{"id":"823568d2-84f1-4e8c-87a8-3124aeba857e","arxiv_id":"2411.13957","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Two new variational multiscale evolve-filter algorithms recover vortex shedding in under-resolved cylinder flow simulations at Re=1000, in both full-order and reduced-order settings.","lead":"This paper proposes two filter-based stabilization algorithms that combine variational multiscale scale separation with evolve-filter regularization for under-resolved flow simulations. The methods are tested on flow past a cylinder at Reynolds number 1000, where they recover vortex shedding that standard evolve-filter smoothing destroys.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'significantly more accurate' claim rests on a single overdiffusive δ and on discarding the one metric that favors EF; a δ-sweep and a phase-shift-insensitive error metric are needed before the abstract claim is supported.","rationale":"The reader's weakest assumption correctly identifies the metric-selection problem: the sole metric that quantitatively favors standard EF (Figure 11) is set aside, while the metrics that favor VMS-EFFC (L2 norm, qualitative vortex recovery) are emphasized. I agree that this is a genuine weakness. My read goes slightly further: the paper selects the value δ=1.59e-3 specifically to make EF overdiffusive (Section 5.2), and it concedes in Remark 6 that other parameter choices can make the relative errors comparable while degrading the average quantities. This means the FOM comparison is a single-calibration demonstration, not yet evidence for an unconditional 'significantly more accurate' claim. The ROM comparison introduces a second, structural bias: all ROM bases are built from VMS-EFFC FOM snapshots, so the VMS-based ROM is expected to match the reference better; a ROM test against DNS, or against a ROM trained on DNS snapshots, would be needed to separate algorithmic merit from data-source alignment. The paper is honest about being a numerical investigation with no numerical analysis yet, and it acknowledges the sensitivity of the parameters, which supports the conditional verdict rather than a rejection. The proposed δ-sweep plus phase-compensated relative error is the minimal experiment that would decide whether the advantage is intrinsic to VMS-EFFC or is an artifact of the chosen operating point.","tokens_in":929,"tokens_out":880,"duration_ms":66213,"concrete_test":"Perform a δ-sweep on the cylinder test with δ = q·1.59e-3 for q ∈ {0.5, 1, 2, 5}. For each method and δ, compute the time-averaged L2 error to the fine-mesh DNS and a phase-compensated relative error Eu_align(t) = min_τ ||u(t) − u_DNS(t+τ)||_{L2} / ||u_DNS(t)||_{L2}. Report the best calibrated δ for each method. If EF's best L2 error is within the spread of VMS-EFFC's, or if the ranking reverses across q where EF is not overdiffusive, then the claimed superiority is calibration-dependent rather than intrinsic. If VMS-EFFC wins on both metrics for all q, the central claim is supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim requires that, under the same large filter radius, VMS-EFFC is materially more accurate than EF in both FOM and ROM simulations. The FOM evidence is built on one deliberately overdiffusive value, δ=1.59e-3 (Section 5.2), and on metrics chosen after the fact: the L2-norm and qualitative vorticity plots favor VMS-EFFC, while the pointwise relative error Eu (Figure 11) favors EF. The paper dismisses Eu as inappropriate without offering a quantitative criterion, and its own summary states that no approach accurately reproduces CD, CL, or relative errors. Remark 6 further concedes that parameter changes that reduce the VMS relative errors make the average quantities worse. Thus the unconditional 'significantly more accurate' conclusion is not yet established: it may be a property of the selected δ and metric rather than of the algorithm. At the ROM level the comparison is also favorable to the new method by construction, because every ROM basis is generated from VMS-EFFC FOM snapshots (step 1 of Algorithms 4-7), so VMS-EFFC-ROM is closer to the reference FOM solution than alternatives that must regularize different dynamics.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two filter-based stabilization variants, VMS-EFFC and VMS-EPFC, that modify the standard evolve-filter (EF) strategy by decomposing the evolved velocity into large and small resolved scales, filtering the small scales, and adding the filtered small scales back to the large scales. The authors test these methods on flow past a cylinder at Re = 1000 with a deliberately overdiffusive filter radius, reporting that VMS-EFFC recovers vortex shedding and matches the DNS L2 norm better than EF, while VMS-EPFC is less accurate. They then extend the methods to reduced-order models, proposing VMS-EFFC-ROM and VMS-EPFC-ROM, and report that VMS-EFFC-ROM outperforms G-ROM and EF-ROM in their tests. The paper is primarily a numerical investigation; it includes no convergence analysis or code release.","tokens_in":33900,"tokens_out":6589,"duration_ms":63571,"significance":"If the central claim is established, the paper would offer a simple, modular correction to the overdiffusive behavior of evolve-filter regularization in both FOM and ROM settings, which would be practically useful. The paper is honest about several limitations, including the acknowledged equivalence of VMS-EFFC to van Cittert approximate deconvolution (Remark 4) and the admitted failure of all methods on some metrics. However, the significance is weakened by the fact that the main algorithmic variant is formally identical to a known deconvolution operator, and the numerical evidence is built on a single filter radius and on a metric selection that is justified only qualitatively.","major_comments":[{"comment":"The conclusion that VMS-EFFC is 'significantly more accurate' is not supported by a consistent accuracy criterion. The pointwise relative error Eu in Figure 11 favors the standard EF, and the manuscript dismisses this metric as 'not appropriate' without a quantitative justification. At the same time, Remark 6 concedes that parameter changes that reduce the VMS relative errors worsen the average quantities, and Section 5.2 states that none of the methods accurately reproduces CD, CL, or the relative errors. The conclusions in Section 7 even claim improved 'relative errors in time,' which contradicts Figure 11. The authors should either adopt a phase-shift-insensitive error metric (e.g., time-aligned errors, error in dominant Fourier modes, or windowed averages) or explicitly restrict the claim to the specific metrics and parameter regime used.","section":"Section 5.2, Figs. 8-11; Section 7"},{"comment":"The FOM comparison is conducted at a single, deliberately overdiffusive filter radius, delta = 1.59e-3, chosen 'to showcase the benefits' of the VMS-based filters. The manuscript itself notes that 'for some values of delta, the EF strategy can be as competitive as the VMS-based algorithm,' but it does not quantify this statement. Without a sweep over delta or a well-defined threshold for what constitutes 'too large' a filter radius, the abstract claim that the new algorithms 'yield significantly more accurate results than the standard EF' is not established as a property of the algorithms rather than a property of the selected parameter. A table or plot showing L2 norm and Eu errors for EF and VMS-EFFC across a range of delta values would be the minimal fix.","section":"Section 5.2, p. 13 and Figs. 4-5"},{"comment":"The ROM comparison is structurally favorable to VMS-EFFC-ROM because every ROM algorithm (G-ROM, EF-ROM, VMS-EFFC-ROM, VMS-EPFC-ROM) builds its reduced basis from snapshots generated by the VMS-EFFC FOM (step 1 in Algorithms 4-7). With such a basis, the VMS-EFFC-ROM is expected to be closer to the reference FOM solution than a ROM that needs to regularize different dynamics. The claim that VMS-EFFC-ROM outperforms EF-ROM as a regularization strategy therefore requires a fairer test, for example repeating the ROM comparison with snapshots from the EF-FOM (or from the plain NSE FOM) and reporting whether the ranking persists. At minimum, the paper should explicitly state and discuss this construction bias.","section":"Section 6, Algorithms 4-7"},{"comment":"Remark 4 shows that VMS-EFFC with delta1 = delta2 is formally identical to the van Cittert approximate deconvolution operator D1 applied to the evolved velocity. This is an important and candid admission, but it undercuts the paper's novelty claim in the Introduction that this is 'the first time the EF and VMS strategies are combined in correction algorithms.' The VMS-EFFC algorithm is then a known deconvolution operation presented in VMS language, and the ROM extension is closely connected to existing approximate deconvolution ROMs. The authors should reposition the contribution to focus on the VMS interpretation, the EPFC variant, and the ROM formulation, and they should discuss the relation to the approximate deconvolution literature earlier and more prominently.","section":"Remark 4, Eq. (4); Section 1"}],"minor_comments":[{"comment":"There are several typos: 'Ganeral setting' appears in Table 2 and Figure 2, 'VSM-EFFC' appears in Remark 9, and 'approch' appears in the same remark. These should be corrected.","section":"Table 2, Figure 2, Remark 9"},{"comment":"The text refers to 'equation (15)' for the inlet boundary condition in the flow-past-a-cylinder problem, but Eq. (15) is the boundary condition for the lid-driven cavity test in Remark 9. The cross-reference should be to Eq. (12).","section":"Section 5.1"},{"comment":"The captions for the center and right panels in the bottom row of Figure 12 are mismatched: the center column is labeled 'VMS-EPFC (y=0.05)' but should be 'VMS-EPFC (y=0.36)', and one of the bottom-right panel labels repeats 'VMS-EPFC (y=0.05)' instead of 'VMS-EFFC (y=0.36)'. This makes the qualitative comparison difficult to follow.","section":"Figure 12 caption"},{"comment":"In Algorithm 7, step 7, the notation shows what appear to be identical quantities on both sides of the assignment for the first set of coefficients, and the index ranges for the zeroed coefficients are written as 'rus + 1, . . . , rus'. The intended truncation of the coefficient vector should be written with distinct symbols (e.g., barred and unbarred coefficients) and with a correct index range.","section":"Algorithm 7, step 7"},{"comment":"The paper does not include a data or code availability statement. Given that the conclusions rely entirely on numerical experiments, providing access to the implementation or at least a detailed reproducibility description would strengthen the manuscript.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The central numerical claim is plausible but currently over-stated relative to the evidence: the FOM comparison uses one overdiffusive delta and a metric choice made after seeing the results, and the ROM comparison trains all methods on snapshots from the winning method. In addition, the formal equivalence of VMS-EFFC to van Cittert approximate deconvolution (Remark 4) substantially reduces the algorithmic novelty; the editor may want to consider whether the remaining contribution (EPFC and the ROM formulations) is sufficient for the journal's scope. I would encourage the authors to add the delta sweep and the fair ROM comparison before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the headline FOM algorithm is not new: VMS-EFFC with equal filter radii is exactly van Cittert approximate deconvolution of order 1, and the authors know it (Remark 4). The novelty that survives is the EPFC variant and the ROM adaptation, neither of which I have seen before in this exact form. Second, the 'significantly more accurate' claim in the abstract is ahead of the evidence: the FOM comparison uses a single deliberately overdiffusive δ, and the one pointwise metric that favors standard EF (relative L2 error, Figure 11) is dismissed as 'not appropriate' without a quantitative justification.\n\nWhat the paper does well: it is clearly written, modular, and unusually candid. Remark 6 admits the test case is a trade-off and that no method recovers drag, lift, or relative errors. The lid-driven cavity experiment in Remark 9 is a genuine second data point where VMS-EFFC beats EF on both L2 norm and relative error. The ROM section reports computational overhead (115% of G-ROM CPU time), which is the kind of practical detail I trust. The equivalence with AD(1) is stated cleanly instead of being hidden.\n\nWhere the soft spots are. The central quantitative claim depends on choosing the L2 norm and qualitative vortex recovery while discarding pointwise relative error. That is not fatal—there are legitimate reasons to prefer average quantities in turbulent flow—but the paper does not provide the standard robustness checks: no δ-sweep, no time-averaged or phase-insensitive error measure, no error bars. The ROM comparison also has a structural bias: the FOM reference and the POD snapshots are all VMS-EFFC, so EF-ROM and G-ROM are tested on data generated by the method they are compared against. That does not invalidate the ROM extension, but it means 'VMS-EFFC-ROM is most accurate' is partly inherited from the FOM choice. Several parameters (γP, ru, δ) are chosen by trial and error, and there is no code.\n\nOverall: a useful, honest numerical study with an overbroad conclusion. The math is straightforward and the algorithms are reproducible from the text. The right response is a serious referee, not a desk reject, but the referee should push for a δ-sweep, a fairer ROM protocol, and a tempered abstract.\n\nRecommendation: send to peer review; require major revisions before acceptance.","headline":"The FOM algorithm is formally van Cittert deconvolution of order 1, and the 'significantly more accurate' claim outruns the single-δ evidence; the ROM extension and EPFC variant are the real contributions, worth a serious but demanding referee.","tokens_in":34388,"tokens_out":3311,"would_cite":true,"duration_ms":32279,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","65N30","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"By filtering only the small resolved scales and adding them back to the large scales, the new VMS-EFFC method restores vortex shedding that an over-diffusive evolve-filter smoothing removes, in full-order and reduced-order simulations.","keywords":["evolve-filter regularization","variational multiscale","differential filter","reduced-order model","convection-dominated flow","Navier-Stokes equations","approximate deconvolution","vortex shedding"],"falsifier":"Re-run the cylinder flow test at $Re = 1000$ with a filter radius small enough that the standard EF is not overdiffusive, since the paper concedes EF can then be competitive; if VMS-EFFC no longer improves the velocity $L^2$ norm or drag coefficient, the claimed superiority does not extend beyond the overdiffusive regime. Alternatively, treat the pointwise relative error in the paper's own Figure 11 as the decisive metric, which would invert the conclusion, since EF has the lower error there.","tokens_in":33446,"feed_emoji":"🌊","tokens_out":8828,"duration_ms":82827,"temperature":0.7,"pith_summary":"The evolve-filter (EF) method is a simple way to stabilize under-resolved convection-dominated flow simulations: evolve the Navier-Stokes equations, then apply a spatial filter to smooth spurious oscillations. When the filter radius is too large, EF becomes overdiffusive and smooths away physically important features such as vortex shedding. This paper proposes fixing that by splitting the evolved velocity into large and small resolved scales, filtering only the small scales, and adding the filtered small scales back to the large scales. The resulting VMS-EFFC and VMS-EPFC algorithms are claimed to be significantly more accurate than standard EF on a flow past a cylinder at Reynolds number 1000, both in the full-order simulation and in reduced-order models built from it. The authors argue that relative pointwise errors are not the right metric for regularized models, and instead use $L^2$ norms, drag and lift coefficients, and qualitative flow patterns to support the claim.","feed_headline":"Correcting filtered small scales restores vortex shedding","feed_subtitle":"The VMS-EFFC method beats standard evolve-filter on Re=1000 cylinder flow, in both full and reduced models.","key_machinery":"The load-bearing mechanism is the correction step: the new velocity is $\\mathbf{u}^{n+1} = \\bar{\\mathbf{w}}^{n+1} + \\widetilde{\\mathbf{w}'}^{\\,n+1}$, where $\\mathbf{w}^{n+1}$ is the evolved velocity, $\\bar{\\mathbf{w}}^{n+1}$ is its large-scale part (obtained by a differential filter of radius $\\delta_1$ in VMS-EFFC, or by an $L^2$-projection in VMS-EPFC), and $\\widetilde{\\mathbf{w}'}^{\\,n+1}$ is the small-scale difference $\\mathbf{w}' = \\mathbf{w} - \\bar{\\mathbf{w}}$ passed through a second differential filter of radius $\\delta_2$. With $\\delta_1 = \\delta_2$, the output equals $D_1(\\mathbf{w}) = 2\\bar{\\mathbf{w}} - \\bar{\\bar{\\mathbf{w}}}$, the first-order van Cittert approximate deconvolution, so the extra filter loop is precisely an approximate deconvolution that restores some of the energy the first filter removed.","core_discovery":"The paper's central discovery is that over-diffusivity in the evolve-filter strategy can be corrected by a scale-separation step: after evolving the velocity, one defines 'large' resolved scales and 'small' resolved scales, applies the differential filter only to the small scales, and forms the new state as the sum of the large scales and the filtered small scales. Two variants are proposed: VMS-EFFC, where the large scales are obtained by a first differential filter, and VMS-EPFC, where a postprocessing $L^2$-projection with grad-div stabilization plays that role. In both variants the filtered small scales serve as a correction that restores features the filter would otherwise destroy. The authors show that when the two filter radii in VMS-EFFC are equal, the algorithm is formally identical to the first-order van Cittert approximate deconvolution operator $D_1(\\mathbf{u}) = 2\\bar{\\mathbf{u}} - \\bar{\\bar{\\mathbf{u}}}$. Numerical tests on flow past a cylinder at $Re = 1000$ indicate that VMS-EFFC is the most accurate strategy among EF, VMS-EPFC, and the ROM variants G-ROM, EF-ROM, VMS-EFFC-ROM, and VMS-EPFC-ROM, based on average quantities and qualitative vortex-shedding behavior.","pith_inferences":["The formal equivalence to $D_1$ suggests a natural extension: replacing the first-order deconvolution by higher-order van Cittert operators $D_n$ could restore even more of the filtered energy; the paper does not test this, but the machinery is in place.","Because the relative-error metric shows EF winning, the practical advantage of VMS-EFFC may lie in statistics and structure rather than pointwise fidelity; applications with phase-sensitive quantities, such as flow control or aeroelastic loads, would need to check whether the phase shift in the lift coefficient matters.","The VMS-EPFC variant's poor pressure reconstruction and divergence levels near $10^{-5}$ hint that the chosen $L^2$-projection is a weak link; a divergence-free or pressure-aware projection could close that gap.","All experiments fix the filter radii at $\\delta_1 = \\delta_2 = 1.59 \\cdot 10^{-3}$, and the paper reports that unequal radii did not help; a systematic sensitivity sweep over $\\delta$ at more Reynolds numbers and in three dimensions would directly test whether the correction idea transfers beyond the two benchmarks examined."],"forward_implications":["At the FOM level, VMS-EFFC provides a parameter-light way to correct overdiffusive filtering: users keep one filter radius and add a second filter of the same radius plus a sum, instead of tuning a relaxation parameter.","At the ROM level, VMS-EFFC-ROM reduces the average relative velocity error roughly sixfold compared with the Galerkin ROM and threefold compared with VMS-EPFC-ROM, while costing about 115% of the G-ROM CPU time.","The formal identity with the first-order van Cittert operator means the correction is a deconvolution step, aligning VMS-EFFC with a whole family of approximate-deconvolution regularizations.","VMS-EPFC performs worse on pressure and divergence, and the authors recommend VMS-EFFC as the FOM model for ROM construction, so the practical pipeline is VMS-EFFC, POD, VMS-EFFC-ROM.","The lid-driven cavity test at $Re = 7500$ shows the qualitative advantage of VMS-EFFC over EF persists beyond the cylinder benchmark, although the paper notes that relative errors remain high."],"supporting_citations":[{"why":"Provides the EF and EFR formulations at both FOM and ROM levels that serve as the baseline for all comparisons, including the grad-div parameter choice and the under-resolved test setting.","marker":"[51]"},{"why":"Defines the van Cittert approximate deconvolution operator $D_1$ that formalizes the VMS-EFFC correction in Remark 4, linking the new method to a known deconvolution family.","marker":"[32]"},{"why":"Supplies the variational multiscale large/small resolved-scale decomposition (direct sum of resolved spaces) that justifies splitting the evolved velocity into large and small scales.","marker":"[42]"},{"why":"Establishes the EF-ROM approach and the use of POD-Galerkin ROMs with filtering, providing the ROM-level comparison baseline for VMS-EFFC-ROM and VMS-EPFC-ROM.","marker":"[18]"},{"why":"Provides the supremizer stabilization used to build stable reduced velocity spaces, which is necessary for all the ROMs compared in the numerical experiments.","marker":"[2]"},{"why":"Documents the overdiffusivity problem of EF and the relaxation strategy (EFR) that the new method aims to replace, motivating the need for a correction-based alternative.","marker":"[6]"}],"fun_headline_variants":["VMS-EFFC beats standard evolve-filter on Re=1000 cylinder flow","VMS-EFFC restores vortex shedding in under-resolved flows","Filtered small scales fix evolve-filter overdiffusivity","VMS-EFFC matches van Cittert deconvolution in special case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's accuracy claim depends on rejecting the pointwise relative velocity error, the one metric in which the standard EF has lower error than VMS-EFFC, in favor of average quantities such as $L^2$ norms and qualitative vortex patterns, which the paper judges to be the appropriate measures for regularized models.","fun_headline_variants_meta":{"raw":{"variants":["VMS-EFFC beats standard evolve-filter on Re=1000 cylinder flow","VMS-EFFC restores vortex shedding in under-resolved flows","Filtered small scales fix evolve-filter overdiffusivity","VMS-EFFC matches van Cittert deconvolution in special case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000941,"raw_usage":{"total_tokens":4082,"prompt_tokens":1067,"completion_tokens":3015,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":683,"completion_tokens_details":{"reasoning_tokens":2935}},"tokens_in":683,"tokens_out":3015,"duration_ms":21829,"temperature":1.0,"reasoning_tokens":2935,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:41:02.000812+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the cylinder flow test at $Re = 1000$ with a filter radius small enough that the standard EF is not overdiffusive, since the paper concedes EF can then be competitive; if VMS-EFFC no longer improves the velocity $L^2$ norm or drag coefficient, the claimed superiority does not extend beyond the overdiffusive regime. Alternatively, treat the pointwise relative error in the paper's own Figure 11 as the decisive metric, which would invert the conclusion, since EF has the lower error there.","supporting_citations":[{"cited_title":"Strazzullo, M","cited_arxiv_id":null,"evidence_quote":"Provides the EF and EFR formulations at both FOM and ROM levels that serve as the baseline for all comparisons, including the grad-div parameter choice and the under-resolved test setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the van Cittert approximate deconvolution operator $D_1$ that formalizes the VMS-EFFC correction in Remark 4, linking the new method to a known deconvolution family."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the variational multiscale large/small resolved-scale decomposition (direct sum of resolved spaces) that justifies splitting the evolved velocity into large and small scales."},{"cited_title":"Ballarin, A","cited_arxiv_id":null,"evidence_quote":"Provides the supremizer stabilization used to build stable reduced velocity spaces, which is necessary for all the ROMs compared in the numerical experiments."},{"cited_title":"Bertagna, A","cited_arxiv_id":null,"evidence_quote":"Documents the overdiffusivity problem of EF and the relaxation strategy (EFR) that the new method aims to replace, motivating the need for a correction-based alternative."}],"review_version":1}