the resulting ccomp with the published zero-fill reconstruction and with ctrue; also report the L2 ratio ||v_m|∂Ω\\Γ|| / ||g_m||_Γ for each m=1,...,N. If the reconstruction shifts by more than the reported 9% relative error, or if the side/top coefficients are not small relative to the Γ data, the zero boundary condition is a dominant systematic error and the global-convergence claim must be restricted to the approximate system (4.8).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6 states that gradient descent 'globally converge[s] to the true solution,' but Theorem 4.1 and estimate (4.10) only connect the minimizer of Jλ,ϵ to v*, the exact solution of the approximate system (4.8), which enforces v*m=0 on ∂Ω\\Γ via (3.15). Remark 3.3 explicitly concedes that this data complementation is 'not rigorous but rather an approximation,' and Remark 3.4 identifies it as a deliberately introduced inaccuracy. In (4.4), the zero boundary condition on ∂Ω\\Γ is enforced as a strongly weighted, effectively hard constraint, so minimizing the functional cannot correct for a nonzero physical scattered field on the unmeasured faces. The error estimate (4.10) has no term for this mismatch, and therefore it cannot support the claimed convergence for Problem 2.1. The same unquantified gap appears one step earlier: the Cauchy data in (3.13)-(3.14) come from the non-convex minimization of (2.10), for which no error bound or convergence guarantee is supplied. The central claim therefore overstates what is established: strict convexity and gradient-descent convergence are shown for an approximating model, not for the stated phaseless inverse problem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a three-stage numerical method for a three-dimensional phaseless inverse scattering problem governed by the Helmholtz equation with multi-frequency, single-point-source data. Stage one retrieves the phase by minimizing a weighted least-squares functional J_k (Eq. 2.10) initialized from a WKB travel-time ansatz. Stage two applies a logarithmic transform and a truncated Fourier expansion in the frequency variable to convert the phased inverse problem into a system of elliptic PDEs with Cauchy boundary data on the bottom face and zero boundary data on the remaining faces (Eqs. 3.16 and 4.2). Stage three minimizes a Carleman-weighted convexified functional J_{\\lambda,\\epsilon} (Eq. 4.4) via gradient descent; Theorem 4.1 states strict convexity and geometric convergence to the unique minimizer, and Eq. (4.10) bounds that minimizer against the solution of the approximate boundary value problem (4.8). Numerical tests on three synthetic scatterers with 10% noise show reconstructions of location and contrast. The central theoretical claim of global convergence is established only for the approximate model (4.8), not for the original phaseless problem (Problem 2.1), because the zero boundary data on unmeasured faces and the uncontrolled phase-retrieval step introduce model errors that are not quantified in the convergence estimate.","tokens_in":18428,"tokens_out":6795,"duration_ms":73289,"significance":"If the claimed global convergence held for the full phaseless problem, this would be a substantial contribution to a hard and practically relevant inverse problem. The manuscript is honest about two key approximations: Remark 3.3 concedes that the zero data complementation on unmeasured faces is \"not rigorous but rather an approximation,\" and Remark 3.4 describes the truncation and data supplementation as deliberate regularization. The Carleman convexification machinery is largely adapted from the authors' prior work (e.g., refs. 13-15, 32, 33), with Theorem 4.1 stated without a full proof. The genuinely new ingredients are the integration of WKB-based phase retrieval with the convexification pipeline and the numerical demonstration. The significance is therefore moderate: the paper offers a plausible and well-motivated heuristic pipeline, but its advertised global-convergence theorem currently applies to an approximating model whose distance from the original phaseless problem is unquantified.","major_comments":[{"comment":"The concluding claim that gradient descent \"globally converge[s] to the true solution\" overstates what is proven. Theorem 4.1 and estimate (4.10) connect the minimizer of J_{\\lambda,\\epsilon} to v*, the exact solution of the approximate system (4.8), which enforces v_m = 0 on \\partial\\Omega\\setminus\\Gamma via (3.15). Remark 3.3 explicitly labels this supplementation an approximation. Since the boundary term on \\partial\\Omega\\setminus\\Gamma in (4.4) carries a weight \\lambda^4, minimizing the functional cannot correct for a nonzero physical scattered field on those faces. The error estimate (4.10) contains no term for this model error, so it does not support convergence to a solution of Problem 2.1. Please either quantify the model error (for example, by bounding the difference between v* and the true v in terms of the true boundary values on \\partial\\Omega\\setminus\\Gamma) or restrict the convergence claim to the approximate model (4.8).","section":"Section 6; Theorem 4.1; Eq. (4.10)"},{"comment":"The phase-retrieval step is non-convex and is solved by fminunc with no proof or error bound. The functional J_k contains a quadratic term in the Laplacian and a quartic term in |v|; the paper states only that among local minimizers the one closest to the WKB initial guess u^(0) is selected. No theorem guarantees that this minimizer approximates the true wave field, nor is any quantitative error bound provided for the reconstructed phase. Because the Cauchy data g_m, h_m in (3.13)-(3.14) are computed from this phase-retrieval output, a failure of phase retrieval would invalidate the entire pipeline. At minimum, the manuscript should include a numerical study reporting quantitative phase-recovery errors against the ground truth, and ideally an error estimate for J_k under the stated WKB assumptions.","section":"Section 2; Eq. (2.10); Section 5.1"},{"comment":"The noise model in (4.9) does not match the actual algorithmic pipeline. The bound compares g_m, h_m to noiseless data g*_m, h*_m of the approximate problem (4.8), but the numerical g_m and h_m are derived from the phase-retrieval minimization, whose error is uncontrolled and not representable as measurement noise of size \\delta in H^1(\\Gamma) \\times L^2(\\Omega). The second norm in (4.9) is written as L^2(\\Omega) while h_m is defined on \\Gamma, which appears to be a typo. Please clarify the noise model and state explicitly whether the phase-retrieval error is included in \\delta; if it is not, the estimate (4.10) cannot be used as an end-to-end accuracy certificate.","section":"Section 4; Eq. (4.9)"},{"comment":"The numerical validation is too narrow to support the claimed robustness. There are only three test cases, all with inclusions at depth z = -0.65 and all with slab-like geometry; the key parameters N = 7, \\lambda = 1.1, and \\epsilon = 10^{-5.75} are tuned on Test 1 with no sensitivity study. No quantitative error metric is reported for the phase-retrieval step (e.g., relative L^2 errors in Re u and Im u), and no comparison with a baseline method is given. In particular, the load-bearing assumption of zero data on \\partial\\Omega\\setminus\\Gamma is not tested; please compute the true scattered-field magnitude on \\partial\\Omega\\setminus\\Gamma for the synthetic examples and show that it is small relative to the measurements on \\Gamma_L, or otherwise quantify the impact of this approximation.","section":"Section 5.3"}],"minor_comments":[{"comment":"The word \"servere\" should be \"severe.\"","section":"Abstract"},{"comment":"The integrand is written as \"Z_\\Omega e^{2\\lambda(z-r)^2} |\\Delta\\phi|^2, dx\"; the stray comma before \"dx\" should be removed.","section":"Lemma 4.1, Eq. (4.5)"},{"comment":"The line \"Choose Carleman parameters x0, \\beta, and \\lambda\" introduces \\beta without any prior or later definition, and x0 is already the source location fixed in Section 2. Please clarify the intended parameters.","section":"Algorithm 1, Step 3"},{"comment":"The reconstruction formula for c^{comp} contains \"dxd\\theta\" in the integrand, but the integration variable is the wave number k; the expression should be consistent with \\frac{1}{k - \\underline{k}} \\int_{\\underline{k}}^{\\overline{k}} \\Re e[\\ldots] \\, dx\\, dk.","section":"Algorithm 1, Step 6"},{"comment":"The proof of Theorem 4.1 invokes a set H in the convexity inequality (4.6), but H is never defined in the manuscript. Presumably H = B(0,M); please define it explicitly.","section":"Section 4, after Eq. (4.6)"},{"comment":"In Test 2 the relative errors of the two inclusions are 1.61% and 16.78%, respectively; the large asymmetry between two similarly shaped inclusions of contrast 5 and 4.5 is not discussed. A brief comment on this variability would improve the presentation.","section":"Section 5.3.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the authors' own prior work for the Carleman convexification machinery, and the novelty resides in the combination with a WKB-based phase retrieval step. The editor may wish to consider whether the paper's contribution is sufficiently substantial for the journal's standards; however, the numerical results are presented in good faith and the approximations are explicitly acknowledged. The main weakness is the mismatch between the global-convergence claim and the approximate model actually analyzed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Le–Nguyen–Nguyen 3D phaseless Helmholtz paper. The new thing is the combination: WKB phase initialization, then a non-convex phase-refinement fit, Fourier dimension reduction, and Carleman convexification. That specific pipeline, applied to the 3D coefficient inverse problem, is not in the prior literature. The numerical tests show it works on three synthetic inclusions with 10% noise, recovering location and contrast respectably. Credit where due: the authors explicitly flag the data complementation on unmeasured faces as an approximation (Remark 3.3), and the Carleman convexification theorem is standard but technically sound.\n\nThe soft spots are not fatal but they are real. The global convergence claim in Section 6 is for the approximate elliptic system (4.8) with v=0 hard-coded on ∂Ω\\Γ. The error estimate (4.10) bounds the minimizer of the Carleman functional against the solution of that approximate system; it has no term for the model error from zero boundary data or from the phase retrieval in (2.10). The phase recovery itself is a non-convex minimization solved with fminunc; no error bound or convergence guarantee is given. So the paper does not actually prove global convergence for Problem 2.1. It proves convergence to a regularized solution of a nearby problem. That is still useful, but it should be said plainly.\n\nThe numerics are a bit thin: three flat inclusions, hand-tuned N, λ, ε chosen by trial-and-error on Test 1, and no comparison with any competing method or sensitivity analysis. The authors call the approach 'robust' but the evidence base is modest.\n\nWho is this for? Anyone working on numerical methods for phaseless inverse scattering, especially the Carleman convexification school. The paper is a serious attempt and deserves a serious referee. I would recommend engaging with it; the referee should ask the authors to temper the claim of global convergence to the true solution, add an explicit error term for the two approximation layers (phase retrieval and boundary data), and expand the numerical study with baselines and parameter sweeps. On the current evidence it is a conditional accept, not a reject.\n\nBest.","headline":"A plausible three-stage pipeline for 3D phaseless inverse scattering, but the 'global convergence' is proven for an approximate system with zero-filled boundary data, not the original phaseless problem.","tokens_in":19035,"tokens_out":2443,"would_cite":false,"duration_ms":24136,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35J25","65N21","35P25","78A46"],"pacs":[],"model":"deepseek-v4-flash","headline":"Phaseless 3D scattering solved by WKB phase recovery, Carleman convexification","keywords":["phaseless inverse scattering","Carleman convexification","WKB approximation","phase retrieval","Helmholtz equation","frequency dimension reduction","global convergence","dielectric constant reconstruction"],"falsifier":"Run the algorithm on synthetic data in which the scattered wave on ∂Ω\\Γ is deliberately made non-negligible (for example, by placing a strong scatterer close to the top face or by moving the source closer to the side), then compare the reconstructed c(x) against the known true profile; a large contrast error concentrated near the top and side regions would show that the zero-boundary approximation, rather than the Carleman convexification step, is the limiting factor.","tokens_in":17961,"feed_emoji":"📡","tokens_out":1921,"duration_ms":21805,"temperature":0.7,"pith_summary":"This paper claims a fully numerical route to reconstruct a 3D dielectric profile from intensity-only backscattered data of a single point source. The authors argue that by first recovering the lost phase through a WKB-informed optimization, then collapsing the frequency dimension with a Fourier expansion, and finally solving the resulting elliptic Cauchy system by Carleman-weighted least squares, one obtains a cost functional that is strictly convex and whose global minimizer approximates the true coefficient. Their stated result is that gradient descent on that functional converges globally, and their numerical tests on noisy synthetic data recover both the location and the contrast of buried scatterers. The reason to care is that phaseless inverse scattering is classically nonlinear and severely ill-posed, and a provably convergent, linearization-free method would make intensity-only imaging practical in settings where phase cannot be measured.","feed_headline":"Intensity-only data recover 3D dielectric profiles","feed_subtitle":"WKB phase retrieval plus Carleman convexification solves phaseless scattering with global convergence.","key_machinery":"The carrying object is the Carleman convexification functional J_{λ,ε}(φ) = Σ_m [∫_Ω $e^{{2λ(z-r)^2}}$ |Σ_n s_{mn} Δφ_n + Σ_{n,l} a_{mnl} ∇φ_n·∇φ_l + Σ_n b_{mn}·∇φ_n|^2 dx + boundary penalty terms on Γ and ∂Ω\\Γ + ε ||φ_m||^2_{H^s(Ω)}], built from the Fourier-truncated logarithmic transform of the wave field. Its job is to convert a non-convex, ill-posed inverse scattering problem into a strictly convex one: the Carleman weight $e^{{2λ(z-r)^2}}$, the boundary data terms, and the ε-regularization together make the second derivative of J_{λ,ε} uniformly positive on the ball B(0,M), so that the unique global minimizer can be found by plain gradient descent. The polynomial-exponential basis {Ψ_n} matters because its first element is not constant, so the principal term Δv_1 survives in the elliptic system, which Legendre or trigonometric bases would lose.","core_discovery":"The paper's central assertion is that phaseless inverse scattering from a single point source can be turned into a strictly convex optimization problem through a three-stage pipeline: (i) reconstruct the complex wave field on a bottom measurement layer by minimizing the functional J_k(v) = ||Δv + $k^{2}$ v||^2 + |||v|^2 - $f^{2}$||^2, initialized with the WKB travel-time guess $u^{{(0)}}$ = f $e^{{ik|x-x0|}}$; (ii) transform u into the logarithmic variable v = $k^{{-2}}$ log(u/u_inc), expand v in the polynomial-exponential basis {Ψ_n} of $L^{2}$(k̲,k̄), truncate at N, and thereby replace the original inverse problem by an elliptic system with Cauchy data on the bottom face and zero data on the remaining faces; and (iii) minimize the Carleman-weighted functional J_{λ,ε} whose weights $e^{{2λ(z-r)^2}}$ enforce strict convexity inside a bounded admissible ball. The Carleman convexification theorem then guarantees that J_{λ,ε} has a unique global minimizer, that gradient descent converges to it from any point in the ball, and that the minimizer approximates the true solution with error O(δ + √ε), where δ is the data noise level. Numerical examples with 10% noise show reconstruction of one and two inclusions with contrast errors around 1.6 to 17 percent, which the authors present as evidence of robustness under single-sided, magnitude-only measurements.","pith_inferences":["The neglected boundary approximation, in which the scattered wave is set to zero on the unmeasured top and side faces of Ω, is the most likely source of systematic error in practice; a testable extension would be to replace v_m = 0 on ∂Ω\\Γ by a Robin-type or extrapolated boundary condition and measure the change in reconstruction error as the source distance d grows.","The method's reliance on multi-frequency data over a finite band [π, 2π] with N = 7 Fourier modes suggests an implicit low-frequency regularization; one could test whether the same pipeline works with a narrower band or with fewer frequencies, which would reveal how much of the contrast recovery comes from the phase retrieval versus the convexification step.","The WKB phase retrieval step and the Fourier truncation are both justified at high frequency, yet the synthetic tests use modest frequencies (k around π to 2π); a natural stress test is to run the same algorithm at lower wave numbers to see where the eikonal phase assumption breaks down in terms of contrast error.","The authors position this as a stepping stone toward the full Maxwell system; a direct extension would apply the same logarithmic and Carleman machinery to vector-valued fields, although the boundary zero-filling approximation would be harder to justify there because of polarization coupling."],"forward_implications":["If the central claim holds, intensity-only backscattering data from a single point source suffice to recover both the geometry and the dielectric contrast of buried objects, without any linearization step.","The O(δ + √ε) error estimate means the reconstruction error can be driven toward the data noise level by sending the regularization parameter ε to zero, making the method a stable regularization rather than a heuristic fit.","Because gradient descent converges globally on the Carleman-weighted functional, the method needs no close initial guess: the travel-time WKB guess merely informs the phase retrieval step, and the convexification step still succeeds from a simplified starting point generated by dropping the nonlinear term.","The framework transfers directly to other coefficient inverse problems that can be rewritten as elliptic systems with Cauchy data, as the authors explicitly intend for Helmholtz-type problems in higher regularity settings.","A practical consequence is that high-frequency imaging systems, where phase is hard or impossible to measure, can rely on magnitude-only sensors placed on a single side of the target domain."],"supporting_citations":[{"why":"Introduces the original Carleman convexification idea for inverse scattering, supplying the strict-convexity mechanism that Theorem 4.1 builds on.","marker":"[20]"},{"why":"Applies convexification to a 3D inverse scattering problem with a moving point source, providing the template for the cost functional and the gradient-descent convergence analysis.","marker":"[15]"},{"why":"Provides the gradient descent convergence result and the admissible-ball framework that Theorem 4.1, part 3, directly cites.","marker":"[33]"},{"why":"Constructs the polynomial-exponential basis {Ψ_n} and proves the invertibility of the matrix S, which is essential for the elliptic system formulation.","marker":"[18]"},{"why":"Supplies the Carleman estimate variant and the proof technique used for the convexity inequality and error estimate in Theorem 4.1 and (4.10).","marker":"[32]"},{"why":"Gives the refined rigorous justification of the WKB ansatz for the Helmholtz equation, which the paper cites as the theoretical basis for the phase initial guess.","marker":"[28]"},{"why":"Justifies the complex logarithm in the change of variables (3.1) and supplies the earlier algorithmic approach that the Fourier-dimension-reduction step follows.","marker":"[24]"},{"why":"Provides the volume integral equation method used to generate the synthetic forward data in the numerical experiments.","marker":"[34]"}],"fun_headline_variants":["Phaseless scattering tamed by Carleman-WKB combo","Single-point phaseless data reconstruct 3D dielectrics","Carleman convexification solves phase-free inverse scattering","WKB ansatz recovers phases for Carleman-stabilized inversion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The scattered wave is assumed to be negligible on the top and side faces of the computational cube, so the boundary data there are set to zero; if the scattered field on those unmeasured faces is not small, the reconstruction carries a systematic error that the paper's error estimate does not cover.","fun_headline_variants_meta":{"raw":{"variants":["Phaseless scattering tamed by Carleman-WKB combo","Single-point phaseless data reconstruct 3D dielectrics","Carleman convexification solves phase-free inverse scattering","WKB ansatz recovers phases for Carleman-stabilized inversion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000351,"raw_usage":{"total_tokens":1983,"prompt_tokens":1081,"completion_tokens":902,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":697,"completion_tokens_details":{"reasoning_tokens":828}},"tokens_in":697,"tokens_out":902,"duration_ms":8498,"temperature":1.0,"reasoning_tokens":828,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:21:19.743407+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the algorithm on synthetic data in which the scattered wave on ∂Ω\\Γ is deliberately made non-negligible (for example, by placing a strong scatterer close to the top face or by moving the source closer to the side), then compare the reconstructed c(x) against the known true profile; a large contrast error concentrated near the top and side regions would show that the zero-boundary approximation, rather than the Carleman convexification step, is the limiting factor.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the original Carleman convexification idea for inverse scattering, supplying the strict-convexity mechanism that Theorem 4.1 builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Applies convexification to a 3D inverse scattering problem with a moving point source, providing the template for the cost functional and the gradient-descent convergence analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the gradient descent convergence result and the admissible-ball framework that Theorem 4.1, part 3, directly cites."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the polynomial-exponential basis {Ψ_n} and proves the invertibility of the matrix S, which is essential for the elliptic system formulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Carleman estimate variant and the proof technique used for the convexity inequality and error estimate in Theorem 4.1 and (4.10)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the refined rigorous justification of the WKB ansatz for the Helmholtz equation, which the paper cites as the theoretical basis for the phase initial guess."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Justifies the complex logarithm in the change of variables (3.1) and supplies the earlier algorithmic approach that the Fourier-dimension-reduction step follows."},{"cited_title":"Lechleiter and D.-L","cited_arxiv_id":null,"evidence_quote":"Provides the volume integral equation method used to generate the synthetic forward data in the numerical experiments."}],"review_version":1}