{"id":"de4199c0-446c-4d71-925d-e1c149de9e78","arxiv_id":"2505.10168","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A geometric-mean effective anisotropy parameter guides space-versus-time coarsening, making space-time multigrid robust enough to solve primal and adjoint systems in a 1D topology optimization of transient heat conduction.","lead":"This paper develops space-time multigrid solvers for transient heat conduction topology optimization, using a new coarsening rule based on the geometric mean of the minimum and maximum diffusivity. The work is relevant to anyone seeking parallel-in-time solvers for design optimization, though the current evidence is limited to one spatial dimension.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The lambda_eff criterion in Eq. 20 was selected by trial and error on the same narrow 1D test family used to validate it, so its transfer to V-cycle coarsening and to optimization-scale designs is the least supported link in the central claim.","rationale":"I read the paper as an empirical methods paper: the contribution is a heuristic coarsening rule plus evidence of robustness in a 1D optimization loop. The reader's CONDITIONAL verdict is fair. My stress-test pass does not find an internal contradiction or a numerical error in the presented experiments; the reported convergence factors, the exclusion of the unstable BP method, and the optimization history appear internally consistent. The load-bearing gap is the selection-validation overlap: Eq. 20 was chosen by inspecting the same six-problem family used to demonstrate it, the demonstration is two-grid only, and the multi-level Algorithm 3 is never compared against alternative lambda definitions. The optimization section narrows the criterion further by freezing the coarsening path using global extrema over chi (Eq. 42), so it does not directly validate the local Eq. 20 either. These are addressable conditions rather than fatal flaws, so I keep the reader's CONDITIONAL verdict. The proposed held-out experiment would settle whether Eq. 20 generalizes to designs outside its training family, which is the main unresolved condition for the paper's central claim.","tokens_in":27000,"tokens_out":5956,"duration_ms":64332,"concrete_test":"Hold out the six training problems and generate 20 new 1D high-contrast designs with multiple alternating conductor/insulator layers of random widths, not of the Eq. 23 ramp form. For each design, using the same BE-FE discretization, pointwise Jacobi smoother (omega = 1/2), and heat load Eq. 15, sweep t_T to vary lambda_eff and record the two-grid convergence factors for pure x-coarsening and pure t-coarsening. Compute the intersection point in lambda_eff for each design. If the spread of intersections over the 20 designs exceeds a factor of 16, or if a fixed lambda_crit = 0.25 mispredicts the better coarsening in more than about 10% of designs, then Eq. 20 is not reliable beyond its training family.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that lambda_eff = sqrt(min_e lambda_e * max_e lambda_e) (Eq. 20) reliably chooses between x- and t-coarsening for high-contrast transient heat conduction. The support is a trial-and-error selection (Section 4) over six 1D problems that all use the same design-field family (Eq. 23, a single linear transition with alpha = 10), the same artificial heat load (Eq. 15), the same pointwise Jacobi smoother (omega = 1/2), and uniform Cartesian meshes. The six problems vary contrast and volume fraction, but they do not sample the multiple alternating small features, greyscale transitions, or evolving material distributions that appear in the Section 7 optimization (Figure 12). Moreover, Figure 5 only demonstrates two-grid convergence-factor intersections for fixed coarsening types; Algorithm 3 applies lambda_eff recursively at every level, and no experiment compares Eq. 20 against the other expressions in Table 2 within a full V-cycle. The Section 7 demonstration further replaces the per-design min/max in Eq. 20 by global extrema over chi in [0,1] (Eqs. 42-44), freezing the coarsening path, so it tests a different and less local criterion. The paper honestly acknowledges these limits in the Conclusion, but the load-bearing condition for the central claim, namely that Eq. 20 predicts coarsening choices outside the training family, is not established by the presented experiments. This is an external-validity gap rather than an internal inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes space-time multigrid (STMG) methods for solving the all-at-once linear systems that arise in density-based topology optimization of transient heat conduction. The central methodological contribution is an effective anisotropy parameter, defined as the geometric mean of the minimum and maximum element diffusivity (Eq. 20), which is used by Algorithm 3 to choose between semi-coarsening in space and semi-coarsening in time for high-contrast heterogeneous diffusivity. The paper also compares four coarse-level reassembly schemes (averaging of conductivity, design field, resistivity, and Galerkin projection) with two interpolation stencils, and introduces a resolution-guarded coarsening strategy (Algorithm 4) intended for designs with small features. The proposed methods are applied as solvers for both the primal and adjoint systems in a one-dimensional, transient-thermal-compliance optimization problem, and converge in every optimization cycle.","tokens_in":27404,"tokens_out":6973,"duration_ms":70464,"significance":"If the proposed effective-anisotropy criterion is valid, it provides a simple, inexpensive way to guide semi-coarsening in STMG for heterogeneous diffusion problems, which is directly relevant to parallel-in-time algorithms for topology optimization. The paper's systematic comparison of rediscretisation methods and its demonstration that STMG works for the adjoint problem, including with a forward-in-time causal prolongation, are useful contributions. The authors are transparent about several limitations, explicitly noting the one-dimensional setting and the trial-and-error origin of the effective anisotropy parameter. However, the central heuristic was selected from the same family of test problems used to validate it, and the optimization demonstration replaces the per-design criterion with a global-extrema variant; these gaps materially limit the strength of the general claim and should be addressed before publication.","major_comments":[{"comment":"The effective anisotropy parameter λeff in Eq. (20) was selected through trial and error among the candidate expressions in Table 2 using the same six test problems (Table 1) that are then used to demonstrate that it \"works well\". All six problems share the design-field family in Eq. (23) with α = 10, the artificial heat load in Eq. (15), the same damped pointwise Jacobi smoother with ω = 1/2, and uniform Cartesian meshes. The claim that λeff reliably determines the coarsening direction is therefore an in-sample statement, and the reported range of intersection points in Figure 5 is not an independent estimate of predictive performance. Because Algorithm 3 applies λeff recursively at every level, this selection-on-validation effect is load-bearing for the central claim. The authors should either test λeff on problems outside this family (for example, multiple alternating features, greyscale transitions, different smoothers or heat loads), or explicitly scope the claim to the tested configuration; the concluding caveat that the authors do not know whether the definition works for other discretisations and multigrid components should be reflected in the abstract and at the point where the criterion is proposed.","section":"Section 4, Eq. (20), Table 2"},{"comment":"The topology-optimization demonstration does not actually use Eq. (20) as stated. In Eqs. (42)-(44), the per-design extrema over element diffusivities are replaced by global extrema of DSIMP over the full range χ ∈ [0,1], which are simply the pure-material diffusivities. This freezes the coarsening path for every optimization cycle and makes the criterion independent of the evolving design. Consequently, the convergence results in Figure 13 validate a different, less local coarsening criterion, not the central per-design λeff rule of Algorithm 3. The manuscript should either recompute λeff from the current design each cycle and report whether the frozen path differs from the adaptive one, or explicitly state that the optimization study evaluates only the frozen-extrema variant.","section":"Section 7, Eqs. (42)-(44)"},{"comment":"All numerical evidence in the paper is restricted to one spatial dimension, uniform Cartesian space-time meshes, and a single smoother (damped pointwise Jacobi with ω = 1/2). The rediscretisation comparison in Section 5.3 attributes the superiority of the resistivity-averaging method to the fact that one-dimensional thermal resistances combine in series, which implies that the ranking of reassembly methods is dimension-dependent. The abstract and intro frame the methods as \"suitable for topology optimisation\" of transient heat conduction, which in practice is a two- or three-dimensional problem. The current evidence does not establish transferability to higher dimensions or to other smoothers, so the claims of suitability should be explicitly scoped to the one-dimensional configuration, or supplemented by at least one two-dimensional test case.","section":"Sections 5-7"}],"minor_comments":[{"comment":"The comparison of candidate expressions for λeff is reported only qualitatively as \"Yes\" or \"No\" without giving the actual ranges of intersection points; reporting the quantitative ranges, as done for the chosen expression in Figure 5, would make the selection argument more convincing.","section":"Section 4, Table 2"},{"comment":"The stencil matrices in Eqs. (9), (25), and (26) appear to be typeset with reversed brackets and unclear orientation, making it difficult to identify which stencil direction corresponds to space and which to time; please fix the LaTeX formatting.","section":"Section 5.1, Eqs. (9), (25), (26)"},{"comment":"The statement that the M ≤ 8 and M = 16 curves are identical for the CR-method is not visible from the figure alone; adding markers or separating the curves would make this claim verifiable.","section":"Section 6.1, Figure 11"},{"comment":"The vertical scale of Figure 13 differs between panels (a) and (b), which reach about 20 cycles, and panel (c), which reaches about 80 cycles; a common scale or explicit annotation would ease cross-panel comparison.","section":"Section 7.3, Figure 13"},{"comment":"The all-at-once matrix in Eq. (5) is large and presented without row-block annotations; marking the Dirichlet row and the C/Δt + K blocks would improve readability.","section":"Section 2.2, Eq. (5)"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its limitations, but the main heuristic was selected on the same test family used to validate it, and all experiments are one-dimensional. The optimization demonstration further substitutes a frozen global-extrema criterion for the per-design λeff rule, so the central claim is not yet supported in the exact setting the paper targets. I would suggest the editor require either broader validation (different smoothers, loads, and at least one 2D case) or a clearly scoped presentation. No code or data repository is provided, which limits reproducibility of the numerical study."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful new thing is the effective anisotropy parameter, λ_eff = √(min λ_e · max λ_e), for choosing between x- and t-coarsening in high-contrast transient heat conduction. Prior STMG work on heterogeneous problems used min-based quantities (Neumüller/Schwalsberger); the geometric mean is a sensible middle ground, and the paper backs it with two-grid convergence-factor curves on six 1D problems spanning k-contrast, c-contrast, and asymmetric volume fractions.\n\nThe systematic comparison of eight rediscretisation combinations (causal vs. bilinear interpolation × K/D/R/P reassembly) is also genuinely new. The results are clean: resistivity-averaging wins for the small-feature 1D problems, consistent with series resistances; causal interpolation beats bilinear when features are easy to resolve; and the BP combination is unstable, worth knowing before you try it. Section 7 pushes the solver inside a 1D topology optimization, converging every cycle for both primal and adjoint systems, and the observation that causal forward-in-time prolongation still works for the backward-in-time adjoint is a useful data point.\n\nThe soft spots are real but not fatal. The λ_eff formula was selected by trial and error on the same six-problem family used to validate it; the paper says so explicitly and lists the rejected candidates in Table 2, which is honest, but it is still selection-on-validation. Everything is one spatial dimension, uniform Cartesian meshes, one smoother (damped Jacobi, ω = 1/2), one synthetic heat load, and no shipped code. On top of that, Section 7 replaces the per-design min/max of Eq. (20) with global extrema over χ ∈ [0,1] (Eqs. 42–44), freezing the coarsening path. So the optimization demo does not actually exercise the local λ_eff rule on the evolving designs, even though the paper implies it does.\n\nNone of this is a takedown. The authors flag the main caveats themselves in the conclusion, the convergence data are shown in sufficient detail to check, and the rediscretisation study stands on its own. What is missing is support for the claim that λ_eff predicts coarsening choices outside the 1D training family: one V-cycle comparison of Eq. (20) against the Table 2 alternatives, and one 2D experiment, would go a long way.\n\nBottom line: this is honest, well-executed empirical work, worth a serious referee. I would send it to review and ask the authors to address the external-validity gap rather than reject. If I worked on parallel-in-time methods for high-contrast diffusion, I would cite it and bring it to a reading group.","headline":"A solid, honest empirical study: the geometric-mean effective anisotropy parameter for semi-coarsening is a genuinely new idea for high-contrast STMG, and the eight-way rediscretisation comparison is useful, but the main heuristic was tuned on the same 1D test family used to validate it, so its external validity is the weak link.","tokens_in":27849,"tokens_out":2233,"would_cite":true,"duration_ms":23057,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65K10","65M55","65M22","65M32","80M10","80M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"One scalar, the geometric mean of min and max diffusivity, reliably picks space- or time-coarsening at each multigrid level; a solver built on it converged for the primal and adjoint systems throughout a 1D topology optimisation.","keywords":["topology optimisation","space-time multigrid","parallel in time","transient","heat conduction","high-contrast","effective anisotropy parameter","semi-coarsening"],"falsifier":"Run the same two-grid STMG method, with the same pointwise Jacobi smoother, on a two-dimensional high-contrast problem containing a thin insulating layer, and measure the spread of $\\lambda_{\\mathrm{eff}}$ values at which the space-coarsening and time-coarsening convergence curves cross. If that crossover band is wider than the roughly one decade ($2^{-3}$ to $2^{-1}$) observed in one dimension, the geometric-mean indicator does not transfer beyond the tested setting; equivalently, repeating the six test problems with a realistic pulsed heat load instead of the synthetic load of Equation (15) would reveal how much of the reliability is load-dependent.","tokens_in":26813,"feed_emoji":"🔥","tokens_out":16744,"duration_ms":133505,"temperature":0.7,"pith_summary":"Topology optimisation of transient heat conduction is expensive because the heat equation must be solved many times while the material layout evolves. This paper proposes space-time multigrid (STMG) methods to make those solves fast and parallelisable in time, and addresses the obstacle that makes STMG hard for design problems: diffusivity contrasts of $10^4$ between conductor and insulator mean the same grid may need time-coarsening in one region and space-coarsening in another. The central proposal is a single indicator, the effective anisotropy parameter $\\lambda_{\\mathrm{eff}} = \\sqrt{\\min_e \\lambda_e \\max_e \\lambda_e}$, which at each multigrid level decides whether to coarsen in space or in time. Across six one-dimensional test problems the crossover between the two choices fell in a narrow band of $\\lambda_{\\mathrm{eff}}$, and the resulting solver converged for both the state and adjoint equations at every cycle of a one-dimensional topology optimisation. Should the indicator transfer to two and three dimensions, the payoff would be a dependable, parallel-in-time solver for design problems that currently take days of computation.","feed_headline":"One number decides how heat-flow solvers coarsen the grid","feed_subtitle":"Space-time multigrid with this rule converged every cycle of a 1D topology optimisation, primal and adjoint.","key_machinery":"The load-bearing object is the effective anisotropy parameter, $\\lambda_{\\mathrm{eff}}$, the geometric mean of the minimum and maximum element-level anisotropy values. It compresses a spatially varying anisotropy field $\\lambda_e = D_e \\Delta t/\\Delta x^2$ that spans orders of magnitude in high-contrast problems into the single number that the coarsening strategy (Algorithm 3) compares against the threshold $\\lambda_{\\mathrm{crit}} = 0.25$: coarsen in time if $\\lambda_{\\mathrm{eff}} < \\lambda_{\\mathrm{crit}}$, otherwise coarsen in space. The second mechanism is the rediscretisation pair: causal, forward-in-time-only prolongation stencils, and coarse-level system matrices reassembled by averaging the thermal resistivity $\\rho = 1/k$ rather than the conductivity, the discrete analogue of combining series resistors, which is why it suits one spatial dimension. A third, optional mechanism (Algorithm 4) forces time-coarsening whenever space-coarsening would drop the spatial element count below a threshold $M$, trading coarse-grid resolution of small features against the solver's sensitivity to large anisotropy.","core_discovery":"On its own terms, the paper establishes that, for the all-at-once backward-Euler finite-element discretisation of transient heat conduction with strongly heterogeneous diffusivity, the question of whether the next multigrid level should coarsen in space or in time can be answered level by level by comparing a single scalar against a fixed threshold. The scalar is the effective anisotropy parameter $\\lambda_{\\mathrm{eff}} = \\sqrt{\\min_e(\\lambda_e) \\max_e(\\lambda_e)}$, where $\\lambda_e = D_e \\Delta t/\\Delta x^2$ is the per-element anisotropy parameter and $D_e = k_e/c_e$ the thermal diffusivity; equivalently $\\lambda_{\\mathrm{eff}} = D_{\\mathrm{eff}} \\Delta t/\\Delta x^2$ with $D_{\\mathrm{eff}}$ the geometric mean of the minimum and maximum diffusivity. The paper reports that this expression was selected by trial and error over six one-dimensional test problems with material contrasts up to $10^4$ in conductivity and/or heat capacity, and that it located the space-versus-time coarsening crossover in the narrow band $2^{-3} \\le \\lambda_{\\mathrm{eff}} \\le 2^{-1}$ across all six problems, beating seven alternative candidate expressions. It further establishes that, among eight rediscretisation combinations, averaging the thermal resistivity $\\rho = 1/k$ when assembling coarse levels performed best in one dimension, because a 1D heat path behaves as resistors in series, and that causal prolongation (transferring information only forwards in time) beat bilinear interpolation for problems without small features. Applied to a one-dimensional topology optimisation, the STMG solver converged in at most 80 cycles in every optimisation cycle, for both the primal system and the adjoint system, even though the adjoint problem's time direction is backwards.","pith_inferences":["The paper's own series-versus-parallel argument implies the ranking of reassembly methods should flip in two or three dimensions: a finned or branched heat sink conducts mostly in parallel, so averaging conductivities or using Galerkin projection should beat averaging resistivities there, a direct, testable extension the authors flag but do not demonstrate.","Because the design-independent version of $\\lambda_{\\mathrm{eff}}$ uses only the two material extremes, the whole coarsening path can be precomputed before an optimisation starts; an extension of this observation is that a general-purpose space-time solver library could choose its semi-coarsening direction without ever reading the design field.","The fact that forward-only prolongation works for the backwards-running adjoint problem suggests the causal stencil is matched to the one-sided time coupling of backward Euler, not to the physical direction of time; for symmetric time integrators such as Crank-Nicolson the same choice may need re-examination.","Because $\\lambda_{\\mathrm{eff}}$ depends only on the extremes of the diffusivity field, it is blind to where the contrast sits; layered test problems may not probe geometries where a high-diffusivity island is surrounded by insulator, and a two-dimensional analogue of problem 8 would show whether the observed resolution heuristic $M \\approx 1/F$ survives."],"forward_implications":["On the six one-dimensional test problems, the crossover between space- and time-coarsening lies within $2^{-3} \\le \\lambda_{\\mathrm{eff}} \\le 2^{-1}$ despite diffusivity contrasts of $10^4$, so a fixed threshold $\\lambda_{\\mathrm{crit}} = 0.25$ is a workable default.","Because $\\lambda_{\\mathrm{eff}}$ can be defined from the material extremes alone, the coarsening path for an entire optimisation can be frozen: prolongation, restriction, and coarse-level storage are built once and reused in every optimisation cycle.","In the 1D optimisation test, the STMG solver converged within at most 80 cycles in every optimisation cycle, for both the primal and the adjoint system, with warm restarts consistently reducing the cycle count.","Causal (forward-in-time-only) prolongation works for the adjoint problem even though adjoint time runs backwards, and this holds also when Galerkin projection embeds the prolongation directly in the coarse-level operators.","For one-dimensional problems with thin features, averaging thermal resistivities on the coarse levels outperforms averaging conductivities, averaging the design field, and Galerkin projection, and the resistivity method's advantage is largest for very small features ($F \\le 1/128$)."],"supporting_citations":[{"why":"Original source of the adaptive semi-coarsening strategy and the threshold concept that Algorithms 2 and 3 generalise.","marker":"[44]"},{"why":"The multigrid implementation that the STMG code modifies, and a standard reference for Galerkin projection in topology-optimisation multigrid.","marker":"[12]"},{"why":"Provides the measured aluminium and epoxy material properties used by test problems 7-9 and the optimisation.","marker":"[56]"},{"why":"Basis of the resistivity-averaging reassembly: the R-method is the one-dimensional analogue of this rediscretisation proposed for electrical conductivity.","marker":"[53]"},{"why":"Source of the penalty powers p_k=3, p_c=2 adopted for the SIMP material interpolation.","marker":"[7]"},{"why":"Supplies the Method of Moving Asymptotes used to update the design field in the optimisation loop.","marker":"[58]"},{"why":"Defines the discrete adjoint method, hence the adjoint system the STMG method is required to solve.","marker":"[59]"},{"why":"Reports space-time multigrid among the fastest parallel-in-time methods, motivating the choice of STMG over alternatives.","marker":"[43]"}],"fun_headline_variants":["Geometric-mean diffusivity rule steers space-time multigrid coarsening","One scalar decides coarsening for transient heat topology optimisation","Physics rule for multigrid levels: geometric mean of diffusivity","Multigrid coarsening key: geometric-mean diffusivity threshold","Space-time multigrid coarsening set by one diffusivity number"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the six one-dimensional test problems, each run with a single synthetic heat load, one smoother (damped pointwise Jacobi, damping factor $1/2$), and uniform Cartesian meshes, are representative enough that the geometric-mean indicator and coarsening rule, selected by trial and error on them, transfer to other loads, smoothers, and dimensions without further tuning.","fun_headline_variants_meta":{"raw":{"variants":["Geometric-mean diffusivity rule steers space-time multigrid coarsening","One scalar decides coarsening for transient heat topology optimisation","Physics rule for multigrid levels: geometric mean of diffusivity","Multigrid coarsening key: geometric-mean diffusivity threshold","Space-time multigrid coarsening set by one diffusivity number"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000825,"raw_usage":{"total_tokens":3719,"prompt_tokens":1171,"completion_tokens":2548,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":787,"completion_tokens_details":{"reasoning_tokens":2453}},"tokens_in":787,"tokens_out":2548,"duration_ms":17421,"temperature":1.0,"reasoning_tokens":2453,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:15:04.259601+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same two-grid STMG method, with the same pointwise Jacobi smoother, on a two-dimensional high-contrast problem containing a thin insulating layer, and measure the spread of $\\lambda_{\\mathrm{eff}}$ values at which the space-coarsening and time-coarsening convergence curves cross. If that crossover band is wider than the roughly one decade ($2^{-3}$ to $2^{-1}$) observed in one dimension, the geometric-mean indicator does not transfer beyond the tested setting; equivalently, repeating the six test problems with a realistic pulsed heat load instead of the synthetic load of Equation (15) would reveal how much of the reliability is load-dependent.","supporting_citations":[{"cited_title":"Horton, S","cited_arxiv_id":null,"evidence_quote":"Original source of the adaptive semi-coarsening strategy and the threshold concept that Algorithms 2 and 3 generalise."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The multigrid implementation that the STMG code modifies, and a standard reference for Galerkin projection in topology-optimisation multigrid."},{"cited_title":"Subramaniam, T","cited_arxiv_id":null,"evidence_quote":"Provides the measured aluminium and epoxy material properties used by test problems 7-9 and the optimisation."},{"cited_title":"Moucha, R","cited_arxiv_id":null,"evidence_quote":"Basis of the resistivity-averaging reassembly: the R-method is the one-dimensional analogue of this rediscretisation proposed for electrical conductivity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the penalty powers p_k=3, p_c=2 adopted for the SIMP material interpolation."},{"cited_title":"Svanberg, The method of moving asymptotes—a new meth od for structural opti- mization, International Journal for Numerical Methods in E ngineering 24 (1987) 359–","cited_arxiv_id":null,"evidence_quote":"Supplies the Method of Moving Asymptotes used to update the design field in the optimisation loop."},{"cited_title":"Michaleris, D","cited_arxiv_id":null,"evidence_quote":"Defines the discrete adjoint method, hence the adjoint system the STMG method is required to solve."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports space-time multigrid among the fastest parallel-in-time methods, motivating the choice of STMG over alternatives."}],"review_version":1}