{"id":"2561cdee-dc36-439c-924c-50ccd2000917","arxiv_id":"2502.07574","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The MsFEM-POD+qMC method computes statistics of ground-state eigenvalues of random Schrödinger operators with a claimed error bound H^6 + rho + s^(-2/p+1) + N^(-alpha).","lead":"This paper introduces a fast solver for the eigenvalue problem of Schrödinger operators with random potentials, combining coarse-scale finite elements with a compressed basis computed from offline samples. If the stated error bounds are correct, simulations of Anderson localization and disorder-averaged quantum spectra would become far cheaper than standard fine-grid approaches.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.4's uniform POD error bound relies on the false scaling ||phi_i|| = O(h^d) and on conflating the reduced solution with the L2 projection; without it, Theorems 5.2 and 5.3 are unproven.","rationale":"Independently of the manuscript's numerical validation of the superconvergence rates, the central new theoretical contribution is the POD error bound. I focused on Lemma 5.4 because it is the only place where the POD term sqrt(rho) is introduced, and both Theorem 5.2 (spatial + POD) and Theorem 5.3 (total error) call it. The proof's two unjustified steps are concrete and testable. The scaling ||phi_i|| <= C h^d is contradicted by the normalization constraint, which ties the basis amplitude to the coarse mesh, and the identification of the reduced solution with the POD projection is not a standard property of POD-reduced optimization. A single numerical experiment that evaluates the lemma's left-hand side at an out-of-sample parameter would settle whether the claimed uniform bound holds; the existing experiments only evaluate aggregate eigenvalue errors at parameters close to the training distribution and never measure this quantity. For this reason I do not see how the central error estimate is proven as written, and I retain the reader's reject verdict. This is not a claim about the method's practical usefulness, only about the validity of the stated theorem.","tokens_in":27017,"tokens_out":11887,"duration_ms":103698,"concrete_test":"Reproduce the 1D setting of Example 6.3 (H = 1/16, h = 1/2048, s = 64, Q = 200, m_i = 3). Choose a test parameter omega* far from all Q snapshots, e.g. omega*_j = 1/2 for all j. Compute the full MsFEM basis phi_i(omega*) from (3.4)-(3.5) and the POD-reduced basis hat(phi)_i(omega*) from (3.15); record the L2 norms ||phi_i(omega*)|| and errors ||phi_i(omega*) - hat(phi)_i(omega*)||. If ||phi_i(omega*)|| scales like H^(1/2) rather than h, and the error is not bounded by C sqrt(rho) for the rho defined by (3.11) with m_i = 3, then Lemma 5.4's uniform bound is false and Theorems 5.2-5.3 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.4 is the linchpin of the POD error analysis: it supplies the C*sqrt(rho) term used in Theorem 5.2 and then Theorem 5.3. Its proof for omega outside the snapshot set Omega_0 = {omega_j} uses the assertion, immediately before Eq. (5.41), that ||phi_i(x,omega_j)|| <= C h^d. This scaling is not correct. The constraint in (3.5) reads \\int phi_i phi^H_j dx = alpha = (1, phi^H_j) with alpha = O(H^d), so the amplitude of phi_i at the coarse node is O(1). Over its localization patch of diameter O(H), the L2 norm is therefore O(H^(d/2)), independent of the fine-mesh size h. With the correct O(H^(d/2)) scale, Eq. (5.41) becomes ||phi_i(omega) - hat(phi)_i(omega)|| <= C ||omega - omega_j||_inf H^(d/2) + C sqrt(rho); since the finite set Omega_0 does not cover [-1/2,1/2]^s in the sup norm, the first term cannot be made uniformly smaller than sqrt(rho). A second gap arises even for omega in Omega_0: the proof sets c_j^i(omega) = (tilde(phi)_i, zeta^i_j), identifying the solution of the reduced optimal problem (3.15) with the L2-projection coefficients, but the energy minimizer over the POD subspace is not the projection. Equation (5.35) therefore does not bound ||phi_i(omega_j) - hat(phi)_i(omega_j)|| as claimed. Since Theorem 5.2 and Theorem 5.3 inherit the sqrt(rho) term from Lemma 5.4, the total error bound is not established. The numerical section does not report rho or test omega away from the Q = 200 snapshots, so it does not supply the missing uniform control.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a multiscale finite element method combined with proper orthogonal decomposition (POD) and quasi-Monte Carlo (qMC) sampling for computing statistics of the minimal eigenvalue and ground state of semiclassical random Schrödinger operators. After truncating the stochastic dimension, an MsFEM approximation is used in the physical domain, POD is used to reduce the cost of constructing the multiscale basis, and randomized lattice rules are used for the stochastic expectation. The main theoretical results are Theorems 5.1–5.3, which claim superconvergent spatial error rates (H^6 for the eigenvalue, H^3 in H1 and H^4 in L2 for the eigenfunction) plus POD and qMC error terms. Numerical experiments validate the deterministic and stochastic convergence rates and explore Anderson localization for spatially random potentials.","tokens_in":27442,"tokens_out":6074,"duration_ms":56536,"significance":"If correct, the combined estimate would be attractive: it would provide a UQ method for random Schrödinger eigenvalue problems with superconvergent spatial accuracy and a POD offline stage whose dimension is independent of the fine mesh. The deterministic MsFEM superconvergence part (Theorem 5.1) is plausible and consistent with earlier LOD/MsFEM results, and the truncation and qMC estimates are carefully imported from Gilbert et al. [15]. However, the new POD error analysis is the load-bearing ingredient for the random case, and it contains a false scaling estimate and a projection-coefficient identification that are not justified. Since Theorems 5.2 and 5.3 inherit these errors, the central claim of the paper is not established. The numerical section also does not report the POD tolerance rho or systematically test online parameters away from the snapshots, so it does not fill the gap.","major_comments":[{"comment":"The uniform bound in Lemma 5.4 for omega outside the snapshot set relies on the assertion, immediately before Eq. (5.41), that ||phi_i(x,omega_j)|| <= C h^d. This scaling is inconsistent with the constraint (3.5), which requires integral phi_i phi_j^H dx = alpha = (1, phi_j^H) = O(H^d). Since phi_i has O(1) amplitude on a patch of diameter O(H), its L2 norm is O(H^{d/2}), independent of the fine-mesh size h. With the correct O(H^{d/2}) scaling, the first term in (5.41) becomes C ||omega - omega_j||_inf H^{d/2}, and because a finite snapshot set Omega_0 cannot cover the continuum in the sup norm, this term cannot be made uniformly smaller than sqrt(rho). Consequently the bound ||phi_i(x,omega) - hat phi_i(x,omega)|| <= C sqrt(rho) in Lemma 5.4, and the sqrt(rho) terms in Theorems 5.2 and 5.3, are not established.","section":"§5.4, Lemma 5.4 and Eq. (5.41)"},{"comment":"For omega in the snapshot set Omega_0, the proof sets c_j^i(omega) = (tilde phi_i, zeta_j^i), i.e., it identifies the coefficients of the solution of the reduced constrained energy problem (3.15) with the coefficients of the L2 projection onto the POD subspace. Equation (5.35) bounds only the L2 projection error; it does not bound the error of the energy minimizer of (3.15). The proof therefore does not justify the claimed equality ||phi_i(x,omega) - hat phi_i(x,omega)|| = ||tilde phi_i(x,omega) - sum_j c_j^i(omega) zeta_j^i(x)|| nor the resulting sqrt(rho) bound.","section":"§5.4, proof of Lemma 5.4 for omega in Omega_0"},{"comment":"The passage from the L2 bound of Lemma 5.4 to the H1 error is not sound. Equation (5.42) states ||nabla phi_i - nabla hat phi_i|| <= C sqrt(rho)/h^2 with no justification; an inverse inequality would give an h^{-1} factor at best, and in any case the stated h^{-2} factor means the bound cannot be uniform as h -> 0. The subsequent line 'Therefore, there exists a constant C such that (5.47)' does not follow from a bound whose right-hand side depends explicitly on h. Since Theorem 5.2 uses the uniform H1 error ||psi_pod_ms - psi_ms||_1 <= C sqrt(rho), this gap affects the eigenvalue and eigenfunction estimates for the random problem.","section":"§5.4, Eqs. (5.42) and (5.47)"},{"comment":"The numerical experiments do not report the POD tolerance rho, and the text explicitly states 'the POD error rho is not discussed here' before Example 6.4. Figure 3 tests one online point outside the snapshot set, but there is no systematic study of the error as a function of rho or as a function of the sup-norm distance from omega to the nearest snapshot. The numerics therefore do not supply the missing uniform control needed for Lemma 5.4 and Theorems 5.2 and 5.3.","section":"§6.2, Example 6.3 and POD error discussion"}],"minor_comments":[{"comment":"Equation (1.9) states the eigenvalue error bound with a sqrt(rho) term, while Theorem 5.3, Eq. (5.53), states the same bound with a rho term; these should be made consistent.","section":"Introduction, Eq. (1.9)"},{"comment":"The reduced optimal problem in Eq. (3.15) uses the bilinear form a(.,.) without specifying whether this is the deterministic form from (3.3) or the random/truncated form A_s(omega;.,.); the notation should be aligned with the random setting used in Algorithm 2.","section":"§3.3, Eq. (3.15)"},{"comment":"The reference to 'Proposition 1 [23]' should be made precise, and the left-hand side of Eq. (5.34) should use the fluctuation tilde phi_i consistently with Eq. (5.35).","section":"§5.4, Eq. (5.34)"},{"comment":"Remark 6.1 correctly notes that for q = 0 the random potentials do not satisfy Assumption 3.1(2)–(3), so the convergence theory does not apply to Example 6.6. This is a useful caveat, but the Anderson-localization simulations in Figures 9 and 10 should be presented as numerical demonstrations outside the theorem rather than as validations of the theoretical error estimates.","section":"§6.3, Remark 6.1"},{"comment":"The notation in Eq. (5.42) mixes the constant C and the mesh size h without indicating which quantities are kept fixed; the constant should be tracked explicitly through the inverse inequality to make the h-dependence clear.","section":"§5.4, Eq. (5.42)"}],"recommendation":"reject","confidential_remarks":"The central POD error analysis is not a local fix: Lemma 5.4 uses an incorrect scaling for the multiscale basis functions and conflates the reduced energy minimizer with the L2 projection. These are load-bearing defects for Theorems 5.2 and 5.3, and the numerical section does not close the gap because it does not report rho or explore online parameters away from the snapshots. The deterministic MsFEM part and the imported qMC/truncation analysis may be salvageable, but the advertised combined estimate for the random problem is not supported in the current manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper combines MsFEM, POD, and qMC for the eigenvalue problem of a semiclassical random Schrödinger operator. The numerical results are encouraging, but the central POD-error lemma has a load-bearing gap, so the headline error bound is not established.\n\nWhat's genuinely new: using a POD-compressed basis to construct random multiscale basis functions for the Schrödinger EVP, so the dofs for basis construction are independent of the fine mesh. The deterministic MsFEM superconvergence estimates (H^6 for the minimal eigenvalue) are carried over from the authors' earlier work and look consistent. The truncation and qMC bounds are imported correctly from Gilbert et al., and the parametric regularity analysis is a reasonable adaptation. The experiments show real efficiency gains and the localization simulations are interesting.\n\nThe soft spot is Lemma 5.4, and it is serious. First, for omega in the snapshot set, the proof identifies the solution of the reduced constrained optimal problem with the L2-projection coefficients. That identification is not justified: the energy minimizer over the POD subspace with the constraint is not the L2 projection, and the paper gives no argument that the difference is controlled by sqrt(rho). Second, for omega outside the snapshots, the proof uses the scaling ||phi_i|| <= C h^d right before (5.41). The constraint (phi_i, phi_j^H) = alpha = O(H^d) fixes the amplitude of phi_i as O(1), so the correct L2 scale is O(H^{d/2}), not O(h^d). With the correct scale, the term C||omega - omega_j||_inf H^{d/2} cannot be made uniformly smaller than sqrt(rho) for a finite snapshot set. Since Theorems 5.2 and 5.3 inherit that term, the main error estimates are unproven.\n\nThe numerics do not close the gap: rho is never reported, and there is no test with omega away from the Q=200 snapshots. Example 6.4 also has a reporting inconsistency—the reference and MsFEM eigenvalues actually differ by about 1.4e-2, not 5.8e-9 as written.\n\nThe paper is not a throwaway. The method is plausible, the numerics suggest it works, and the POD analysis can probably be repaired. The deterministic part and the qMC/truncation parts are solid. The right outcome is major revision with the POD error analysis redone, not desk rejection.\n\nRecommendation: send it to a referee who knows the POD/elliptic eigenvalue literature, with a specific request to scrutinize Lemma 5.4. If the lemma is fixed, the paper becomes publishable; as it stands, the central claim is not proven.","headline":"The method is practical and the numerics are good, but Lemma 5.4 uses a false scaling for the multiscale basis, so the central error bound is unproven.","tokens_in":27990,"tokens_out":6550,"would_cite":false,"duration_ms":53511,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","65N25","65D30","65N30","81Q05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A POD-based multiscale finite element method computes the expected minimal eigenvalue of a semiclassical random Schrödinger operator with root-mean-square error $O(H^6 + \\rho + s^{-2/p+1} + N^{-\\alpha})$, and the expected value of a…","keywords":["eigenvalue problem","semiclassical random Schrödinger operator","proper orthogonal decomposition","multiscale model reduction","quasi-Monte Carlo","convergence analysis","Anderson localization"],"falsifier":"Fix a coarse mesh $H$ and a smooth 1D potential, compute the constrained multiscale basis function $\\phi_i$ on fine meshes $h = 1/64, 1/128, 1/256$, and measure $\\|\\phi_i\\|_{L^2}$; the proof of Lemma 5.4 requires this norm to shrink like $h$, while the constraint $\\int_D \\phi_i \\phi_j^H dx = \\alpha$ suggests it should stay at a scale set by $H$. If the norm does not shrink with $h$, the step leading to (5.41) is false and the claimed $\\sqrt{\\rho}$ POD error for $\\omega$ outside the snapshot set is unsupported. A complementary check is to fix $H$, $\\rho$, and $m_i$, take $\\omega$ outside the snapshot set, and observe whether $\\|\\phi_i(\\omega)-\\hat\\phi_i(\\omega)\\|$ stays bounded as $h\\to 0$ as the theorem predicts or grows as the alternative scaling would predict.","tokens_in":26755,"feed_emoji":"⚛️","tokens_out":14281,"duration_ms":99723,"temperature":0.7,"pith_summary":"This paper builds a reduced-order numerical method for the eigenvalue problem of a semiclassical Schrödinger operator whose random potential is written as an infinite series of independent uniform random variables. After truncating the series to $s$ terms, the method solves the resulting parametric eigenvalue problem on a coarse mesh using multiscale finite element basis functions, then compresses those basis functions with a proper orthogonal decomposition so that constructing the basis for a new random sample costs only a few POD coefficients, independent of the fine mesh. Statistics of the eigenvalue and ground state are collected with a quasi-Monte Carlo lattice rule. The central result is the combined root-mean-square error bound of Theorem 5.3: $O(H^6 + \\rho + s^{-2/p+1} + N^{-\\alpha})$ for the expected minimal eigenvalue and $O(H^3 + \\sqrt{\\rho} + s^{-2/p+1} + N^{-\\alpha})$ for a linear functional of the ground state, where $H$ is the coarse mesh size, $\\rho$ the POD tolerance, $s$ the truncation dimension, and $N$ the number of qMC samples. If correct, this means high-accuracy eigenvalue statistics can be obtained with online degrees of freedom equal to the coarse-mesh nodes rather than the fine-mesh resolution.","feed_headline":"Random Schrödinger spectra computed with sixth-order spatial accuracy","feed_subtitle":"A POD-based multiscale method keeps online cost independent of the fine mesh while retaining superconvergence.","key_machinery":"The load-bearing object is the constrained multiscale basis function $\\phi_i$, defined as the minimizer of the energy $a(\\phi,\\phi) = \\frac{\\epsilon^2}{2}\\int_D |\\nabla\\phi|^2 dx + \\int_D v_0 \\phi^2 dx$ subject to $\\int_D \\phi \\phi_j^H dx = \\alpha \\delta_{ij}$ for every coarse finite element basis function $\\phi_j^H$, with $\\alpha = (1,\\phi_j^H)$ chosen to remove mesh-size dependence. The POD stage replaces each random snapshot $\\phi_i(x,\\omega_j)$ by its mean $\\zeta_i^0$ plus a few fluctuation modes $\\zeta_i^1,\\dots,\\zeta_i^{m_i}$; the orthogonality identity $(\\zeta_i^k,\\phi_j^H)=0$ for $k\\ge 1$ means that only the single constraint for $j=i$ survives in the reduced optimal problem (3.15), so constructing $\\hat\\phi_i(\\omega)$ online costs only $m_i+1$ POD coefficients. This reduction preserves the MsFEM orthogonality—the multiscale space is the $a$-orthogonal complement of the kernel of the coarse $L^2$ projection—which is the mechanism behind the $H^6$ eigenvalue superconvergence in Theorem 5.1.","core_discovery":"The paper claims that the combined MsFEM-POD-qMC approximation achieves the superconvergent error rates stated in Theorem 5.3: under Assumption 3.1 (uniformly bounded potentials, $\\ell^p$-summability of the component norms, and $W^{1,\\infty}$ regularity), the root-mean-square error of the expected minimal eigenvalue is bounded by $C(H^6 + \\rho + s^{-2/p+1} + N^{-\\alpha})$, and the root-mean-square error of the expected value of any continuous linear functional of the ground state is bounded by $C(H^3 + \\sqrt{\\rho} + s^{-2/p+1} + N^{-\\alpha})$, with $\\alpha = \\min\\{1-\\delta, 1/p - 1/2\\}$. The proof splits the total error into four additive parts: the MsFEM discretization error, whose key deterministic estimate is $|\\lambda_{\\mathrm{ms}} - \\lambda| \\le C H^6$ (Theorem 5.1); the dimension-truncation error $s^{-2/p+1}$; the qMC integration error $N^{-\\alpha}$; and a POD error contribution $\\sqrt{\\rho}$ (or $\\rho$ for the eigenvalue) arising from the low-dimensional approximation of the random multiscale basis functions. Numerical experiments in Section 6 confirm the predicted rates and show that the method reproduces Anderson-localized eigenfunctions for white-noise-like potentials when the coarse mesh satisfies $H < \\epsilon$.","pith_inferences":["Because the reduction acts on the basis functions rather than on the eigenpairs, the same POD-MsFEM construction should extend to higher eigenpairs and to nonlinear eigenvalue problems such as Gross-Pitaevskii ground states, provided the uniform spectral-gap assumption used in the parametric regularity analysis holds for the eigenvalue of interest.","The asymmetry in the error bound (linear $\\rho$ for the eigenvalue, $\\sqrt{\\rho}$ for the eigenfunction) suggests that a looser POD tolerance can be used when only eigenvalue statistics are wanted; one could adaptively choose $m_i$ per coarse node instead of the fixed $m_i=3$ used in the experiments.","The $q=0$ white-noise experiments fall outside the assumptions of the theorem (Assumption 3.1(2)-(3) fail), so the paper's claim that $H<\\epsilon$ suffices there is better read as a separate conjecture; a systematic study as $s\\to\\infty$ with $H$ scaled by $\\epsilon$ would test it.","If the $H^6$ rate is generic for constrained multiscale spaces, the same POD reduction could be applied to other self-adjoint elliptic eigenvalue problems with affine parametric coefficients, not only Schrödinger operators."],"forward_implications":["The expected minimal eigenvalue of the random Schrödinger operator can be approximated to sixth order in the coarse mesh size $H$ while the online linear system has only $N_H$ (the number of coarse nodes) degrees of freedom; fine-mesh work is confined to the offline POD snapshot generation.","For a fixed POD tolerance $\\rho$ and truncation dimension $s$, increasing the number of qMC samples $N$ reduces the error at rate $N^{-\\alpha}$, which is almost first order when the potential components decay fast enough ($p \\le 2/3$), compared with $N^{-1/2}$ for Monte Carlo.","Because the total error is additive in $H^6$, $\\rho$, $s^{-2/p+1}$, and $N^{-\\alpha}$, a user can balance the four contributions independently when choosing the coarse mesh, POD rank, truncation dimension, and sample count.","For random potentials with decaying high-frequency components ($q>1$ in the examples), the method needs no constraint linking $H$ to the semiclassical parameter $\\epsilon$; for the $q=0$ white-noise limit, the experiments indicate that $H<\\epsilon$ suffices to resolve localized eigenfunctions.","The numerical experiments show the method reproducing Anderson-localized eigenfunctions in 1D and 2D at a fraction of the fine-FEM cost, with 29.72 seconds versus 341.17 seconds in the reported 1D comparison."],"supporting_citations":[{"why":"Supplies the dimension-truncation and qMC error bounds (Propositions 5.1 and 5.2) that the total error estimate inherits.","marker":"[15]"},{"why":"Establishes the MsFEM superconvergence for Schrödinger eigenvalue problems that Theorem 5.1 adapts.","marker":"[27]"},{"why":"Provides the duality-based finite element eigenvalue error estimates used in Lemma 5.1.","marker":"[37]"},{"why":"Gives the POD snapshot-error identity used in Lemma 5.4 to bound the basis approximation error by $\\sqrt{\\rho}$.","marker":"[23]"},{"why":"The Clément-type quasi-interpolation argument that fixes the constraint scale $\\alpha$ and makes the basis construction mesh-size independent.","marker":"[26]"},{"why":"Demonstrates analogous superconvergence rates in localized orthogonal decomposition spaces, supporting the shared orthogonal-decomposition mechanism behind Theorem 5.1.","marker":"[17]"},{"why":"The algebraic perturbation analysis for POD-MsFEM elliptic problems that the proof of inequalities (5.45)-(5.47) follows.","marker":"[29]"}],"fun_headline_variants":["Sixth-order superconvergence for random Schrodinger spectra","POD-MsFEM solver: mesh-independent basis for random Schrodinger EVPs","Random Schrodinger eigenvalues with H^6 accuracy and qMC speed","Model reduction makes semiclassical random Schrodinger EVPs practical","Superconvergent solver for semiclassical random Schrodinger operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the reduced basis works for random potentials not included in the offline snapshot set assumes that each constrained multiscale basis function has $L^2$ norm of order $h^d$, where $h$ is the fine mesh size; if the constraint $\\int_D \\phi_i \\phi_j^H dx = \\alpha$ actually sets the scale through the coarse mesh $H$, then the perturbation term in the argument cannot be made uniformly smaller than the POD tolerance $\\rho$, and the POD error bound for such potentials would fail.","fun_headline_variants_meta":{"raw":{"variants":["Sixth-order superconvergence for random Schrodinger spectra","POD-MsFEM solver: mesh-independent basis for random Schrodinger EVPs","Random Schrodinger eigenvalues with H^6 accuracy and qMC speed","Model reduction makes semiclassical random Schrodinger EVPs practical","Superconvergent solver for semiclassical random Schrodinger operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000984,"raw_usage":{"total_tokens":4227,"prompt_tokens":1051,"completion_tokens":3176,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":3093}},"tokens_in":667,"tokens_out":3176,"duration_ms":21760,"temperature":1.0,"reasoning_tokens":3093,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:17:16.130023+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a coarse mesh $H$ and a smooth 1D potential, compute the constrained multiscale basis function $\\phi_i$ on fine meshes $h = 1/64, 1/128, 1/256$, and measure $\\|\\phi_i\\|_{L^2}$; the proof of Lemma 5.4 requires this norm to shrink like $h$, while the constraint $\\int_D \\phi_i \\phi_j^H dx = \\alpha$ suggests it should stay at a scale set by $H$. If the norm does not shrink with $h$, the step leading to (5.41) is false and the claimed $\\sqrt{\\rho}$ POD error for $\\omega$ outside the snapshot set is unsupported. A complementary check is to fix $H$, $\\rho$, and $m_i$, take $\\omega$ outside the snapshot set, and observe whether $\\|\\phi_i(\\omega)-\\hat\\phi_i(\\omega)\\|$ stays bounded as $h\\to 0$ as the theorem predicts or grows as the alternative scaling would predict.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the dimension-truncation and qMC error bounds (Propositions 5.1 and 5.2) that the total error estimate inherits."},{"cited_title":"Li and Z","cited_arxiv_id":null,"evidence_quote":"Establishes the MsFEM superconvergence for Schrödinger eigenvalue problems that Theorem 5.1 adapts."},{"cited_title":"Sun and A","cited_arxiv_id":null,"evidence_quote":"Provides the duality-based finite element eigenvalue error estimates used in Lemma 5.1."},{"cited_title":"Kunisch and S","cited_arxiv_id":null,"evidence_quote":"Gives the POD snapshot-error identity used in Lemma 5.4 to bound the basis approximation error by $\\sqrt{\\rho}$."},{"cited_title":"Li and Z","cited_arxiv_id":null,"evidence_quote":"The Clément-type quasi-interpolation argument that fixes the constraint scale $\\alpha$ and makes the basis construction mesh-size independent."},{"cited_title":"Henning and A","cited_arxiv_id":null,"evidence_quote":"Demonstrates analogous superconvergence rates in localized orthogonal decomposition spaces, supporting the shared orthogonal-decomposition mechanism behind Theorem 5.1."},{"cited_title":"Ma, W.-K","cited_arxiv_id":null,"evidence_quote":"The algebraic perturbation analysis for POD-MsFEM elliptic problems that the proof of inequalities (5.45)-(5.47) follows."}],"review_version":1}