{"id":"3f523771-ae96-429f-8776-fd48241f4945","arxiv_id":"2504.19421","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a coupled diffusion fluorescence model, the source is unique and Lipschitz stable from terminal data, and a two-stage regularized inversion from discrete noisy sensors has explicit probabilistic convergence rates.","lead":"Diffusion-based fluorescence imaging is modeled by two coupled heat-like equations, and this paper recovers the unknown fluorophore absorption from one noisy final-time image. The authors prove uniqueness and stability for clean data and give probabilistic error bounds for reconstructions from discrete noisy sensors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No existence proof for the noisy fixed point qσ in D; P1 reconstructions can leave the admissible set, so Theorems 3.2 and 3.4 rest on an unverified hypothesis.","rationale":"The reader's weakest_assumption identifies exactly the gap I would treat as load-bearing: the noisy fixed point qσ is assumed to exist in D, and no argument shows that the P1 reconstructions (fσ, Sfσ) inherit the admissible-set conditions of Assumption 2.1. My reading of Section 2 confirms that D and the positivity of -Δg + pg are used essentially in Lemma 2.6 and Theorem 2.2. Once fσ + pSfσ replaces -Δg + pg in (3.4), neither the monotonicity of Kσ nor the stability estimate for (qσ, q∗) follows from the given hypotheses. The paper's numerical experiments are suggestive but not dispositive, since they do not test the boundary of the admissible set. This is a fixable gap rather than a demonstrated falsehood, so the reader's CONDITIONAL verdict is appropriate. I would not change the verdict.","tokens_in":24822,"tokens_out":5872,"duration_ms":63000,"concrete_test":"Run the numerical setting of Example 4.2 with n = 500 and the 10% noise level used in Figure 11, for a fixed noise realization. Solve (3.2) for fσ and Sfσ, compute qσ0 = (fσ + pSfσ)/ue(x,T;0), and check pointwise whether q0 ≤ qσ0 ≤ M, where q0 = (-Δg + pg)/ue(x,T;0). Then iterate (3.3) and record whether the iterates remain in D and whether a fixed point is reached. Repeat over 100 independent noise draws and report the fraction of draws for which qσ0 ∈ D and all iterates stay in D; if that fraction is not 1, or is not bounded by a probability matching the tail rate in Theorem 3.4, the theorem's existence hypothesis is violated in the paper's own experimental regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The probabilistic claims in Theorems 3.2 and 3.4 are stated for qσ, a fixed point of the noisy iteration (3.3), but the paper never proves that such a fixed point exists in the domain D defined by (2.2). This is not a minor technicality: D is precisely the set on which the deterministic monotone-convergence and stability results of Section 2 operate, and D is defined through the exact quantities -Δg and g. Starting from P1, one only has qσ0 = (fσ + p Sfσ)/ue(x,T;0). For qσ0 to lie in D one needs the pointwise inequalities (-Δg + p g)/ue(x,T;0) ≤ qσ0 ≤ M, equivalently fσ + p Sfσ ≥ -Δg + pg and fσ + p Sfσ ≤ M ue(x,T;0). The unconstrained minimization (3.2) imposes no such bounds on fσ, and the stated Gaussian/sub-Gaussian noise model does not rule out realizations for which these inequalities fail. If qσ0 leaves D, the sign analysis of Lemma 2.6 does not carry over to Kσ: the proof requires ∂tum(q2) + fσ + pSfσ ≥ 0, replacing the exact positivity of -Δg + pg guaranteed by Assumption 2.1(c) with an unverified property of the reconstruction. Consequently Theorem 2.2, whose hypotheses include q̂, q̃ ∈ D and whose proof uses those sign properties, cannot be invoked for the pair (qσ, q∗) without an additional argument. The advertised exponential tails and expectation rates therefore rest on an existence-and-regularity assumption that is neither stated as an assumption nor proved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the recovery of the fluorophore absorption coefficient q in a coupled excitation–emission diffusion system from the terminal-time measurement g(x)=u_m(x,T). Section 2 defines a monotone operator K in (2.1) on a domain D and proves, under Assumption 2.1 and a smallness condition, that the fixed-point iteration (2.7) converges monotonically to the unique fixed point in D (Theorem 2.1), and derives two conditional Lipschitz stability estimates (Theorem 2.2). Section 3 treats discrete noisy observations by splitting the problem into P1, a Tikhonov recovery of f*=-Δg and Sf*=g via (3.2), and P2, a fixed-point iteration with the noisy operator Kσ in (3.4). Theorems 3.1 and 3.3 give expectation and exponential-tail error estimates for P1; Theorems 3.2 and 3.4 state analogous convergence rates for qσ relative to the exact q*. Section 4 reports two-dimensional numerical experiments for smooth and discontinuous sources.","tokens_in":25156,"tokens_out":21519,"duration_ms":214122,"significance":"If established, the deterministic part is a clean conditional-stability result for a nonlinear coupled-system inverse problem, and the stochastic part would provide practically useful explicit rates with exponential tails. The paper has clear strengths: the monotone fixed-point construction is elegant, the smallness condition in Theorem 2.2 is explicit, the use of empirical-process tools is appropriate, and the numerical experiments are extensive and reproduce the predicted P1 rates. However, the advertised P2 estimates in Theorems 3.2 and 3.4 rest on an unproved existence, regularity, and membership hypothesis for the noisy fixed point qσ, and the derivation combining the deterministic and stochastic estimates is incomplete. The central claims of the paper are therefore not yet supported as stated.","major_comments":[{"comment":"The theorems are stated for a fixed point qσ of the noisy iteration (3.3), but neither the existence of such a fixed point nor its membership in the domain D of (2.2) is proved. The initial value qσ0=(fσ+p(x)Sfσ)/u_e(x,T;0) is only known to lie in H^s(Ω); for s=0 (and for s=1 when d=3) H^s is not embedded in C(Ω̄), so qσ0 may fail to be continuous, while D is a subset of C(Ω). Even when qσ0 is continuous, membership in D would require the pointwise inequalities (-Δg+pg)≤fσ+pSfσ≤M u_e(x,T;0), and the unconstrained minimization (3.2) imposes no such bounds. Moreover, the sign analysis in Lemma 2.6 for Kσ would require ∂t u_m(x,T;q2)+fσ+pSfσ≥0, a property that is not inherited from Assumption 2.1(c). Consequently Theorem 2.2, whose hypotheses include q̂,q̃∈D and whose proof uses those sign properties, cannot be invoked for the pair (qσ,q*). The paper also does not prove that the iteration (3.3) converges to qσ, so the output of Algorithm 2 is not covered by the theorem. This is a load-bearing gap: the expectation and exponential-tail estimates in Theorems 3.2 and 3.4 hold only conditionally on an unstated existence-and-regularity hypothesis. A proof of high-probability containment in D, or an explicit assumption combined with a modified algorithm that enforces it, is required.","section":"§3.2–3.3, Theorems 3.2 and 3.4"},{"comment":"Even granting that a fixed point qσ∈D exists, the sentence 'combining Theorem 2.2 and Theorem 3.1' omits the key transfer step. To apply Theorem 2.2 one must first show that Gqσ = u_m(x,T;qσ) equals Sfσ, i.e., the noisy analogue of Lemma 2.5; this identity is not stated or proved. One must then bound the quantities Δ(Gqσ-Gq*) and Gqσ-Gq* in the norms appearing in Theorem 2.2 in terms of the right-hand sides of (3.8)–(3.11). Theorem 3.1 directly controls E‖fσ-f*‖² in (H^1)* and E‖Sfσ-Sf*‖²_n, and the proof contains auxiliary H^1 estimates, but the passage from these quantities to E‖qσ-q*‖_{(H^1)*} or E‖qσ-q*‖_{L2} is not written. Without this derivation, Theorem 3.2 is not established; the same comment applies to Theorem 3.4.","section":"§3.2, Theorem 3.2"},{"comment":"The right-hand sides of the two estimates in Theorem 3.2 contain ‖f*‖²_{L2(Ω)} and ‖f*‖²_{H^1(Ω)}, respectively, but taking square roots of (3.10) and (3.11) and applying Theorem 2.2 yields factors proportional to ‖f*‖_{L2(Ω)} and ‖f*‖_{H^1(Ω)}, not their squares. This is not merely a typo: the balancing conditions already use ‖f*‖^{-1}, and the tail version in Theorem 3.4 uses the linear quantity ρ0=‖f*‖_{H^s(Ω)}+σ n^{-1/2}. The statement should be corrected to use the first power of the norm, and the constants in the proof should be checked accordingly.","section":"§3.2, Theorem 3.2 statement"}],"minor_comments":[{"comment":"Line 5 of Algorithm 2 says 'Solve 3.2 for fσ', but the iteration in (3.3) requires solving the forward coupled system (1.1)–(1.2) with q=qσ_j. As written, the algorithm re-solves the P1 problem and never updates the forward solutions, so it is not reproducible.","section":"§4.1, Algorithm 2, line 5"},{"comment":"In the proof, the semi-metric for the sub-Gaussian process {(e,Sv)_n} is written d(u,v)=δ n^{-1/2}‖Su−Sv‖_n, whereas Lemma 3.7 defines d(u,v)=σ n^{-1/2}‖Su−Sv‖_n; the subsequent integrals use σ, so the displayed semi-metric is a typo that should be corrected.","section":"§3.3, Proof of Theorem 3.3"},{"comment":"The proof states a condition 'n^{d/2/(2+s)}λ≥1' that does not match the theorem's condition 'n^{(4+2s)/d}λ≥1'. The exponent should be harmonized, since the validity of the L2-error estimate depends on the sampling/inverse inequality connecting the empirical norm with the Sobolev norm.","section":"§3.2, Proof of Theorem 3.1"},{"comment":"The caption of Figure 4 says that E(Err 2) is plotted against λ^{1/3}, but the text and Theorem 3.1 give the rate λ^{1/4} for the (H^1)* error when s=0; please correct the caption.","section":"§4.2, Figure 4 caption"},{"comment":"There are several small typographical issues: 'Kronerker delta' should be 'Kronecker delta', 'Orilicz' should be 'Orlicz', and 'Inf p(x)' should be 'inf p(x)'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The probabilistic part relies heavily on Lemmas 3.2, 3.3, 3.7, and 3.8 imported from the authors' prior work [11], so the genuinely new content is the deterministic monotone-operator analysis and its formal combination with the stochastic P1 estimates. The main risk is the missing existence and regularity of the noisy fixed point qσ; if this can be repaired by an explicit high-probability admissibility argument or by modifying the algorithm, and if the scaling in Theorem 3.2 is corrected, the paper would be considerably stronger."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know up front. The deterministic half of this paper—the monotone fixed point operator, the uniqueness theorem, and the two stability estimates under Assumption 2.1—looks credible and internally consistent. The stochastic half, as written, does not deliver what it advertises: Theorems 3.2 and 3.4 are stated for a fixed point qσ of the noisy iteration (3.3), but the paper never proves such a fixed point exists in the admissible set D, and it never shows the P1 reconstructions fσ, Sfσ satisfy the pointwise bounds that D and the sign analysis require. Without that, Theorem 2.2 cannot be invoked for the pair (qσ, q∗), and the exponential-tail and expectation rates rest on an unstated hypothesis.\n\nWhat is genuinely new is the application of the stochastic machinery of [11] to a nonlinear coupled diffusion fluorescence model: the empirical-process estimates, the peeling argument, and the two-stage inversion scheme are not in the literature for this model, and the numerical experiments are informative. The paper is also honest about borrowing Lemmas 3.2, 3.3, 3.7 and 3.8 from [11], and those are published results, so this is not hidden circularity. The deterministic proof chain—fixed point equivalence, monotone convergence, energy estimates—has no obvious holes, though it inherits strong conditions (p ≥ Δg/g, positivity of boundary derivatives and a smallness condition). Those are structural limitations, not errors.\n\nThe soft spots are all in Section 3. The stress-test note is correct and load-bearing, not a nitpick. Starting from P1 you only have qσ0 = (fσ+pSfσ)/ue(x,T;0); the unconstrained minimization (3.2) imposes no lower bound fσ+pSfσ ≥ -Δg+pg and no upper bound, so qσ0 can leave D. The sign estimate in Lemma 2.6 requires ∂tum(q2)+fσ+pSfσ ≥ 0, which replaces exact positivity guaranteed by Assumption 2.1(c) with an unverified property of a noisy reconstruction. Theorem 3.2 is also asserted as a combination of Theorems 2.2 and 3.1 without writing the key step that transfers the P1 error to the q error. There are typos and some dimensionally confusing constants, but those are minor by comparison.\n\nWho should read this: anyone working on fluorescence optical tomography or on regularized inversion from discrete noisy sensors for nonlinear parabolic inverse problems. It deserves a serious referee, not a desk reject, because the deterministic core is valuable and the gap is fixable—by adding an explicit admissibility assumption on the reconstructed data, or by projecting fσ+pSfσ into the right set, plus a written derivation of the fixed-point transfer. My recommendation: send it to review, but with a referee report that asks for those changes before the probabilistic rates are accepted.","headline":"A credible deterministic inverse-source analysis; the stochastic sections need a proof that the noisy fixed point exists in the admissible set before the rates can be trusted.","tokens_in":25679,"tokens_out":3427,"would_cite":false,"duration_ms":34654,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","65J20","65M60","65N21","65N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Discrete noisy terminal measurements still determine the fluorophore absorption coefficient through a two-stage regularized inversion, with convergence in expectation and exponential tails.","keywords":["inverse source problem","coupled diffusion equations","fluorescence optical tomography","uniqueness","Lipschitz stability","regularization","stochastic convergence","fixed-point iteration"],"falsifier":"Take the smooth-source test case ($b=(x+y)^2 t+5$, $p=x+y+10$), choose $n=100$ sensors and noise $\\sigma=0.01$ so that the P1 reconstruction $Sf^\\sigma$ is not guaranteed to stay positive, and check whether the iteration (3.3) still has a fixed point in $D$ and whether the empirical probability $P(\\|q^\\sigma-q^*\\|_{(H^1)^*} > (\\lambda^{1/4}+\\lambda^{1/2}\\|p\\|_{\\infty})\\rho_0(1+z))$ respects the bound $Ce^{-Cz^2}$ for $z=2$; a violation would show the theorem's hypotheses fail in a regime the paper does not exclude.","tokens_in":24543,"feed_emoji":"🔬","tokens_out":9849,"duration_ms":87913,"temperature":0.7,"pith_summary":"The paper addresses a nonlinear inverse source problem in the coupled diffusion system that models fluorescence optical tomography: from the final-time emission field $u_m(x,T)$, recover the absorption coefficient $q(x)$ of the fluorophores. It establishes, under stated positivity and size conditions, that this recovery is unique and Lipschitz stable in both $L^2$ and the dual Sobolev norm $(H^1(\\Omega))^*$, by exhibiting a monotone fixed-point operator whose fixed points are exactly the sources. Since real sensors deliver only discrete pointwise and noisy values, the paper then splits the inversion into two sub-problems — a regularized elliptic reconstruction of $g$ and $-\\Delta g$, followed by the fixed-point iteration — and proves quantitative convergence of the noisy reconstruction to the exact source, both in expectation and with an exponential tail, with explicit dependence on the regularization parameter, noise variance, number of sensors, and spatial dimension. Two-dimensional experiments illustrate that the predicted rates $\\lambda^{1/4}$ and $\\lambda^{1/6}$ appear numerically.","feed_headline":"Fluorophore source recovered from noisy sensors at proven rates","feed_subtitle":"Two-step regularized inversion turns scattered noisy data into expectation and exponential-tail convergence.","key_machinery":"The load-bearing object is the operator $Kq = (\\partial_t u_m(x,T;q) - \\Delta g + p g)/u_e(x,T;q)$, whose fixed points coincide exactly with sources $q$ satisfying $u_m(x,T;q)=g$ (Lemma 2.5). The paper shows $K$ is monotone nondecreasing on its domain $D$ (Lemma 2.6) and Lipschitz continuous in $L^2$ (Lemma 2.8), so the iteration $q_0=(-\\Delta g+pg)/u_e(x,T;0)$, $q_{n+1}=Kq_n$ increases monotonically to the unique fixed point (Theorem 2.1). For noisy data, P1 is an elliptic optimal-control problem with penalty $\\lambda\\|f\\|^2_{H^s(\\Omega)}$ and empirical data misfit; its analysis rests on the spectral growth of the discrete eigenvalue problem $(\\psi,v)_{H^s}=\\rho(S\\psi,Sv)_n$, which via Weyl-type bounds $\\rho_k \\ge C k^{2(2+s)/d}$ converts $n$ and $\\lambda$ into effective noise filtering. The exponential-tail bound rests on viewing $(e,Su)_n$ as a sub-Gaussian empirical process and invoking covering-entropy estimates for Sobolev balls.","core_discovery":"On the paper's own terms, the central discovery is that discrete noisy terminal measurements still determine the absorption coefficient $q^*$ through a two-stage scheme with quantitative error control. Stage P1 solves the Tikhonov-regularized elliptic optimal control problem $$f^\\$\\sigma$ = \\arg\\min_{f\\in H^s(\\$\\Omega$)} \\frac{1}{n}\\sum_{i=1}^n (Sf(x_i)-g_i^\\$\\sigma$)^2 + \\$\\lambda$\\|f\\|^2_{H^s(\\$\\Omega$)},\\quad s\\in\\{0,1\\},$$ to reconstruct $f^*=-\\Delta g$ and $g=Sf^*$; stage P2 runs the fixed-point iteration $$q^\\sigma_0 = \\frac{f^\\$\\sigma$ + p\\,Sf^\\$\\sigma$}{u_e(x,T;0)},\\qquad q^\\sigma_{n+1} = \\frac{\\partial_t u_m(x,T;q^\\sigma_n) + f^\\$\\sigma$ + p\\,Sf^\\$\\sigma$}{u_e(x,T;q^\\sigma_n)}.$$ Under Assumption 2.1 and the smallness condition $\\sqrt{T M_b (M+1)}/(m_Q\\sqrt{C_p})<1$, Theorem 3.2 gives $E\\|q^\\sigma - q^*\\|_{(H^1(\\Omega))^*} \\le C(\\lambda^{1/4}+\\lambda^{1/2}\\|p\\|_{L^\\infty})\\|f^*\\|^2_{L^2}$ for $s=0$, with the analogous $L^2$ rate $\\lambda^{1/6}$ for $s=1$, and Theorem 3.4 gives the exponential-tail bound $P(\\|q^\\sigma - q^*\\|_{(H^1(\\Omega))^*} > (\\lambda^{1/4}+\\lambda^{1/2}\\|p\\|_{L^\\infty})\\rho_0(1+z)) \\le C e^{-Cz^2}$ for $s=0$, where $\\lambda^{1/2+d/8}=O(\\sigma n^{-1/2}\\rho_0^{-1})$ and $\\rho_0=\\|f^*\\|_{H^s}+\\sigma n^{-1/2}$. The numerical experiments display the predicted linear dependence of expected errors on $\\lambda^{1/4}$ and $\\lambda^{1/6}$.","pith_inferences":["An extension the paper leaves implicit is that any inverse problem whose forward map factorizes into a data-completion step followed by a monotone fixed-point step could inherit the same two-stage probabilistic estimates, since the analysis separates the stochastic regularization (P1) from the deterministic inversion (P2).","The smallness condition $\\sqrt{T M_b (M+1)}/(m_Q\\sqrt{C_p})<1$ suggests the rates degrade as the fluorophore absorption $M$ approaches the background absorption scale; testing the transition where this constant crosses 1 would map the practical limits of the stability theorem.","The numerical observation that H1 penalization wins for smooth sources and L2 penalization for discontinuous ones has no rigorous selection rule in the paper; a data-driven or oracle rule for choosing $s$ would be a natural next question.","Algorithm 1's convergence is demonstrated experimentally but not proven; proving that the self-consistent $\\lambda$ iteration converges would close the loop between the rate theory and the practical implementation."],"forward_implications":["If the theorems hold, a practitioner can choose the regularization parameter explicitly as $\\lambda^{1/2+d/8}=O(\\sigma n^{-1/2})$ for $s=0$ and predict the achieved source error before running the inversion.","The exponential-tail statement means the probability of a large reconstruction error decays like $e^{-Cz^2}$, so averaging over independent experiments drives the error down quickly.","The H1-penalized variant ($s=1$) yields convergence of the source in $L^2$ at rate $\\lambda^{1/6}$, while the L2-penalized variant yields convergence in the dual norm $(H^1)^*$ at rate $\\lambda^{1/4}$; the numerics indicate smooth sources favour $s=1$ and discontinuous sources favour $s=0$.","The deterministic stability estimate supplies the bridge: stochastic error in reconstructing $g$ and $-\\Delta g$ is amplified by the explicit constant $C=\\sqrt{C_p}/(m_Q\\sqrt{C_p}-\\sqrt{T M_b (M+1)})$, so the whole pipeline inherits quantitative control from the two sub-problems.","The self-consistent Algorithm 1 estimates $\\lambda$ without knowing $f^*$ or the noise variance $\\sigma$ and converges in a few iterations in the experiments, making the theoretical rates reachable in practice."],"supporting_citations":[{"why":"Supplies the regularized inverse-source formulation for pointwise noisy data, including the sampling-error inequalities and the spectral lemmas used in P1.","marker":"[11]"},{"why":"Provides the parabolic regularity and maximum principle that yield positivity and bounds for $u_e$ and $u_m$, which define the monotone operator's domain.","marker":"[15]"},{"why":"Provides the Weyl-law eigenvalue growth used to convert the discrete spectral problem into rates in $n$ and $\\lambda$.","marker":"[16]"},{"why":"Supplies the sub-Gaussian tail inequality and maximal inequality for empirical processes used in the exponential-tail estimates.","marker":"[36]"},{"why":"Supplies the covering-entropy estimate for Sobolev balls used to control the empirical process increments.","marker":"[9]"},{"why":"Provides the coupled-diffusion fluorescence imaging model that motivates the excitation-emission system and the inverse source problem.","marker":"[31]"}],"fun_headline_variants":["Noisy sensor data still yields source recovery with proven rates","Two-step inversion recovers source from noisy data with explicit error bounds","Quantitative recovery of absorption coefficient from noisy terminal data","Exponential-tail convergence for source inversion under noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rate and tail theorems assume that the fixed point $q^\\sigma$ of the noisy iteration exists inside the admissible set $D$ and that the first-stage reconstructions satisfy all the positivity and size conditions imposed on the exact data, which the paper does not prove.","fun_headline_variants_meta":{"raw":{"variants":["Noisy sensor data still yields source recovery with proven rates","Two-step inversion recovers source from noisy data with explicit error bounds","Quantitative recovery of absorption coefficient from noisy terminal data","Exponential-tail convergence for source inversion under noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00061,"raw_usage":{"total_tokens":2945,"prompt_tokens":1153,"completion_tokens":1792,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":769,"completion_tokens_details":{"reasoning_tokens":1726}},"tokens_in":769,"tokens_out":1792,"duration_ms":11977,"temperature":1.0,"reasoning_tokens":1726,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:53:39.744391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the smooth-source test case ($b=(x+y)^2 t+5$, $p=x+y+10$), choose $n=100$ sensors and noise $\\sigma=0.01$ so that the P1 reconstruction $Sf^\\sigma$ is not guaranteed to stay positive, and check whether the iteration (3.3) still has a fixed point in $D$ and whether the empirical probability $P(\\|q^\\sigma-q^*\\|_{(H^1)^*} > (\\lambda^{1/4}+\\lambda^{1/2}\\|p\\|_{\\infty})\\rho_0(1+z))$ respects the bound $Ce^{-Cz^2}$ for $z=2$; a violation would show the theorem's hypotheses fail in a regime the paper does not exclude.","supporting_citations":[{"cited_title":"Stochastic co nvergence of regularized solutions and their finite element approximations to invers e source problems","cited_arxiv_id":null,"evidence_quote":"Supplies the regularized inverse-source formulation for pointwise noisy data, including the sampling-error inequalities and the spectral lemmas used in P1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the parabolic regularity and maximum principle that yield positivity and bounds for $u_e$ and $u_m$, which define the monotone operator's domain."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Weyl-law eigenvalue growth used to convert the discrete spectral problem into rates in $n$ and $\\lambda$."},{"cited_title":"van der Vaart and J.A","cited_arxiv_id":null,"evidence_quote":"Supplies the sub-Gaussian tail inequality and maximal inequality for empirical processes used in the exponential-tail estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the covering-entropy estimate for Sobolev balls used to control the empirical process increments."},{"cited_title":"On fluorescence imaging: The diffusion equation model and recov ery of the absorption coefficient of fluorophores","cited_arxiv_id":null,"evidence_quote":"Provides the coupled-diffusion fluorescence imaging model that motivates the excitation-emission system and the inverse source problem."}],"review_version":1}