{"id":"72235d99-e4b8-430b-a788-5755652e2df0","arxiv_id":"2412.07929","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Dirichlet-Neumann averaged Gaussian fields are claimed to have isotropic periodized covariance, but a zero-mode normalization error breaks the exact covariance identity.","lead":"The paper proposes a fast method for sampling Gaussian random fields on grids by averaging fields with Dirichlet and Neumann boundary conditions, which is meant to remove the padding overhead of circulant embedding. The central derivation contains a normalization error in the zero-frequency basis mode, so the main theorem that the averaged field has the claimed periodized covariance is false as stated.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Basis (2.5) is not orthonormal: the constant mode has L2 norm sqrt(2) per dimension, so Proposition 2.2's covariance carries an extra 2^{#zero components} factor and the DNA field is not the claimed periodization; Eq. (2.19) also has the wrong kappa exponent.","rationale":"The reader's weakest assumption identifies exactly the load-bearing defect: the basis in (2.5) is not orthonormal because of the constant mode, and the proof of Proposition 2.2 depends on that orthonormality. I verified the egregious case d=1: the covariance computed from the printed definitions differs from \\varphi^{(\\pi)}_{2\\alpha,n}(x-y) by an extra \\hat{\\varphi}(0)/(2\\alpha) term, so Proposition 2.2 is false even in the simplest case. In higher dimensions the error introduces mode-dependent weights 2^{#zeros}, which not only break the claimed periodisation identity but also make the covariance anisotropic. Since Proposition 2.7 and Corollary 2.8 inherit the same non-orthonormal basis via (2.22), the SPDE connection is equally unsupported. The separate sign error in (2.19) further undermines the SPDE scaling, although the main theorem already fails on its own. Both flaws are consistent with the reader's assessment: they are mechanical and correctable, so a revised version with properly normalised DCT/DST basis functions and corrected SPDE scaling might well be viable, but the present manuscript's central claims are not established. The REJECT verdict should stand.","tokens_in":25034,"tokens_out":17945,"duration_ms":152534,"concrete_test":"Recompute the d=1 covariance from (2.5)-(2.7) with \\alpha=1, Matérn \\nu=0.5, \\ell=1 (so \\hat{\\varphi}(0)=2). The true E[u_{1,n}(x)u_{1,n}(y)] equals (1/2)\\sum_{m\\in\\mathbb{Z}_n} \\hat{\\varphi}(m/2)e^{i\\pi m(x-y)} + 1, not the claimed (1/2)\\sum_{m\\in\\mathbb{Z}_n} \\hat{\\varphi}(m/2)e^{i\\pi m(x-y)}; the marginal variance is thus 1 larger than \\varphi^{(\\pi)}_{2,n}(0). Sampling the DNA field on a fine grid and estimating the pointwise variance will show this +1 offset, confirming the normalization error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Proposition 2.2, whose proof relies on the basis e^b_{\\mu,\\alpha} in (2.5) being orthonormal in L2(D_\\alpha). It is not: for \\mu_j=0, e^{(0)}_{\\mu_j}(x_j/\\alpha)=1 has L2 norm \\sqrt{\\alpha} on (0,\\alpha), so the prefactor (2/\\alpha)^{1/2} gives norm \\sqrt{2}, not 1. Consequently, every mode with a zero component is overweighted by a factor 2^{#\\{j: \\mu_j=0\\}} relative to an orthonormal basis. Tracing the proof of Proposition 2.2, the covariance gains an extra term compared to the claimed \\varphi^{(\\pi)}_{2\\alpha,n}(x-y). In d=1 the true covariance is \\alpha^{-1}[\\lambda_{\\alpha,0} + \\sum_{m=1}^n \\lambda_{\\alpha,m} \\cos(\\pi \\alpha^{-1} m(x-y))], whereas the claimed periodization is (2\\alpha)^{-1}[\\lambda_{\\alpha,0} + 2\\sum_{m=1}^n \\lambda_{\\alpha,m} \\cos(...)]; the zero-frequency weight is off by a factor 2, so even in one dimension Proposition 2.2 is false as stated. In d>1 the overweighting depends on which coordinates are zero, breaking the identification with the single sum over \\mathbb{Z}^d_n in (2.10)-(2.11) and destroying isotropy. The same non-orthonormality invalidates (2.22) and hence Proposition 2.7 and Corollary 2.8. Independently, the SPDE scaling in (2.19) has the wrong exponent of \\kappa: from (1.6) and \\beta=\\nu/2+d/4 one obtains \\eta_{\\mu,\\alpha}^{-2\\beta}=C_\\nu^{-1}\\kappa^{-2\\nu}\\hat{\\varphi}((2\\alpha)^{-1}\\mu), not \\kappa^{+2\\nu}; this changes the white-noise scaling in (2.17a). Both flaws are mechanical and likely fixable, but the printed central theorem is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new sampling framework, Dirichlet-Neumann Averaging (DNA), which generates isotropic Gaussian random fields on a hypercube by averaging independent fields with homogeneous Dirichlet and homogeneous Neumann boundary conditions on the faces, using tensor-product cosine/sine bases evaluated by DCTs and DSTs. The central claim is Proposition 2.2, asserting that the covariance of the averaged field is the periodization phi^(pi)_{2 alpha,n} of the target isotropic covariance, with no padding and a doubled effective scale. The paper then derives exponential error bounds for Matérn covariances (Lemma 2.3, Lemma 2.4, Proposition 2.5), links the construction to the Whittle-Matérn SPDE in Proposition 2.7, and states in Corollary 2.8 that averaging SPDE solutions with these boundary conditions yields genuinely isotropic fields without domain extension. Numerical experiments compare the DNA covariance error with circulant embedding and with oversampled SPDE sampling, including non-cuboid domains.","tokens_in":25416,"tokens_out":14268,"duration_ms":128994,"significance":"The underlying idea is attractive and, once the technical errors are corrected, could be a genuinely useful alternative to circulant embedding and to oversampling in SPDE-based samplers. The manuscript's strengths are the concrete algorithmic proposal (DCT/DST evaluation with no padding), the explicit non-asymptotic error estimates for Matérn covariances, and the numerical evidence across several kernels and domains. However, the main theoretical results as printed are not correct: the central identity in Proposition 2.2 is invalidated by a normalization error in the basis (2.5), and the SPDE scaling in Proposition 2.7 contains an incorrect power of kappa. Both defects appear mechanical and repairable, but they currently undermine Proposition 2.2, Lemma 2.4, Proposition 2.7, and Corollary 2.8, as well as the claimed interpretation of the numerical experiments.","major_comments":[{"comment":"The basis functions e^b_{mu,alpha} in (2.5) are not orthonormal in L2((0,alpha)^d). For mu_j=0, the one-dimensional factor is identically 1, so its L2 norm on (0,alpha) is sqrt(alpha); with the prefactor (2/alpha)^{1/2} its norm becomes sqrt(2), not 1. Consequently every mode with at least one zero component is overweighted by a factor 2^{# {j : mu_j = 0}} relative to an orthonormal basis. Tracing the proof of Proposition 2.2, the covariance of the DNA GRF (2.7) is not the claimed periodization phi^(pi)_{2 alpha,n}. In d=1, the true averaged covariance is alpha^{-1} lambda_{alpha,0} + alpha^{-1} sum_{m=1}^n lambda_{alpha,m} cos(pi alpha^{-1} m (x-y)), whereas the claimed phi^(pi)_{2 alpha,n}(x-y) equals (2 alpha)^{-1} lambda_{alpha,0} + alpha^{-1} sum_{m=1}^n lambda_{alpha,m} cos(...); the zero-frequency term is off by a factor of 2. In d>1, the overweighting depends on which coordinates vanish, so the covariance is not the scalar periodization and is not isotropic. The reindexing in (2.10)-(2.11) is also not a bijection from pairs (mu,q) with mu in N^d_n to Z^d_n, because zero components are counted once per available sign. The same non-orthonormality invalidates the use of (2.11) in the proof of Lemma 2.4 and the orthonormality assumption in equations (2.21)-(2.22) of Proposition 2.7. The issue is repairable by using an index-dependent normalization: 1/sqrt(alpha) for the constant mode and sqrt(2/alpha) for the positive modes in each dimension. The paper should be revised to either adopt this corrected basis or explicitly recompute all covariance identities with the printed basis.","section":"Section 2.1, Eq. (2.5) and Proposition 2.2"},{"comment":"The exponent of kappa in equation (2.19) is wrong. From the Matérn Fourier transform (1.6) and the identity beta = nu/2 + d/4, a direct calculation gives eta_{mu,alpha}^{-2 beta} = (kappa^2 + pi^2 alpha^{-2} ||mu||_2^2)^{-(nu+d/2)} = C_nu^{-1} kappa^{-2 nu} hat{varphi}((2 alpha)^{-1} mu), not C_nu^{-1} kappa^{+2 nu} hat{varphi}((2 alpha)^{-1} mu). The sign of the kappa exponent is load-bearing because it determines the white-noise scaling in the SPDE (2.17a): to make the coefficients of u^b_alpha in (2.18) equal to sqrt(hat{varphi}((2 alpha)^{-1} mu)), the right-hand side of (2.17a) should be sqrt(C_nu) kappa^{+nu} W, not sqrt(C_nu) kappa^{-nu} W. This affects the proof of Proposition 2.7 through equations (2.20)-(2.23) and any numerical implementation of the SPDE-based DNA variant. The authors should correct the sign and re-derive the SPDE constant consistently.","section":"Section 2.2, Eq. (2.19) and Eq. (2.17a)"},{"comment":"The numerical experiments do not resolve the normalization ambiguity in the manuscript. The paper reports empirical covariances and errors for the DNA method, but it does not state which normalization was used in the DCT/DST implementation. If the implementation used standard orthonormal DCT/DST normalizations, the experiments correspond to the corrected basis rather than to the basis defined in (2.5); if it used the printed normalization, the zero-frequency overweighting should be visible in quantities such as the marginal variance. Either way, the experiments as reported cannot validate Proposition 2.2 as printed. The authors should specify the transform normalization, rerun the experiments with the corrected basis, and, if the printed basis is retained, recompute the theoretical covariance against which the errors are measured.","section":"Section 3, Figures 3-5 and Table 1"}],"minor_comments":[{"comment":"The notation in (1.14) should use complex conjugation: the condition should read xi_{-mu} = overline{xi_mu} and mathbb{E} xi_mu overline{xi_eta} = delta_{mu eta}; as written, xi_{-mu} = xi_mu is inconsistent with mathbb{E} xi_mu xi_eta = delta_{mu eta} for eta = -mu.","section":"Section 1, Definition 1.4"},{"comment":"In the proof of Lemma 2.4, the sentence 'analogously to the Proof of Lemma 2.4' should refer to the shell-summation argument in the proof of Lemma 2.3, not to the current lemma.","section":"Section 2.1, proof of Lemma 2.4"},{"comment":"There is a typo, 'homoegeneous', in the discussion of the basis functions satisfying boundary conditions in Section 1.","section":"Section 1"},{"comment":"The manuscript would benefit from a reproducibility statement describing the DCT/DST normalization, the random number generation, and the software used for the numerical experiments, since the central theoretical claims depend sensitively on the normalization of the basis.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"Both main theorems are false as printed, but I do not recommend rejection because the defects are localized and repairable: an index-dependent normalization fixes Proposition 2.2 and its consequences, and a corrected kappa exponent fixes Proposition 2.7. The larger risk is that the numerical section may have been implemented with the corrected normalization while the theory uses the printed one; the authors need to reconcile these explicitly. If the authors are unwilling or unable to make these corrections, the paper should not be published."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one for the idea, not the theorem. Averaging Dirichlet and Neumann boundary-condition fields to cancel boundary anisotropy is genuinely new as far as I know, and the DCT/DST implementation without padding plus the SPDE connection is nice packaging. The numerical experiments are honest: they show the averaged field has flat marginal variance, and they don't oversell the non-cuboid cases; Figure 7 even says the basic idea breaks down in non-convex corners. Citation pattern is fine; the cited work really only proves that single boundary conditions fail, so the averaging step is not stolen from there.\n\nThe soft spot is load-bearing and it is in the first main claim. Basis (2.5) is not orthonormal: the constant mode has L2 norm sqrt(2) per dimension, not 1. So the proof of Proposition 2.2 breaks at the reindexing step (2.10)-(2.11). With the printed definitions the true covariance is not the claimed periodization; zero-frequency components are overweighted by a factor 2^{#zero components}, and for d>1 that destroys isotropy. The same non-orthonormality invalidates (2.22), so Proposition 2.7 and Corollary 2.8 do not follow. Separately, (2.19) has the wrong kappa exponent: from (1.6) and beta = nu/2 + d/4 you get kappa^{-2nu}, not kappa^{+2nu}, which changes the white-noise scaling in (2.17a).\n\nNone of this looks like a deep conceptual failure; it reads as a mechanical normalization slip. With the standard orthonormal DCT/DST scaling the construction probably works, and the numerics are consistent with that. But as printed the central theorem is false, and Lemmas 2.3-2.5 analyze a field that is not the one actually constructed. That is a reject as-is.\n\nWould I engage? Yes. This deserves a serious referee. The idea is worth the community's time, and the fix is plausibly one revision away. I would send it to review with an explicit request to correct the normalization, reprove Propositions 2.2 and 2.7, and double-check the kappa scaling before acceptance. I would not cite it in its current form until a corrected version appears.","headline":"The DNA averaging idea is genuinely new and worth referee time, but the central covariance identity is false as printed because the basis (2.5) is not orthonormal; the paper needs a mechanical but load-bearing correction.","tokens_in":26044,"tokens_out":1874,"would_cite":false,"duration_ms":44903,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G15","60G60","60G10","60-08","65C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Averaging Dirichlet- and Neumann-boundary Gaussian fields yields an isotropic field whose covariance is the target, periodised at doubled scale.","keywords":["Gaussian random fields","isotropic covariance","Dirichlet-Neumann averaging","circulant embedding","SPDE sampling","Matérn covariance","discrete cosine and sine transforms","boundary conditions"],"falsifier":"Take the DNA field $u_{\\alpha,n}$ on a hypercube with $\\alpha=1$ and large $n$, estimate the marginal variance $\\mathbb{E}[u_{\\alpha,n}(x)^2]$ at an interior point by Monte Carlo, and compare it with the claimed value $\\varphi^{(\\pi)}_{2\\alpha,n}(0)$. Under the basis scaling in (2.5), the constant mode has $L^2$ norm $2^{d/2}$ rather than 1, so the measured variance will deviate from the formula by a known constant factor; this single comparison settles whether the covariance identity holds as written.","tokens_in":24743,"feed_emoji":"🎲","tokens_out":11910,"duration_ms":102225,"temperature":0.7,"pith_summary":"This paper proposes a way to sample Gaussian random fields with an isotropic covariance on a regular grid without padding the domain or altering the covariance. The idea is to draw several independent fields, one for each combination of homogeneous Dirichlet and Neumann boundary conditions on the faces of a hypercube, and average them with equal weights. The paper proves that the averaged field's covariance is exactly a periodisation of the target isotropic covariance at twice the scale parameter, so the field is isotropic on the original domain. For Matérn covariances it derives explicit error bounds that decay exponentially in the scale parameter, and numerical experiments indicate that the remaining covariance error is small for typical applications. In the SPDE-based sampling setting, the same averaging is shown to produce genuinely isotropic fields without oversampling.","feed_headline":"Averaged boundary conditions yield isotropic Gaussian samples","feed_subtitle":"Averaging sine- and cosine-based fields cancels boundary artifacts and doubles the effective scale, shrinking covariance error.","key_machinery":"The object that carries the argument is the DNA field $u_{\\alpha,n}=2^{-d/2}\\sum_{b\\in\\{0,1\\}^d}u^b_{\\alpha,n}$, averaged over all $2^d$ tensor-product bases of cosine and sine functions on $(0,\\alpha)^d$. The load-bearing identity is the trigonometric product rule $e^{(r)}_k(\\tau_1)e^{(r)}_k(\\tau_2)=\\frac12(e^{(0)}_k(\\tau_1-\\tau_2)+(-1)^r e^{(0)}_k(\\tau_1+\\tau_2))$, combined with the binomial sum $\\sum_{b\\in\\{0,1\\}^d}\\prod_j(a_j+(-1)^{b_j}b_j)=2^d\\prod_j a_j$; together they kill every term depending on $x+y$ and leave only the pairwise differences $x-y$. After relabelling the individual frequency signs $q\\in\\{-1,1\\}^d$, the surviving sum is exactly the Fourier representation of the periodised covariance $\\varphi^{(\\pi)}_{2\\alpha,n}$. In the SPDE version the same eigenfunctions diagonalise $\\kappa^2-\\Delta$ with eigenvalues $\\eta_{\\mu,\\alpha}=\\kappa^2+\\pi^2\\alpha^{-2}\\|\\mu\\|_2^2$, which match the Matérn spectral density through $\\eta_{\\mu,\\alpha}^{-2\\beta}=C_\\nu^{-1}\\kappa^{2\\nu}\\hat\\varphi((2\\alpha)^{-1}\\mu)$.","core_discovery":"The central claim is that averaging $2^d$ independent random fields, built from tensor products of cosine modes (Neumann) and sine modes (Dirichlet) on a hypercube, cancels the anisotropic $x+y$ terms in their covariance functions exactly. The covariance of the averaged field reduces to $\\varphi^{(\\pi)}_{2\\alpha,n}(x-y)$, a periodisation of the target isotropic covariance $\\varphi$ with the period doubled and a controlled truncation. For Matérn covariances the paper proves a total error bound of the form $\\|\\varphi^{(\\pi)}_{2\\alpha,n}-\\varphi\\|_\\infty \\le C(\\alpha^{2\\nu+d}n^{-2\\nu}+\\alpha^{\\nu-1/2}e^{-2\\vartheta\\kappa\\alpha})$, so increasing the scale parameter and the truncation together makes the error arbitrarily small. For $d\\le3$ and Matérn smoothness $\\nu\\ge1/2$, the same averaged construction is identified with solutions of the fractional SPDE $(\\kappa^2-\\Delta)^\\beta u = \\sqrt{C_\\nu}\\kappa^{-\\nu}W$ under mixed Dirichlet/Neumann boundary conditions; Corollary 2.8 states that this averaged SPDE solution is genuinely isotropic with no domain extension.","pith_inferences":["The boundary-cancellation identity is independent of the constant-mode normalisation, so renormalising the constant term in the DCT basis would preserve the averaging idea while changing the exact covariance formula; this is the first repair to test if the identity is to be used as stated.","The mechanism suggests a local generalisation: on domains with symmetry or local convexity, averaging only the normal and tangential boundary conditions may restore isotropy, as the disc experiment hints, but non-convex corners will need a different construction.","DNA produces a periodised covariance, not the exact target, so it is best suited to high-throughput or Bayesian-inference settings where small covariance bias is acceptable; for applications requiring exact covariances, circulant embedding with adequate padding remains the safer choice.","The doubled-scale observation transfers to any existing periodisation-based analysis: a DNA sampler should match the accuracy of a periodic-boundary sampler with half the padding, which could simplify parameter choice in production codes."],"forward_implications":["On a uniform grid, DNA sampling needs only DCTs and DSTs at the target resolution, with no padding or domain extension; the effective periodisation scale is doubled relative to periodic boundary conditions, so the covariance error is smaller for the same computational cost.","For Matérn fields with $\\nu\\ge1/2$, the paper's explicit bounds show the covariance error can be driven to zero by choosing the scale parameter $\\alpha$ and truncation $n$ together.","In the SPDE setting, averaging $2^d$ Dirichlet/Neumann solutions on the original domain removes the need to oversample, which addresses memory and communication bottlenecks in massively parallel simulations.","Because the DNA covariance is genuinely isotropic for any $\\alpha$, users of SPDE-based samplers no longer need to balance domain-extension size against covariance error; the remaining error is dominated by finite-element discretisation across typical mesh widths.","Covariance functions that would force circulant embedding to use extreme padding, such as smooth Matérn, Gaussian, and Cauchy kernels, can be sampled with small error at $\\alpha=1$, according to the paper's experiments."],"supporting_citations":[{"why":"Supplies the circulant-embedding baseline that DNA is compared against and whose padding cost motivates the construction.","marker":"[13]"},{"why":"Provides the Matérn spectral density and the padding-size analysis for circulant embedding that DNA seeks to avoid.","marker":"[21]"},{"why":"Identifies the boundary-induced anisotropy in SPDE sampling and the oversampling remedy that DNA replaces.","marker":"[25]"},{"why":"Establishes the SPDE representation of Matérn fields and the folded-covariance viewpoint used in Proposition 2.7.","marker":"[27]"},{"why":"Provides the periodisation-analysis framework and the Bessel-function bound used in the exponential error estimate.","marker":"[4]"},{"why":"Establishes the Fourier-based simulation framework for stationary Gaussian processes that the periodised-field definition builds on.","marker":"[42]"},{"why":"Supplies the well-posedness and eigenfunction calculus for fractional elliptic SPDEs used in Proposition 2.7.","marker":"[8]"},{"why":"Supplies regularity and convergence results for Whittle–Matérn fields used in the SPDE discretisation argument.","marker":"[11]"}],"fun_headline_variants":["Sine-cosine averaging cancels GP boundary error","DNA sampling: isotropic Gaussian fields minus padding","Averaged field trick makes Gaussian samples isotropic","Dirichlet-Neumann averaging: exact isotropy, no domain extension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the sine/cosine basis used in the expansion is orthonormal, including the constant term; under the stated scaling the constant term is not unit-length, and the proof's relabelling of frequencies relies on that orthonormality.","fun_headline_variants_meta":{"raw":{"variants":["Sine-cosine averaging cancels GP boundary error","DNA sampling: isotropic Gaussian fields minus padding","Averaged field trick makes Gaussian samples isotropic","Dirichlet-Neumann averaging: exact isotropy, no domain extension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000651,"raw_usage":{"total_tokens":3025,"prompt_tokens":1023,"completion_tokens":2002,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":1939}},"tokens_in":639,"tokens_out":2002,"duration_ms":12850,"temperature":1.0,"reasoning_tokens":1939,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:27:43.998074+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the DNA field $u_{\\alpha,n}$ on a hypercube with $\\alpha=1$ and large $n$, estimate the marginal variance $\\mathbb{E}[u_{\\alpha,n}(x)^2]$ at an interior point by Monte Carlo, and compare it with the claimed value $\\varphi^{(\\pi)}_{2\\alpha,n}(0)$. Under the basis scaling in (2.5), the constant mode has $L^2$ norm $2^{d/2}$ rather than 1, so the measured variance will deviate from the formula by a known constant factor; this single comparison settles whether the covariance identity holds as written.","supporting_citations":[{"cited_title":"Dietrich and Garry N","cited_arxiv_id":null,"evidence_quote":"Supplies the circulant-embedding baseline that DNA is compared against and whose padding cost motivates the construction."},{"cited_title":"Graham, Frances Y","cited_arxiv_id":null,"evidence_quote":"Provides the Matérn spectral density and the padding-size analysis for circulant embedding that DNA seeks to avoid."},{"cited_title":"‘Analysis of Boundary Effects on PDE-Based Sampling of Whittle–Matérn Random Fields’","cited_arxiv_id":null,"evidence_quote":"Identifies the boundary-induced anisotropy in SPDE sampling and the oversampling remedy that DNA replaces."},{"cited_title":"‘An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: the Stochastic Partial Differential Equation Approach’","cited_arxiv_id":null,"evidence_quote":"Establishes the SPDE representation of Matérn fields and the folded-covariance viewpoint used in Proposition 2.7."},{"cited_title":"Graham, Van Kien Nguyen and Robert Scheichl","cited_arxiv_id":null,"evidence_quote":"Provides the periodisation-analysis framework and the Bessel-function bound used in the exponential error estimate."},{"cited_title":"Wood and Grace Chan","cited_arxiv_id":null,"evidence_quote":"Establishes the Fourier-based simulation framework for stationary Gaussian processes that the periodised-field definition builds on."},{"cited_title":"‘Numerical Solution of Fractional Elliptic Stochastic PDEs with Spatial White Noise’","cited_arxiv_id":null,"evidence_quote":"Supplies the well-posedness and eigenfunction calculus for fractional elliptic SPDEs used in Proposition 2.7."},{"cited_title":"Cox and Kristin Kirchner","cited_arxiv_id":null,"evidence_quote":"Supplies regularity and convergence results for Whittle–Matérn fields used in the SPDE discretisation argument."}],"review_version":1}