{"id":"071f80db-86fc-428b-84eb-9ceec4c53d84","arxiv_id":"2506.17575","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stability and stochastic convergence rates are proven for a Tikhonov regularization method that reconstructs the initial velocity of a fractional wave equation from scattered, noisy terminal point measurements.","lead":"Scientists recovered the initial velocity of a fractional wave process from noisy measurements taken at scattered points at a later time, using a Tikhonov regularization method with convergence guarantees. The paper is worth reading because it quantifies how more measurement points and better parameters reduce reconstruction error under stochastic noise for a class of ill-posed backward problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's proof needs E[M_n^{2/β}] from a second-moment bound; for 0<β<1 Jensen goes the wrong way, and the same missing higher-moment/correlation gap affects Theorem 4.","rationale":"The paper has a solid core: the Mittag-Leffler stability estimates in Theorems 1–2, the construction of the finite-dimensional subspace, and the numerical experiments are valuable and appear technically coherent. My concern is not with the fractional-wave modeling or the Tikhonov discretization but with the probabilistic step that converts second-moment bounds into the high-order moments appearing in the stated rates. Since Theorem 3 is stated for all β>0 while its proof invokes Theorem 2 (which restricts β to [0,1]), the regime 0<β<1 is exactly where the high-moment step is needed and not available. The same class of gap affects Theorem 4, contrary to the reader's impression that only Theorem 3 is impacted. These are not cosmetic issues: with only finite-variance noise, a random variable can have finite second moment but infinite p-th moment for p>2, so the left-hand expectations could be infinite while the right-hand sides are finite. The problem is addressable by adding explicit higher-moment or boundedness assumptions on the noise, or by a genuinely different argument, which is why a conditional verdict remains appropriate rather than a claim that the results are false. The reader's weakest assumption correctly identified the Theorem 3 moment gap; I extend it to the correlation/factorization step and to Theorem 4, hence partial agreement.","tokens_in":18548,"tokens_out":16010,"duration_ms":152365,"concrete_test":"Run the scalar/one-point analogue of the setup: take X=R, S=I, so (4.4) is a_n^* = (1/n)Σ_i m_i / (1+ρ_n), and M_n = |a_n^* - a^*|. Choose β=1/2 and iid noise e_i with E e_i=0 and E e_i^2=σ^2 but E|e_i|^4=∞ (e.g. a t-distribution with 4 degrees of freedom). Compute E|a_n^* - a^*|^6 directly from the noise law. If this moment is infinite or fails the claimed bound, the step 'E[M_n^{2/β}] ≤ C A^{1/β}' is false without extra assumptions; if it unexpectedly holds, repeat with a heavier-tailed finite-variance law. Independently, re-derive the factorization of E[M_n^{2/β}( n^{-4/d}||a_n^*-a^*||_{L2}^2 + ... )] in the Theorem 3 proof and check whether any inequality justifies separating the correlated factors.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised L2-rate in Theorem 3 is not derived by the given proof unless an unstated higher-moment or boundedness assumption is added. In the proof, after defining M_n = ||(-∆)^β(a_n^* - a^*)||_{L2}, the argument requires E[M_n^{2/β}] ≤ C( ||a^*||_X^2 + σ^2/(n ρ_n^{1+d/[4(1+β)]}) )^{1/β}. Lemma 4.5 only provides E[M_n^2] ≤ C A. For 0<β<1, p=2/β>2 and Jensen gives E[M_n^p] ≥ (E M_n^2)^{p/2}; the reverse bound is false without boundedness or higher-moment control, and none is stated (only E e_i=0, E e_i^2≤σ^2). The proof also factors E[M_n^{2/β}( n^{-4/d}||a_n^*-a^*||_{L2}^2 + ... )] into E[M_n^{2/β}] times the expectation of the second factor, which silently assumes a correlation bound. The same gap appears in Theorem 4: a deterministic M_n is chosen with M_n ≥ C E||a_n^*-a^*||_{L2}^2, but the proof then effectively uses E[||a_n^*-a^*||_{L2}^4] ≤ C M_n^2 and E[||a_n^*-a^*||_{L2}^2 ||S(a_n^*-a^*)||_{L2}^2] ≤ M_n E||S(...)||_{L2}^2, which require fourth-moment or almost-sure bounds not supplied by Lemma 4.5. Thus the H^{-1} theorem is not insulated from the missing-moment problem. Additionally, the final displayed bound in Theorem 3 has an exponent mismatch: the proof yields ρ_n A^{1+1/β}, not ρ_n A^{2/β}.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the backward problem for a fractional wave equation with Caputo derivative of order α∈(1,2), aiming to recover the initial velocity a_1 from terminal observations. The authors use asymptotic and root-distribution properties of Mittag-Leffler functions to establish conditional Hölder stability (Theorems 1 and 2), with a relaxed condition on the terminal time for α∈(1,4/3]. They then propose a Tikhonov regularization method based on scattered point measurements corrupted by independent identically distributed noise, and claim stochastic convergence rates in L^2 and H^{-1} norms (Theorems 3 and 4) that exhibit explicit dependence on the number of sensors n. A fixed-point iteration for selecting the regularization parameter is also presented, followed by numerical experiments for several test problems.","tokens_in":18965,"tokens_out":15292,"duration_ms":134643,"significance":"If the main convergence theorems were valid, the paper would provide the first stochastic convergence rates for the backward fractional wave problem with scattered point observations, with explicit dependence on the number of sensors, and would extend the stability analysis to the range α∈(1,4/3] without restrictions on the terminal time. The stability estimates in Theorems 1–2 and the spectral lower bounds in Lemmas 4.2–4.4 are largely standard and appear correct, and the numerical experiments with a practical fixed-point parameter selection are informative. However, the advertised rates in Theorems 3 and 4 are not established by the proofs as written: the missing higher-moment and correlation estimates are load-bearing. The results may be recoverable under additional assumptions or with a repaired argument, but the paper in its current form does not prove its central claims.","major_comments":[{"comment":"In the proof of Theorem 3, after defining M_n = ||(-∆)^β(a_n^*-a*)||_{L^2}, the assertion E[M_n^{2/β}] ≤ C( ||a*||_X^2 + σ^2/(n ρ_n^{1+d/[4(1+β)]}) )^{1/β} is used. Lemma 4.5 provides only the second-moment estimate E[M_n^2] ≤ C A. For 0<β<1, the exponent 2/β exceeds 2, so Jensen's inequality gives E[M_n^{2/β}] ≥ (E[M_n^2])^{1/β}, and the reverse bound requires an additional boundedness or higher-moment assumption on M_n; none is stated in the noise assumptions (only E e_i=0 and E e_i^2≤σ^2). The proof also splits expressions of the form E[M_n^{2/β}( n^{-4/d}||a_n^*-a*||_{L^2}^2 + ... )] as E[M_n^{2/β}] times the expectation of the second factor, which silently assumes a correlation bound. Consequently, the L^2 convergence rate in Theorem 3 is not derived by the given proof.","section":"Section 4.3, proof of Theorem 3"},{"comment":"The same missing-moment problem affects Theorem 4. The proof chooses a deterministic M_n ≥ C E||a_n^*-a*||^2_{L^2}, but then passes from the pointwise conditional stability bound ||a||^2_{H^{-1}} ≤ C ||a||_{L^2} ||S(a)||_{L^2} to a bound on E||a_n^*-a*||^4_{H^{-1}} of the form C M_n (n^{-4/d} M_n + ρ_n ||a*||^2 + σ^2/(n ρ_n^{d/4})). This step effectively requires an estimate such as E[||a_n^*-a*||^2_{L^2} ||S(a_n^*-a*)||^2_{L^2}] ≤ M_n E||S(a_n^*-a*)||^2_{L^2}, which is not supplied by Lemma 4.5. Thus the H^{-1} theorem is not insulated from the higher-moment gap.","section":"Section 4.3, proof of Theorem 4"},{"comment":"The final display in Theorem 3 states E||a_n^*-a*||^{2+2/β} ≤ C ( ||a*||^2_X + σ^2/(n ρ_n^{1+d/[4(1+β)]}) )^{2/β} ( n^{-4(1+β)/d} + ρ_n ). The proof, however, yields a first term proportional to A^{2/β} n^{-4(1+β)/d} and a second term proportional to ρ_n A^{1+1/β}, where A denotes the bracket. For β≠1, ρ_n A^{1+1/β} cannot be absorbed into ρ_n A^{2/β} with a constant independent of A. The statement of Theorem 3 must be corrected to match the proof, or the proof must be revised accordingly.","section":"Theorem 3, displayed bound"},{"comment":"Lemma 4.4 asserts that the space V_n is orthonormal with respect to (Sψ_i,Sψ_j)_n = δ_ij. However, the basis constructed in Lemma 4.3 satisfies (Sψ_i)(x_j)=δ_ij, so with the definition (u,v)_n = n^{-1}Σ u(x_i)v(x_i) one obtains (Sψ_i,Sψ_j)_n = n^{-1}δ_ij, not δ_ij. Since Lemma 4.5's estimate of the noise term, leading to σ^2/n Σ(1+ρ μ_k^{(n)})^{-1}, relies on the representation and normalization from Lemma 4.4, this inconsistency is load-bearing; the proof needs rescaling (e.g., replacing ψ_i by sqrt(n)ψ_i) and the subsequent rates must be re-derived. As written, the factor of n in the discrete norm is not consistently tracked.","section":"Lemma 4.4 and Lemma 4.5"}],"minor_comments":[{"comment":"The proof ends with 'This completes the proof of the lemma', but the statement being proven is a corollary; the wording should be adjusted.","section":"Corollary 2.1, proof"},{"comment":"The symbol β is used both as the smoothing order in X=D((-∆)^β) and as the power in the error exponent in Theorem 3; this overloading is confusing and should be disambiguated.","section":"Throughout"},{"comment":"In the expression (ϕ−P_V[ϕ_i]) the index is missing; it should read (ϕ_i - P_V[ϕ_i]).","section":"Lemma 4.3, proof"},{"comment":"The reported noise levels '≈24%, 15%' for σ=0.2 and ||Sa_1||_{L∞}≈0.95,1.30 appear to be slightly off (one obtains approximately 21% and 15%); please verify the values or clarify the computation.","section":"Example 1, Section 5"},{"comment":"The displayed optimal regularization parameter and optimal error should be re-derived after the exponent mismatch in Theorem 3 is fixed; the current formulas rely on the unproven bound.","section":"Remark 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains two rather separate strands: a standard conditional-stability analysis for the fractional-wave backward problem, and a stochastic scattered-data discretization analysis. The stability strand is sound and the numerical experiments are useful. The stochastic strand, however, does not currently prove the advertised rates because of the missing higher-moment and correlation estimates in Theorems 3 and 4. These gaps are repairable in principle (for instance by assuming bounded noise or finite moments of sufficiently high order, or by a different decomposition that avoids the product factorization), but the repair requires nontrivial reworking and may weaken the claimed results. I recommend major revision rather than rejection. The novelty claim of 'first stochastic convergence rates' should be checked carefully against the related works [1] and [11] once the technical gaps are closed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has one new stability result worth keeping, but its headline L2 convergence rate isn't actually proven. The proof of Theorem 3 needs a higher-moment bound that does not follow from the second-moment estimate in Lemma 4.5, and the same kind of gap shows up in Theorem 4. As written, the advertised rates are unsupported.\n\nWhat's new and solid: the stability estimates for the backward problem with alpha in (1,2) are built from known Mittag-Leffler asymptotics, and the alpha in (1,4/3] case avoids terminal-time exclusions using positivity of E_{alpha,2}. That is a nice observation and the argument checks out. The problem setting — scattered point measurements with noise treated as random variables — is a practical and underexplored one, and the regularization scheme plus the iterative parameter-choice algorithm are clearly described and the numerical examples are sensible.\n\nThe soft spot is specifically in the stochastic convergence analysis. Lemma 4.5 provides second-moment bounds. Theorem 3 then needs E[M_n^{2/beta}] with M_n = ||(-Delta)^beta(a_n^* - a^*)||. For 0<beta<1, the exponent 2/beta is greater than 2, so Jensen's inequality gives a lower bound, not an upper bound. The proof also factors a product of expectations without stating a correlation or almost-sure bound. The stress-test note is right that Theorem 4 has a similar issue: a deterministic M_n is chosen as a mean-squared bound, but the proof then uses fourth moments and product moments that don't follow from Lemma 4.5. There is also an exponent mismatch in the last displayed line of Theorem 3: the proof gives rho_n A^{1+1/beta}, not rho_n A^{2/beta}.\n\nThese are specific and likely fixable with added assumptions (e.g., bounded noise or higher-moment control), but they are not cosmetic. The paper's central advertised result is the L2 rate, and that result is currently conditional on an unstated assumption.\n\nBottom line: this deserves a serious referee. The stability part is valuable and the stochastic setting is worthwhile; a careful revision addressing the moment gap would make the paper publishable. I'd send it to review, with a note pointing at Theorem 3's proof.","headline":"Solid stability results for backward fractional wave problems, but the advertised L2 convergence rate in Theorem 3 rests on an unproved higher-moment bound; worth a careful revision and a serious referee.","tokens_in":19469,"tokens_out":2189,"would_cite":false,"duration_ms":21303,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35R09","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Tikhonov regularization of scattered noisy point measurements recovers the initial velocity of a fractional wave equation, with stochastic error bounds that improve as the number of sensors grows.","keywords":["fractional wave equation","backward problem","scattered point measurements","Tikhonov regularization","stochastic convergence","Mittag-Leffler function","inverse problem"],"falsifier":"Fix a smooth target $a^*$, choose $\\beta=1/4$, draw the noise $e_i$ from a distribution with finite variance but heavy tails, and estimate $\\mathbb{E}[\\|a_n^*-a^*\\|_{L^2(\\Omega)}^{2+2/\\beta}]$ over many realizations for increasing $n$; if this empirical moment fails to follow the predicted $O(n^{-4(1+\\beta)/d}+\\rho_n)$ scaling, or if the step $\\mathbb{E}[M_n^{2/\\beta}]$ diverges, then the $L^{2}$ rate of Theorem 3 is not valid as stated.","tokens_in":18329,"feed_emoji":"📡","tokens_out":8196,"duration_ms":80812,"temperature":0.7,"pith_summary":"The paper treats the backward problem for the fractional wave equation on a bounded domain: recover the unknown initial velocity $a_1$ from terminal observations $u(\\cdot,T)$ when measurements are available only at $n$ scattered points and are corrupted by stochastic noise. The central claim is that a Tikhonov regularization over the spectral space $D((-\\Delta)^\\beta)$ converges with explicit rates, with the expected $L^2$ error of moment $2+2/\\beta$ bounded by a power of the regularization parameter plus $n^{-4(1+\\beta)/d}$, and an analogous $H^{-1}$ bound for rough initial data. A supporting stability theory uses the asymptotic and root-distribution properties of Mittag-Leffler functions to control division by the solution kernel, avoiding the non-uniqueness caused by real zeros. The paper also supplies an iterative fixed-point rule that selects the regularization parameter without knowing the noise level or the norm of the solution. If valid, these estimates quantify how additional sensors improve the inversion, which classical full-domain regularization analyses cannot express.","feed_headline":"Noisy point sensors yield proven recovery for fractional wave problems","feed_subtitle":"Error bounds now track sensor count, noise level, and regularization, so more noisy sensors provably sharpen the reconstruction.","key_machinery":"The load-bearing object is the Mittag-Leffler function $E_{\\alpha,2}(z)$, whose value at $z=-\\lambda_n T^\\alpha$ appears in the denominator of the coefficient formula $(a_1,\\varphi_n)=(u(\\cdot,T),\\varphi_n)/(T\\,E_{\\alpha,2}(-\\lambda_n T^\\alpha))$. Its asymptotic lower bound turns division by the solution kernel into a polynomial amplification that the Tikhonov penalty absorbs, while the fact that the function has only finitely many real zeros supplies the exceptional terminal times to exclude. On the sampling side, the machinery is the quasi-uniform discrete norm inequality $\\|u\\|_{L^2(\\Omega)}^2\\le C(\\|u\\|_n^2+n^{-2k/d}\\|u\\|_{H^k(\\Omega)}^2)$, which converts scattered point evaluations into Sobolev control and yields the explicit dependence on the sensor count $n$ in the convergence rates.","core_discovery":"Using the eigenfunction expansion $u(x,t)=\\sum_{n\\ge 1} t\\,E_{\\alpha,2}(-\\lambda_n t^\\alpha)(a_1,\\varphi_n)\\varphi_n(x)$, the forward map $S$ has coefficients controlled from below and above through Mittag-Leffler lower bounds of the form $|E_{\\alpha,2}(-\\lambda T^\\alpha)|\\ge C(1+\\lambda T^\\alpha)^{-1}$. The paper proves that the minimizer $a_n^*$ of $\\min_{a\\in X}\\|(Sa)(x)-m\\|_n^2+\\rho_n\\|a\\|_X^2$, with $m_i=(Sa^*)(x_i)+e_i$ and i.i.d. zero-mean noise of variance at most $\\sigma^2$, satisfies, for $a^*\\in D((-\\Delta)^\\beta)$ with $\\beta>0$, $$\\mathbb{E}\\big[\\|a_n^*-a^*\\|_{$L^{2}$(\\$\\Omega$)}^{2+2/\\$\\beta$}\\big] \\le C\\Big(\\|a^*\\|$_X^{2}$+\\frac{\\$sigma^{2}$}{n\\$rho_n^{{1+d/[4(1+\\beta)]}}$}\\Big)^{2/\\$\\beta$}\\Big($n^{{-4(1+\\beta)/d}}$+\\rho_n\\Big),$$ and for $a^*\\in L^2(\\Omega)$, $$\\mathbb{E}\\big[\\|a_n^*-a^*\\|_{$H^{{-1}}$(\\$\\Omega$)}^4\\big] \\le C\\big($n^{{-4/d}}$+\\rho_n\\big)\\Big(\\|a^*\\|_{$L^{2}$(\\$\\Omega$)}^2+\\frac{\\$sigma^{2}$}{n\\$rho_n^{{1+d/4}}$}\\Big)^2.$$ These bounds hold for $\\alpha\\in(1,4/3]$ at every terminal time, and for $\\alpha\\in(4/3,2)$ when $T^\\alpha$ avoids the finite exceptional set coming from real zeros of $E_{\\alpha,2}$; the key structural step is a discrete eigenvalue lower bound $\\mu_k^{(n)}\\ge C k^{4(1+\\gamma)/d}$ on the sampling space, which transfers the smoothing of the forward operator into the explicit $n$-dependence.","pith_inferences":["The proof of Theorem 3 calls for the moment bound $\\mathbb{E}[M_n^{2/\\beta}]$ with $M_n=\\|(-\\Delta)^\\beta(a_n^*-a^*)\\|_{L^2(\\Omega)}$, but only a second-moment bound for $M_n$ is established; assuming bounded noise or a sufficiently high moment on $e_i$ would close this gap, and the stated rate should survive such an added hypothesis.","Because the convergence rates depend on the spatial dimension only through $d$ and the exponent $4(1+\\beta)/d$, the framework suggests a quantitative sensor-placement rule: quasi-uniform scattered points suffice, and the benefit of each additional sensor decays more slowly in lower dimensions.","The same discrete eigenvalue estimates could support recovering both initial values $a_0$ and $a_1$ from terminal observations at two time levels, which the authors list as future work; the stochastic analysis would need a two-parameter version of the sampling-space eigenvalue bound."],"forward_implications":["For $\\alpha\\in(1,4/3]$, the stability and convergence results hold for every terminal time $T>0$; for $\\alpha\\in(4/3,2)$, they hold for all times outside a countable exceptional set determined by the zeros of $E_{\\alpha,2}$.","With the optimal regularization parameter, the expected $L^2$ reconstruction error is of order $n^{-4(1+\\beta)/d}+\\rho_n$, so increasing the number of scattered sensors provably reduces the error even when the observation noise is large.","The iterative fixed-point algorithm selects the regularization parameter without prior knowledge of the noise level $\\sigma$ or the norm of the true initial value, and in the numerical experiments it reaches a stable parameter within a few iterations.","For rough initial data in $L^2(\\Omega)$, the $H^{-1}$ quartic error bound gives stable recovery in a weak norm, with the same explicit dependence on $n$ and $\\rho_n$, which covers initial profiles that are not smooth enough for the $L^2$ rate."],"supporting_citations":[{"why":"Supplies the well-posedness and eigenfunction expansion for the forward fractional wave problem used throughout the stability and regularization analysis.","marker":"[24]"},{"why":"Provides the Mittag-Leffler lower bound in Lemma 2.2 and the backward-problem framework for the diffusion-wave case with possible real zeros.","marker":"[5]"},{"why":"Supplies the variational form and stochastic-convergence argument behind the proof of Lemma 4.5.","marker":"[1]"},{"why":"Gives the discrete-versus-Sobolev norm inequalities of Lemma 4.1 that convert scattered point measurements into Sobolev control.","marker":"[29]"},{"why":"Supplies the monotone fixed-point iteration concept behind Algorithm 1 for finding the optimal regularization parameter.","marker":"[11]"},{"why":"Represents the stability and regularization benchmark for backward diffusion-wave problems that this work extends to scattered stochastic measurements.","marker":"[35]"}],"fun_headline_variants":["Noisy scattered sensors provably recover fractional wave data","Stochastic error bounds for backward fractional wave equations","Scattered noisy measurements enable fractional wave inversion","Optimal regularization for fractional wave from scattered point noise","Fractional wave backward stable under stochastic scattered noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $L^{2}$-rate proof in Theorem 3 relies on the step $\\mathbb{E}[M_n^{2/\\beta}]\\le C(\\|a^*\\|_X^2+\\sigma^2/(n\\rho_n^{1+d/[4(1+\\beta)]}))^{1/\\beta}$, where $M_n=\\|(-\\Delta)^\\beta(a_n^*-a^*)\\|_{L^2(\\Omega)}$; Lemma 4.5 supplies only the second-moment estimate, and for $\\beta<1$ the exponent $2/\\beta$ exceeds $2$, so the advertised rate needs a higher-moment or boundedness condition on the noise that is not stated.","fun_headline_variants_meta":{"raw":{"variants":["Noisy scattered sensors provably recover fractional wave data","Stochastic error bounds for backward fractional wave equations","Scattered noisy measurements enable fractional wave inversion","Optimal regularization for fractional wave from scattered point noise","Fractional wave backward stable under stochastic scattered noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001815,"raw_usage":{"total_tokens":7213,"prompt_tokens":1083,"completion_tokens":6130,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":699,"completion_tokens_details":{"reasoning_tokens":6058}},"tokens_in":699,"tokens_out":6130,"duration_ms":41551,"temperature":1.0,"reasoning_tokens":6058,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:08:10.102244+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a smooth target $a^*$, choose $\\beta=1/4$, draw the noise $e_i$ from a distribution with finite variance but heavy tails, and estimate $\\mathbb{E}[\\|a_n^*-a^*\\|_{L^2(\\Omega)}^{2+2/\\beta}]$ over many realizations for increasing $n$; if this empirical moment fails to follow the predicted $O(n^{-4(1+\\beta)/d}+\\rho_n)$ scaling, or if the step $\\mathbb{E}[M_n^{2/\\beta}]$ diverges, then the $L^{2}$ rate of Theorem 3 is not valid as stated.","supporting_citations":[{"cited_title":"Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems","cited_arxiv_id":null,"evidence_quote":"Supplies the well-posedness and eigenfunction expansion for the forward fractional wave problem used throughout the stability and regularization analysis."},{"cited_title":"Backward problems in time for frac- tional diffusion-wave equation.Inverse Problems, 36(12):124003, 2020","cited_arxiv_id":null,"evidence_quote":"Provides the Mittag-Leffler lower bound in Lemma 2.2 and the backward-problem framework for the diffusion-wave case with possible real zeros."},{"cited_title":"Stochastic Convergence Analysis of Inverse Potential Problem","cited_arxiv_id":"2410.14106","evidence_quote":"Supplies the monotone fixed-point iteration concept behind Algorithm 1 for finding the optimal regularization parameter."},{"cited_title":"Backward diffusion-wave problem: Stability, regular- ization, and approximation.SIAM Journal of Scientific Computing, 44(5):A3183– A3216, 2022","cited_arxiv_id":null,"evidence_quote":"Represents the stability and regularization benchmark for backward diffusion-wave problems that this work extends to scattered stochastic measurements."}],"review_version":2}