{"id":"55e3f8a2-50af-4c84-8dfc-456f55f6d9fb","arxiv_id":"2507.01789","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"A multi-frequency regularization method and a spectral covariance method are proposed to recover source terms in stochastic Helmholtz and finite-jump Lévy-driven wave equations from final-time observations.","lead":"This paper proposes a multi-frequency Tikhonov regularization scheme for source strength recovery in a one-dimensional stochastic Helmholtz equation without attenuation, and a spectral method to reconstruct two source terms in stochastic wave equations driven by finite-jump Lévy processes from final-time data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Helmholtz inversion is implemented with kernel cos²(2ν|x−y|), while the stochastic model yields kernels cos²(ν|x−y|) and cos(2ν|x−y|); the numerical experiments therefore do not test the claimed physics.","rationale":"The paper has two advertised contributions: a computational remedy for the ill-posed stochastic Helmholtz inverse source problem, and a reconstruction method for the Lévy-driven stochastic wave equation. The Helmholtz contribution is the more load-bearing because the abstract's claim of developing a methodology to reconstruct the source hinges on the numerical scheme of Section 3 being a valid discretization of the actual forward map. I re-derived (3.11) and (3.12) from (3.8); they are correct. The subsequent substitution of cos²(2ν|x−y|) is not justified and is not a discretization of either kernel. Definition 2 then formalizes a forward operator that does not match the stochastic Helmholtz equation. Therefore the uniqueness, stability, and convergence theorems in Section 3.1.2 concern a different operator, and the reported reconstructions do not establish the paper's central claim. The Lévy section also has a genuine issue: formula (4.15) determines only products g_k g_l, so the sign of g is not identified, contradicting Theorem 8(2); Remark 7 admits this and says the code performs phase correction using prior information. This is a further correctness problem, but the Helmholtz kernel mismatch is sufficient on its own to invalidate the main positive claim. I do not see a need to change the reader's verdict: REJECT remains appropriate. If the numerical test I propose were to show that the mismatch is immaterial in practice, the Helmholtz objection would weaken, but based on the text there is no such evidence.","tokens_in":25064,"tokens_out":4738,"duration_ms":53663,"concrete_test":"Rerun the Helmholtz experiment exactly as in Section 3.1.2 but generate Hobs(xi, νk) using the physical stochastic Helmholtz model, i.e. via H1 from (3.11) or H2 from (3.12) with the true µ, and invert with the Definition 2 operator A using kernel cos²(2ν|xi−yj|). If the reconstruction error remains at the reported level (≈1e-3 for single-mode µ), the mismatch is harmless; if the inversion fails or the error grows by orders of magnitude, the numerical validation is invalid. Also inspect the data-generation code to determine which kernel was used to create Hobs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.1.1, the authors correctly derive from u(x) = (1/2iν)∫_M e^{iν|x−y|}√µ(y) dW(y) that H1(x,ν,µ) = E|Re(2iνu(x))|² = ∫_M cos²(ν|x−y|)µ(y) dy (3.11) and H2(x,ν,µ) = E|Re(2iνu(x))|² − E|Im(2iνu(x))|² = ∫_M cos(2ν|x−y|)µ(y) dy (3.12). The very next step, however, replaces these by the discretization H(x,ν,µ) ≈ ∆y Σ_j cos²(2ν|x−y_j|)µ(y_j), and Definition 2 defines the forward operator Tν with kernel cos²(2ν|x−y|). No argument shows that (3.11) or (3.12) equals this kernel; cos²(2νr) = (1+cos(4νr))/2 is not an approximation of cos²(νr) or cos(2νr). Consequently, if the observed data Hobs is generated from the true stochastic Helmholtz equation, it is not related to the discretized operator used in the inversion, and Theorems 2–6 (compactness, ill-posedness, uniqueness, stability, convergence) do not apply to the actual inverse problem. If, instead, Hobs was generated directly from cos²(2ν|x−y|), the numerical experiments solve a synthetic integral equation that never tests the stochastic Helmholtz inverse source problem. Either way, the central claim that multi-frequency data fusion and Tikhonov regularization mitigates the ill-posedness of the stochastic Helmholtz inverse source problem is unsupported. This concern is independent of the Lévy section, where Theorem 8(2)'s uniqueness claim also conflicts with the sign ambiguity admitted in Remark 7.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper treats two inverse source problems. For the one-dimensional stochastic Helmholtz equation without attenuation, with random source f = sqrt(mu) W', it proposes to reconstruct the strength mu from second-moment data of the wave field at several frequencies, using a feature function, multi-frequency data fusion, and Tikhonov regularization. For the stochastic wave equation driven by a finite-jump Lévy process, it proves a stability estimate for the mild solution and derives spectral formulas for recovering the deterministic source f and the spatial factor g from final-time data u(x,T). The paper contains theoretical statements on ill-posedness, uniqueness, stability, and convergence, together with numerical experiments for both problems.","tokens_in":1694,"tokens_out":1686,"duration_ms":109341,"significance":"If the central claims were correct, the paper would offer a genuinely new computational route to the no-attenuation Helmholtz inverse source problem and one of the first reconstruction schemes for a Lévy-driven stochastic wave equation. The direct stability estimate for the Lévy problem (Theorem 7) appears sound and is a useful contribution. However, the Helmholtz numerical framework as implemented does not invert the operator derived from the stochastic model, and the uniqueness statement for g in the Lévy section is overclaimed, as the paper itself partially acknowledges in Remark 7. In its current form the manuscript does not support its principal advertised conclusions.","major_comments":[{"comment":"Equations (3.11) and (3.12) correctly derive H1(x,ν,µ) = ∫ cos²(ν|x−y|)µ(y)dy and H2(x,ν,µ) = ∫ cos(2ν|x−y|)µ(y)dy. The next display, however, approximates H by Δy Σ cos²(2ν|x−y_j|)µ(y_j), and Definition 2 defines the forward operator Tν with kernel cos²(2ν|x−y|). Since cos²(2νr) is not an approximation of cos²(νr) or cos(2νr), there is no chain of reasoning connecting the physical data to the operator being inverted. Consequently, the numerical reconstructions in Section 3.1.3 and the theorems built on Definition 2 (Theorems 2–6) concern a different integral operator, and the paper's central claim that multi-frequency fusion mitigates the ill-posedness of the no-attenuation stochastic Helmholtz inverse problem is unsupported as written.","section":"3.1.1, Definition 2, 3.1.3"},{"comment":"Formula (4.15) determines the products g_k g_l, so for every solution g the function −g gives exactly the same covariance data. Theorem 8(2) therefore cannot be correct as stated: the source g is not uniquely determined by Cov(u_k(T), u_l(T)). Remark 7 concedes the sign ambiguity and indicates that the code applies a phase correction using prior information, but this limitation is not reflected in the abstract, in the statement of Theorem 8, or in the numerical claims. The final-time covariance data can at best determine g up to a global sign unless additional information is supplied.","section":"4.2.2, Theorem 8(2), Remark 7"},{"comment":"The theorem states the stability constant C(α) = 1/√α, but the continuous proof derives ∥µ_{α,1}−µ_{α,2}∥ ≤ (1/α)(Σ_k ∥T_{ν_k}∥)δ. The discrete proof first obtains (σ_max(A)/α)δ and then asserts the SVD bound δ/(2√α), so the rate 1/√α is not established by the arguments given. The statement and proof need to be aligned, since the quantitative stability estimate is one of the advertised theoretical results.","section":"Theorem 5"},{"comment":"The numerical reconstruction of f is inconsistent with formula (4.14). Equation (4.14) divides by ∫_0^T h(τ) sin((T−τ)√λ_k)/√λ_k dτ, while the numerical section writes \"the theoretical relationship\" with ar U_k equal to f_k times Σ_j A_k(t_j)Δt and then defines f^rec_k = ar U_k. Unless that time integral equals 1, which is not stated, the computed f_k are not the modal coefficients of f. This needs to be corrected or the normalization made explicit.","section":"4.2.3"},{"comment":"The claimed proof that I_{kl} ≠ 0 for every algebraic T is not valid. The Lindemann–Weierstrass/Vandermonde argument concerns a family of algebraically independent parameters T_j, not a fixed algebraic T, and it does not establish the desired non-degeneracy. Moreover, for T = π (an algebraic number), I_{kl} = 0 for all k ≠ l, so the statement \"for any algebraic T, I_{kl} ≠ 0\" is false. The numerical setting uses T = 1, so the non-degeneracy of the covariance kernel used in the reconstruction is not proved.","section":"4.2.2, proof of Theorem 8(2)"}],"minor_comments":[{"comment":"Several occurrences of \"2viu\" should read \"2iνu\"; the same typo appears in the sentences defining H1 and H2.","section":"3.1.1"},{"comment":"Definition 2 states that Tν maps L²([0,1]) into L²([0,1]), but the numerical design places observation points on I = [−1.2,−0.2] ∪ [1.2,2.2]. The theoretical framework should either use a data space L²(I) or explain how the kernel and observations are extended.","section":"Definition 2 vs. 3.1.1"},{"comment":"The null-space example µ(y) = cos(8νy) is not shown to satisfy Tνµ = 0 at ν = π/2; the displayed integral does not vanish by the argument given. Replace it with a correct construction, e.g., functions orthogonal to 1, cos(4νy), and sin(4νy).","section":"Theorem 2(2)"},{"comment":"The text says the jump time interval follows a Poisson distribution; in a Poisson process the number of jumps in an interval is Poisson-distributed while the interarrival times are exponential. The wording should be corrected.","section":"Assumption 2"},{"comment":"The figure captions are highly repetitive and do not identify which subplot corresponds to which true strength function or which noise level. Please expand the captions and label the axes.","section":"Figures 1–6"}],"recommendation":"reject","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has two halves with very different fates. The Lévy half is a genuine contribution: the inverse source problem for stochastic wave equations driven by finite-jump Lévy processes hasn't been treated before, and the direct stability estimate in Theorem 7 is derived cleanly via spectral decomposition and appears correct. The reconstruction formulas (4.14)–(4.15) are reasonable, and Remark 7 correctly identifies the sign ambiguity in g, though that admission undercuts the literal uniqueness claim in Theorem 8(2). The multi-frequency Tikhonov framework for the no-attenuation Helmholtz case is also new in this setting, but the implementation has a load-bearing problem: equations (3.11)–(3.12) derive kernels cos²(ν|x−y|) and cos(2ν|x−y|), and the text then replaces these with cos²(2ν|x−y|) with no justification. The numerical experiments generate data from that cos²(2ν|x−y|) operator and invert it, so they validate a synthetic integral equation, not the stochastic Helmholtz inverse source problem. Theorems 2–6 are proved for the wrong operator. This is not a minor typo; it breaks the Helmholtz claims. The Lévy numerics are more honest (they simulate the forward model and then invert), and the reported errors are plausible, with the sign correction flagged as extra prior information. The writing is rough and there are typos, but that's not the issue. Bottom line: the Lévy part deserves publication after revisions, and the Helmholtz part needs to either fix the kernel consistency or be reframed as a numerical study of a different integral equation. As it stands the paper shouldn't be accepted, but it's not a waste of time—the Lévy half is worth reading. I'd send it to a referee if I were the editor, mainly to get the Lévy part in shape and force the Helmholtz issue to be addressed.","headline":"The Lévy half is a real, salvageable contribution, but the Helmholtz numerical experiments invert an operator that does not match the derived stochastic model, breaking the paper's central claim.","tokens_in":26057,"tokens_out":3768,"would_cite":false,"duration_ms":42058,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35R60","60H15","65J20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the two source terms of a stochastic wave equation driven by a finite-jump Lévy process can be reconstructed from the final-time wave field, and that multi-frequency Tikhonov regularization mitigates the ill-posed…","keywords":["stochastic wave equation","inverse source problem","stochastic Helmholtz equation without attenuation","finite-jump Lévy process","ill-posedness","Tikhonov regularization","multi-frequency data fusion","final-time data"],"falsifier":"Generate synthetic observations by solving the actual stochastic Helmholtz equation (2.1)--(2.2) with a known compactly supported $\\mu$, then feed the second-moment data into the paper's multi-frequency inversion with the $\\cos^2(2\\nu|x-y|)$ kernel; if the recovered $\\mu$ differs from the truth beyond the reported error levels, the numerical scheme is not inverting the physical forward map. For the Lévy equation, run the covariance-only reconstruction (4.15) on a sign-changing $g$ without phase correction; the paper's Remark 7 predicts it will return $|g|$ rather than $g$.","tokens_in":24826,"feed_emoji":"🌊","tokens_out":13303,"duration_ms":132909,"temperature":0.7,"pith_summary":"The paper tries to show that two ill-posed stochastic inverse source problems become numerically tractable when the right statistical features of the wave field are used. For the one-dimensional stochastic Helmholtz equation without attenuation, it argues that second moments alone give only the average source strength, but a feature function built from real and imaginary parts, combined with multi-frequency data fusion and Tikhonov regularization, can recover the strength point by point. For the wave equation driven by a finite-jump Lévy process, it claims that final-time data $u(x,T)$ determine the deterministic source $f$ through the mean of the modal coefficients and the random source $g$ through their covariance, up to the sign of $g$. If these claims hold, sources in settings with jump-type stochastic fluctuations, such as seismic or financial-volatility models, could be characterized from a single final observation.","feed_headline":"Recover both wave sources from final-time data","feed_subtitle":"Modal means and covariances of u(x,T) separate f and g; multi-frequency regularization covers the Helmholtz case.","key_machinery":"The machinery is mode-by-mode statistical inversion. In the Helmholtz half, the operative object is the integral operator $T_\\nu$ with kernel $K_\\nu(x,y)=\\cos^2(2\\nu|x-y|)$; the paper proves this operator is compact and self-adjoint, so a single frequency is ill-posed, while a dense set of frequencies $\\{4\\nu_k\\}$ forces the null space to be trivial and makes the Tikhonov-regularized fit well-posed and $O(\\sqrt{\\delta})$-convergent. In the Lévy half, the Laplace eigenbasis diagonalizes the wave equation, so each mode $u_k(t)$ solves $u_k''+\\lambda_k u_k=f_k h(t)+g_k\\dot L_t$; the mean $\\mathbb E[u_k(T)]$ and covariance $\\mathrm{Cov}(u_k,u_l)$ become explicit linear expressions in $f_k$ and $g_kg_l$, inverted through (4.14)--(4.15). The covariance kernel $I_{kl}=\\int_0^T \\sin(k(T-\\tau))\\sin(l(T-\\tau))/(kl)\\,d\\tau$ is shown to be nonzero for algebraic $T$, which is what carries the uniqueness claim for $g$.","core_discovery":"On the paper's own terms, the discovery is that the obstruction to inverse source recovery in these stochastic settings is not the randomness itself but the choice of statistical data and the numerical treatment of the resulting operators. For the no-attenuation Helmholtz equation $\\Delta u+\\nu^2 u=f$ with $f=\\sqrt{\\mu}\\,W'$, the variance $\\mathbb E|u(x)|^2$ depends only on $\\int_M \\mu\\,dy$, so the paper defines the feature functions $H^1(x,\\nu,\\mu)=\\mathbb E|\\mathrm{Re}(2\\mathrm i\\nu u)|^2=\\int_M \\cos^2(\\nu|x-y|)\\mu(y)\\,dy$ and $H^2=\\mathbb E|\\mathrm{Re}(2\\mathrm i\\nu u)|^2-\\mathbb E|\\mathrm{Im}(2\\mathrm i\\nu u)|^2=\\int_M \\cos(2\\nu|x-y|)\\mu(y)\\,dy$, then discretizes with kernel $\\cos^2(2\\nu|x-y|)$ and inverts with multi-frequency Tikhonov regularization. For the Lévy-driven equation $u_{tt}-\\Delta u=f(x)h(t)+g(x)\\dot L_t$, the paper establishes the stability estimate (4.13) for the mild solution and derives the reconstruction formulas (4.14) and (4.15): the modal means give $f_k$ when the drift is zero, and the modal covariances give $g_kg_l$ up to sign, with uniqueness of $f$ under monotonicity conditions and of $g$ for algebraic $T$.","pith_inferences":["One could test whether the Helmholtz inversion still works with the physically derived kernels $\\cos^2(\\nu|x-y|)$ and $\\cos(2\\nu|x-y|)$ in place of $\\cos^2(2\\nu|x-y|)$; the paper's uniqueness and regularization theorems do not directly cover those operators.","Because the covariance data are insensitive to the sign of $g$, an immediate extension is to couple the final-time covariance with third-order moments or a known deterministic source term to break the sign symmetry; the paper notes the ambiguity but does not resolve it.","The same spectral-covariance scheme should extend to infinite-jump Lévy processes by truncating or renormalizing the jump measure; the paper lists this as future work, and the stability estimate suggests control is retained when $\\lambda_p\\sigma_j^2$ stays finite.","The dense-frequency condition in the Helmholtz half is stated for $\\{4\\nu_k\\}$ dense in $[0,\\infty)$; in higher dimensions the same idea would need a different oscillatory-integral argument, since the one-dimensional $|x-y|$ kernel structure does not transpose directly."],"forward_implications":["Under the dense-frequency uniqueness theorem, the multi-frequency Helmholtz inverse problem has a unique regularized solution that converges to the true strength at rate $O(\\sqrt{\\delta})$ as the data noise $\\delta\\to0$.","The numerical experiments indicate that 3--4 frequency data points suffice for single-Fourier-mode strengths and about 10 for multi-mode strengths, compared with the 16 frequencies used in the attenuated case.","In the Lévy-driven equation, the modal means recover the deterministic source $f$ exactly when the drift $b=0$, and the modal covariances recover the random source $g$ up to sign, with relative $L^2$ errors of order $10^{-3}$ for simple modes and $10^{-2}$ for multi-mode reconstructions.","The stability estimate (4.13) bounds the expected space-time energy of the mild solution by the $L^2$ norms of $f$ and $g$, so the direct problem is mean-square well posed whenever the Gaussian-Poisson jump parameters are finite."],"supporting_citations":[{"why":"Supplies the regularity lemma for the microlocal isotropic Gaussian random field used to justify the Helmholtz forward solution.","marker":"[29]"},{"why":"Supplies the boundedness lemma for the volume potential and the attenuated-case numerical baseline (16 frequencies) that the new Helmholtz scheme is compared with.","marker":"[28]"},{"why":"Supplies the final-time-data mean/covariance reconstruction strategy and the uniqueness proof template for $f$ used in Theorem 8.","marker":"[20]"},{"why":"Supplies the Lévy--Itô decomposition and Campbell's theorem used to split the mild solution and prove the stability estimate (4.13).","marker":"[3]"},{"why":"Supplies the Lindemann--Weierstrass theorem used to prove covariance-based uniqueness of $g$ for algebraic $T$.","marker":"[4]"},{"why":"Supplies the compact self-adjoint operator facts used to establish single-frequency ill-posedness of the Helmholtz operator.","marker":"[31]"},{"why":"Supplies the regularization convergence and stability framework used in Theorems 5 and 6 for the Tikhonov solutions.","marker":"[18]"}],"fun_headline_variants":["Final-time covariance data recover stochastic wave sources","New framework solves stochastic wave source recovery from final data","Jump-type noise models yield deterministic source reconstruction","Modal statistics unmask sources in stochastic wave equations","Covariance of final wave field identifies both source terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Helmholtz reconstruction presupposes that the feature function with kernel $\\cos^2(2\\nu|x-y|)$ used in the discretization and Definition 2 is the forward map that generated the observations; the paper's own derivation from the wave equation yields kernels $\\cos^2(\\nu|x-y|)$ and $\\cos(2\\nu|x-y|)$ instead, so the numerical operator and the physical model may be inconsistent. The Lévy reconstruction additionally presupposes that the Lévy parameters $(b,\\sigma,\\lambda_p,\\sigma_j^2)$ are known and that prior information fixes the sign of $g$, since formula (4.15) determines only the product $g_kg_l$ and hence $|g|$.","fun_headline_variants_meta":{"raw":{"variants":["Final-time covariance data recover stochastic wave sources","New framework solves stochastic wave source recovery from final data","Jump-type noise models yield deterministic source reconstruction","Modal statistics unmask sources in stochastic wave equations","Covariance of final wave field identifies both source terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1556,"prompt_tokens":1084,"completion_tokens":472,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":700,"completion_tokens_details":{"reasoning_tokens":400}},"tokens_in":700,"tokens_out":472,"duration_ms":6474,"temperature":1.0,"reasoning_tokens":400,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:43:54.275111+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate synthetic observations by solving the actual stochastic Helmholtz equation (2.1)--(2.2) with a known compactly supported $\\mu$, then feed the second-moment data into the paper's multi-frequency inversion with the $\\cos^2(2\\nu|x-y|)$ kernel; if the recovered $\\mu$ differs from the truth beyond the reported error levels, the numerical scheme is not inverting the physical forward map. For the Lévy equation, run the covariance-only reconstruction (4.15) on a sign-changing $g$ without phase correction; the paper's Remark 7 predicts it will return $|g|$ rather than $g$.","supporting_citations":[{"cited_title":"Inverse random source scattering for the helmholtz equation with attenuation","cited_arxiv_id":null,"evidence_quote":"Supplies the regularity lemma for the microlocal isotropic Gaussian random field used to justify the Helmholtz forward solution."},{"cited_title":"An inverse random source problem for the one-dimensional helmholtz equation with attenuation","cited_arxiv_id":null,"evidence_quote":"Supplies the boundedness lemma for the volume potential and the attenuated-case numerical baseline (16 frequencies) that the new Helmholtz scheme is compared with."},{"cited_title":"An inverse source problem for the stochastic wave equation","cited_arxiv_id":"2101.04744","evidence_quote":"Supplies the final-time-data mean/covariance reconstruction strategy and the uniqueness proof template for $f$ used in Theorem 8."},{"cited_title":"Transcendental number theory","cited_arxiv_id":null,"evidence_quote":"Supplies the Lindemann--Weierstrass theorem used to prove covariance-based uniqueness of $g$ for algebraic $T$."},{"cited_title":"Methods of modern mathematical physics: Functional anal- ysis, volume 1","cited_arxiv_id":null,"evidence_quote":"Supplies the compact self-adjoint operator facts used to establish single-frequency ill-posedness of the Helmholtz operator."}],"review_version":1}