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REVIEW 4 major objections 4 minor 2 cited by

One-shot Parareal Approach for Topology Optimisation of Transient Heat Flow

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Pith's one-line read A single warm-started Parareal iteration per design update speeds transient heat-conduction topology optimization by 4.95x on 16 threads, with final objectives within 2% of the sequential reference.

desk verdict Honest, well-written one-shot Parareal paper with a real but only partially tested sensitivity-accuracy claim; deserves review, not rejection. read the letter →

arxiv 2411.19030 v1 pith:IXNA6WMZ submitted 2024-11-28 cs.CE

Magnus Appel , Joe Alexandersen This is my paper · ORCID
classification cs.CE MSC 65K1065M5565Y0568W1065M2265M3280M1080M50
keywords topologyoptimisationtransientheatconductionflowparallel-in-timePararealone-shotapproachadjointsensitivityanalysisMethodofMovingAsymptotes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to make transient heat-conduction topology optimization cheaper by using the parallel-in-time method Parareal, then finds that letting Parareal converge before each design update gives no useful speedup. The proposed fix is a one-shot approach: run exactly one warm-started Parareal iteration per optimization cycle, using the previous design's solution as the initial guess, and let the optimizer work with the resulting approximate objective and sensitivities. On a two-dimensional test problem, the method reaches a peak speedup of 4.95x on 16 threads and produces final designs whose objective values lie within roughly 2% of the sequential reference, with qualitatively similar topologies. The paper also identifies the sequential evaluation of coarse propagators as the main bottleneck, and shows that a zero-coarse-propagator variant is faster but unstable.

What carries the argument

The load-bearing mechanism is the one-shot, warm-started Parareal iteration with augmented state variables for the objective and its gradient. Parareal splits the time axis into blocks, runs expensive fine propagators in parallel, and applies a sequential correction from a cheap coarse propagator; the paper modifies it to save every intermediate temperature field so the adjoint equation can be evaluated. It then appends a cumulative objective θ_n to the primal state and a cumulative sensitivity g_n to the adjoint state, so one Parareal pass returns the total objective Θ = θ_Nt and the total gradient ∇Θ = g_1. At each design step, the initial guess is the previous design's solution, turning one iteration into an approximate correction of a nearby state rather than a cold start.

What would settle it

Compute exact sequential sensitivities at a late optimization iteration and compare them with the one-shot Parareal sensitivity estimate g1; if the relative error exceeds the loosest tolerance ($10^{-2}$) that the preliminary tests required, the convergence of the optimizer is not supported. Alternatively, run the same test with a longer terminal time or a faster oscillating heat load; if the final design's true objective moves outside the ±2% band relative to sequential optimization, the central comparability claim fails.

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Extended reading notes

Core claim

The central discovery is that a single warm-started Parareal iteration per optimization cycle, rather than a converged time-parallel solve, is enough for gradient-based topology optimization of transient heat conduction to reach designs comparable to the sequential benchmark. The paper modifies Parareal to store intermediate fine time states so that adjoint sensitivity analysis is possible, augments the primal and adjoint states with cumulative objective and cumulative sensitivity variables, and feeds the resulting estimates to the optimizer. On the test problem, the one-shot method reached a peak speedup of 4.95x with 16 threads, and every final design's true objective fell within ±2% of the sequential reference, with similar material distributions. The comparison method, Parallel Local-in-Time, reached 11.8x with 32 threads but was unstable when the coarse time interval became too small relative to the physics.

Load-bearing premise

The method's accuracy depends on the assumption that one warm-started Parareal iteration per optimization step yields objective and sensitivity estimates good enough for the optimizer to converge, even though the intermediate fine temperature states that feed the sensitivities are acknowledged to be quite poor, and the assumption is tested on one problem only.

Editorial extensions

If this is right

  • One-shot Parareal can cut wall-clock time by about a factor of five on 16 threads for transient heat-conduction topology optimization, making it a practical alternative to fully sequential time stepping.
  • The optimizer does not need a converged time-parallel solve; one warm-started Parareal iteration per design update is enough for the optimizer to converge to a comparable local minimum.
  • Final designs vary slightly across thread counts because the approximations depend on the number of coarse intervals, but all tested cases stayed within roughly 2% of the sequential reference objective.
  • Further speedup depends mainly on making the coarse propagators cheaper; replacing them with zero-cost propagators is faster but loses stability.
  • If both the one-shot method and the sequential reference are stopped after the same wall-clock time, the one-shot method is likely to give better designs because each of its iterations is much cheaper.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same recipe should transfer to other time-dependent PDE-constrained design problems, such as transient linear elasticity or diffusion-reaction, where a backward-Euler primal/adjoint pair exists; testing it there would show whether the one-shot accuracy assumption depends on the coarse propagator's error structure.
  • The ±2% objective closeness is a single-problem result; with faster load oscillations or longer time horizons, the one-shot sensitivity estimates could degrade and the final designs might diverge more from the sequential reference.
  • A route indicated but not tested in the paper is to coarsen the coarse propagator in space or parallelize it spatially, which directly attacks the identified bottleneck and could push speedup toward the PLT level while retaining stability.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a one-shot Parareal method for topology optimization of transient heat conduction. Parareal is modified to store intermediate temperature states, cumulative objective and sensitivity variables are introduced, and a single warm-started Parareal iteration is used per MMA design update. The method is tested on a 2D transient heat problem with a p=20 power-mean objective; the authors report a peak speedup of 4.95x on 16 threads, final objectives within ±2% of the sequential reference, and qualitatively similar designs. A comparison with Parallel Local-in-Time (PLT) topology optimization shows PLT is unstable for large thread counts, and a bottleneck analysis attributes most of the one-shot Parareal overhead to the sequential coarse propagators.

Significance. The contribution is timely and potentially useful: time parallelization of transient topology optimization is an open practical problem, and the one-shot warm-started formulation is a natural and clearly explained extension of Parareal. The adjoint and cumulative-state derivations in Section 4 are algebraically consistent, and no fitted parameters are used. The paper is transparent about the known weakness (the intermediate fine-time states are poor, Figure 3) and about the MATLAB timing artifacts (Appendix A), and the comparison with PLT gives a useful reference point. The central limitation is that the accuracy of the one-shot objective/sensitivity estimates during the optimization loop is not measured at all; the empirical claims rest on a single test problem. If the claims hold, the method is a worthwhile addition, but the validation currently needs strengthening.

major comments (4)
  1. [Section 4.3, Eq. (4.7)] The central claim that one warm-started Parareal iteration per MMA update yields designs comparable to sequential optimization rests on an undocumented accuracy assumption. Section 4.3 states that the modified Parareal returns intermediate fine-time states that are 'quite poor' (Figure 3), and Eq. (4.7) assembles ∇Θ from the full set of intermediate temperatures. Section 5 measures objective and sensitivity errors only for a fixed design with a cold-start, multi-iteration Parareal; it does not measure the error of the one-shot warm-started estimates inside the actual optimization loop. The authors should report the relative errors in Θ and ∇Θ as a function of optimization iteration for at least one case (e.g., Nτ=16), or provide an error bound, to substantiate that the estimates remain accurate enough for MMA to make progress. Without this, the connection between the reported design quality and the Parareal estimates is an empirical assumption.
  2. [Section 6.2] The generalizability of the result is not yet established. The method is tested on exactly one problem: a single square domain, one oscillatory heat load, and a p=20 power-mean objective. The authors note in Section 6.2 that the final designs differ with Nτ and that the problem is multimodal; with no repeated runs or additional test cases, the observed ±2% objective agreement could be specific to the smoothing properties of the p=20 power mean and the chosen load. A second test case with a different domain, load, or objective, as well as a few runs with different initial designs or problem parameters, are needed to support the claim that the one-shot method reliably produces comparable designs.
  3. [Section 6.2 and Appendix A] The headline speedup of 4.95x is based on single wall-clock measurements, and the authors themselves state in Appendix A that the timing work-around 'adds a bit of uncertainty' to the sequential reference time. Because the speedup is a central quantitative claim, at least several timing repeats per thread count should be reported, along with the mean or median and spread. This is especially important since the speedup curve in Figure 7 is non-monotonic and the method converges to different local minima depending on Nτ.
  4. [Section 6.2, final paragraph] The paper states that if both methods are stopped after the same wall-clock time, the one-shot Parareal method is likely to return better results than the reference method, but this is not demonstrated. The objective histories in Figure 10 are plotted against optimization iteration, not wall-clock time, and the one-shot method lags up to 4% behind the reference for the first 50 iterations. The authors should provide an objective-versus-wall-clock-time comparison, or at least a quantitative argument using the measured per-iteration times, before making this practical claim.
minor comments (4)
  1. [Section 4.4] There is a typo: the text says 'the coarse propagator for the primal problem, Gadj' but the context is the adjoint problem; it should read 'the coarse propagator for the adjoint problem'.
  2. [Algorithm 4.2] The terminal condition for the cumulative sensitivity is not stated explicitly; the algorithm should specify that g_{Nt+1} = 0 before the backward sweep.
  3. [Section 5.2] The preliminary tests use a fixed design obtained after a completed sequential optimization; a one-sentence reminder that this design is not representative of the early one-shot iterations would help readers interpret the relevance of the measured errors.
  4. [Reproducibility] No code or data availability statement is provided; making the MATLAB implementation and test case available would substantially improve reproducibility and would allow independent verification of the speedup and design comparisons.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the one-shot Parareal estimates are derived from the same discrete state and adjoint equations as the sequential reference, not fitted or imported via self-citation; the acknowledged in-loop accuracy limitations are empirical risk, not definitional circularity.

full rationale

Walking the derivation chain, I find no step in which a predicted quantity is defined in terms of itself or in which a fitted or renamed input is presented as a prediction. The discrete primal and adjoint problems (Eqs. (2.7), (4.1)-(4.7)) are the same equations used by the sequential reference, and the Parareal estimates of Theta and gradient Theta are produced by propagating those equations with the cumulative-objective and cumulative-sensitivity states (Eqs. (4.11)-(4.16)); the one-shot loop (Algorithm 6.1) replaces exact solves with one warm-started Parareal correction. There are no data-fitted parameters that force the reported 4.95x speedup or the final objectives within +/-2%; parameters such as Ntau, beta, eta, r_fil, p, k0 and kmin are fixed problem inputs or swept values, and speedup and objective values are measured against a separately computed sequential reference. The admitted weakness, that Section 4.3 states the modified Parareal's intermediate fine-time states are 'quite poor' and Section 6.2 attributes different local minima to Ntau-dependent estimates, is an accuracy and robustness limitation of using one Parareal iteration, not circularity: the estimates are not equal to the reference output by construction, and convergence is checked by evaluating the true objective (Figure 10). The only co-author citation ([30]) is background on transient heat-sink effects; all load-bearing algorithmic references ([16], [20], [24], [26]) are external prior work. The PLT comparison is an independent benchmark, explicitly re-derived as the zero-coarse-propagator case in Eqs. (7.1)-(7.2), and the instability of PLT for large Ntau is an empirical observation, not a self-citation-forced conclusion. Even Appendix A's timing correction makes the sequential comparison fairer rather than manufacturing speedup. I therefore find no significant circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

No model parameters are fitted to target data; parameters are standard topology optimization inputs, algorithm settings, or problem setup. Ntau is swept as the parallelism level. The central claim depends on these choices, but they are not calibrated to force the reported speedup or design quality. The auxiliary cumulative states are paper-specific algorithmic devices rather than physical postulates.

free parameters (5)
  • Objective power p = 20
    Chosen to approximate the max temperature norm; affects objective and sensitivity magnitudes and therefore the optimized designs.
  • Coarse time-point count Ntau (thread count) = varied over divisors of 480 from 2 to 32
    The central speedup and final design depend strongly on Ntau; it is swept, not fitted to a target result.
  • Projection/filter parameters (beta, eta, rfil) = beta=32, eta=0.5, rfil=0.03
    Standard topology optimization filter and projection settings chosen without calibration; they influence design quality and convergence.
  • Maximum area fraction amax = 0.3
    Volume constraint of the test problem; a standard problem parameter.
  • Material and SIMP penalty parameters (k0, kmin, c0, cmin, pk, pc) = 3, 0.03, 1, 0.5, 3, 2
    From Table 1; these define the physics of the test problem and are not fitted.
assumptions (5)
  • standard math Parareal converges to the sequential fine-propagator solution after enough iterations (standard Parareal theory).
    Invoked throughout Section 4.2 to justify the correction formula and the use of the sequential solution as reference; cited to Lions et al. and Gander-Vandewalle.
  • standard math Adjoint sensitivity formula gradTheta = -1/Nt Lambda^T partialR/partialChi is valid under the state constraint M S = Q.
    Equations (4.1)-(4.4); standard Lagrangian/adjoint derivation for differentiable discrete PDE constraints.
  • domain assumption Warm start: design changes slowly between MMA updates, so previous primal and adjoint solutions are good initial guesses for one Parareal iteration.
    Section 6, paragraph 2; this is load-bearing because the one-shot approach does only one Parareal iteration per optimization step.
  • ad hoc to paper One Parareal iteration with this warm start gives objective and sensitivity estimates accurate enough for MMA to converge to a good design.
    Not proven; the entire empirical claim rests on this, and it is tested on one problem only.
  • domain assumption The chosen heat source q(x,t) = 0.5(1-t)[1+cos(50t)] is representative enough to test the method.
    Section 5.1; the oscillatory load is designed to challenge coarse propagators, but generality is not established.
invented entities (2)
  • Cumulative objective state theta_n
    purpose: Auxiliary state solved alongside temperature in Parareal so the objective is available at every coarse time point without post-processing.
    Defined in Eq. (4.11)-(4.13); it is an algorithmic construct with no external falsifiable handle.
  • Cumulative sensitivity state g_n
    purpose: Auxiliary state solved backward in time alongside adjoint temperature so total sensitivities are available from one Parareal solve.
    Defined in Eq. (4.15)-(4.17); it is an algorithmic construct with no external falsifiable handle.

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Pith. "Pith review of One-shot Parareal Approach for Topology Optimisation of Transient Heat Flow." pith.science (2026). https://pith.science/paper/IXNA6WMZ

@misc{pith2026241119030,
  author       = {Pith},
  title        = {Pith review of: One-shot Parareal Approach for Topology Optimisation of Transient Heat Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXNA6WMZ}},
  note         = {Machine review of arXiv:2411.19030}
}
abstract

This paper presents a method of performing topology optimisation of transient heat conduction problems using the parallel-in-time method Parareal. To accommodate the adjoint analysis, the Parareal method was modified to store intermediate time steps. Preliminary tests revealed that Parareal requires many iterations to achieve accurate results and, thus, achieves no appreciable speedup. To mitigate this, a one-shot approach was used, where the time history is iteratively refined over the optimisation process. The method estimates objectives and sensitivities by introducing cumulative objectives and sensitivities and solving for these using a single iteration of Parareal, after which it updates the design using the Method of Moving Asymptotes. The resulting method was applied to a test problem where a power mean of the temperature was minimised. It achieved a peak speedup relative to a sequential reference method of $5\times$ using 16 threads. The resulting designs were similar to the one found by the reference method, both in terms of objective values and qualitative appearance. The one-shot Parareal method was compared to the Parallel Local-in-Time method of topology optimisation. This revealed that the Parallel Local-in-Time method was unstable for the considered test problem, but it achieved a peak speedup of $12\times$ using 32 threads. It was determined that the dominant bottleneck in the one-shot Parareal method was the time spent on computing coarse propagators.

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Forward citations

Cited by 2 Pith papers

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  1. Large-Scale Topology Optimisation of Time-dependent Thermal Conduction Using Space-Time Finite Elements and a Parallel Space-Time Multigrid Preconditioner

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    Treating time as a spatial dimension and preconditioning the all-at-once system with space-time multigrid yields up to 52x speed-up for transient thermal topology optimization, demonstrated up to 4.2 billion degrees o...

  2. Space-Time Multigrid Methods Suitable for Topology Optimisation of Transient Heat Conduction

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    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.

Reference graph

Works this paper leans on

30 extracted references · 19 canonical work pages · cited by 2 Pith papers

  1. [1]

    N. Aage, E. Andreassen, and B. S. Lazarov , Topology optimization using petsc: An easy- to-use, fully parallel, open source topology optimization framework, Structural and Multi- disciplinary Optimization, 51 (2015), pp. 565–572. ONE-SHOT PARAREAL METHOD OF TOPOLOGY OPTIMISATION 25

    work page 2015

  2. [2]

    N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund , Giga-voxel computational mor- phogenesis for structural design , Nature, 550 (2017), pp. 84–86

    work page 2017

  3. [3]

    Alexandersen, O

    J. Alexandersen, O. Sigmund, and N. Aage , Large scale three-dimensional topol- ogy optimisation of heat sinks cooled by natural convection , International Journal of Heat and Mass Transfer, 100 (2016), pp. 876–891, https://doi.org/https://doi.org/ 10.1016/j.ijheatmasstransfer.2016.05.013, https://www.sciencedirect.com/science/article/ pii/S0017931015307365

    doi:10.1016/j.ijheatmasstransfer.2016.05.013 2016

  4. [4]

    Amir, One-shot procedures for efficient minimum compliance topolo gy optimization , Struc- tural and Multidisciplinary Optimization, 67 (2024), p

    O. Amir, One-shot procedures for efficient minimum compliance topolo gy optimization , Struc- tural and Multidisciplinary Optimization, 67 (2024), p. 39 , https://doi.org/10.1007/ s00158-024-03763-5

    work page 2024

  5. [5]

    Borrvall and J

    T. Borrvall and J. Petersson , Large-scale topology optimization in 3d using paral- lel computing , Computer Methods in Applied Mechanics and Engineering, 19 0 (2001), pp. 6201–6229, https://doi.org/https://doi.org/10.1016/S0045-7825(01)00216-X, https:// www.sciencedirect.com/science/article/pii/S004578250100216X

    doi:10.1016/s0045-7825(01)00216-x 2001

  6. [6]

    Bourdin, Filters in topology optimization , International Journal for Numerical Methods in Engineering, 50 (2001), pp

    B. Bourdin, Filters in topology optimization , International Journal for Numerical Methods in Engineering, 50 (2001), pp. 2143–2158, https://doi.org/https://doi.org/10.1002/nme.116, https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.116, https://arxiv.org/abs/https:// onlinelibrary.wiley.com/doi/pdf/10.1002/nme.116

    doi:10.1002/nme.116 2001

  7. [7]

    T. E. Bruns and D. A. Tortorelli , Topology optimization of non-linear elastic structures and compliant mechanisms , Computer methods in applied mechanics and engineering, 19 0 (2001), pp. 3443–3459

    work page 2001

  8. [8]

    Dbouk , A review about the engineering design of optimal heat transf er systems using topology optimization , Applied Thermal Engineering, 112 (2017), pp

    T. Dbouk , A review about the engineering design of optimal heat transf er systems using topology optimization , Applied Thermal Engineering, 112 (2017), pp. 841– 854, https://doi.org/https://doi.org/10.1016/j.applthermaleng.2016.10.134, https://www. sciencedirect.com/science/article/pii/S135943111632645X

    doi:10.1016/j.applthermaleng.2016.10.134 2017

Show all 30 references
  1. [9]

    J. D. Deaton and R. V. Grandhi , A survey of structural and multidisciplinary contin- uum topology optimization: post 2000 , Structural and Multidisciplinary Optimization, 49 (2014), pp. 1–38

    2014

  2. [10]

    X. Du, M. Sarkis, C. E. Schaerer, and D. B. Szyld , Inexact and truncated parareal-in-time krylov subspace methods for parabolic optimal control prob lems, Electronic Transactions on Numerical Analysis, 40 (2013), pp. 36–57

    2013

  3. [11]

    Emmett and M

    M. Emmett and M. Minion, Toward an efficient parallel in time method for partial differe ntial equations, Communications in Applied Mathematics and Computational Science, 7 (2012), pp. 105–132

    2012

  4. [12]

    Friedhoff, R

    S. Friedhoff, R. D. Falgout, T. Kolev, S. MacLachlan, and J. B. Schroder, A multigrid- in-time algorithm for solving evolution equations in paral lel, tech. report, Lawrence Liver- more National Lab.(LLNL), Livermore, CA (United States), 2 012

  5. [13]

    M. J. Gander, F. Kwok, and J. Salomon , Paraopt: A parareal algorithm for optimality sys- tems, SIAM Journal on Scientific Computing, 42 (2020), pp. A2773– A2802, https://doi. org/10.1137/19M1292291, https://doi.org/10.1137/19M1292291, https://arxiv.org/abs/ https://doi.org/10....

    2020 doi

  6. [14]

    M. J. Gander and S. Vandewalle , Analysis of the parareal time-parallel time-integration method, SIAM Journal on Scientific Computing, 29 (2007), pp. 556–57 8, https://doi.org/ 10.1137/05064607X, https://doi.org/10.1137/05064607X

    2007 doi

  7. [15]

    G¨otschel and M

    S. G¨otschel and M. L. Minion , Parallel-in-time for parabolic optimal control problems u s- ing pfasst , in Domain Decomposition Methods in Science and Engineerin g XXIV, P. E. Bjørstad, S. C. Brenner, L. Halpern, H. H. Kim, R. Kornhuber, T. Rahman, and O. B. Widlund, eds., C...

    2018

  8. [16]

    J. K. Guest, A. Asadpoure, and S.-H. Ha , Eliminating beta-continuation from heaviside projection and density filter algorithms , Structural and Multidisciplinary Optimization, 44 (2011), pp. 443–453

    2011

  9. [17]

    J. K. Guest, J. H. Pr´evost, and T. Belytschko, Achieving minimum length scale in topology optimization using nodal design variables and projection f unctions, International journal for numerical methods in engineering, 61 (2004), pp. 238–25 4

    2004

  10. [18]

    Horton and S

    G. Horton and S. Vandewalle , A space-time multigrid method for parabolic partial differ- ential equations , SIAM Journal on Scientific Computing, 16 (1995), pp. 848–86 4

    1995

  11. [19]

    /suppress Laniewski-Wo/suppress l/suppress lk and J

    L. /suppress Laniewski-Wo/suppress l/suppress lk and J. Rokicki, Adjoint lattice boltzmann for topology optimiza- tion on multi-gpu architecture , Computers & Mathematics with Applications, 71 (2016), pp. 833–848, https://doi.org/https://doi.org/10.1016/j.camwa.2015.12.043, ht...

    2016 doi

  12. [20]

    Lions, Y

    J.-L. Lions, Y. Maday, and G. Turinici , R´ esolution d’edp par un sch´ ema en temps parar´ eel, Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathe matics, 332 (2001), 26 MAGNUS APPEL AND JOE ALEXANDERSEN pp. 661–668, https://doi.org/https://doi.org/10.1016/S0764-4...

    2001 doi

  13. [21]

    G. I. N. Rozvany , On symmetry and non-uniqueness in exact topology optimizat ion, Struc- tural and Multidisciplinary Optimization, 43 (2011), pp. 2 97–317, https://doi.org/10.1007/ s00158-010-0564-0

    2011

  14. [22]

    N. R. G. S. G ¨unther and J. B. Schroder , A non-intrusive parallel-in-time approach for simultaneous optimization with unsteady pdes , Optimization Methods and Soft- ware, 34 (2019), pp. 1306–1321, https://doi.org/10.1080/10556788.2018.1504050, https:// doi.org/10.1080/105567...

    2019

  15. [23]

    Steinbach and H

    O. Steinbach and H. Yang , Comparison of algebraic multigrid methods for an adaptive space–time finite-element discretization of the heat equat ion in 3d and 4d , Numeri- cal Linear Algebra with Applications, 25 (2018), p. e2143, https://doi.org/https://doi. org/10.1002/nla.214...

    2018 doi

  16. [24]

    Svanberg , The method of moving asymptotes—a new method for structural optimiza- tion, International Journal for Numerical Methods in Engineeri ng, 24 (1987), pp

    K. Svanberg , The method of moving asymptotes—a new method for structural optimiza- tion, International Journal for Numerical Methods in Engineeri ng, 24 (1987), pp. 359– 373, https://doi.org/https://doi.org/10.1002/nme.1620240207, https://onlinelibrary.wiley. com/doi/abs/10.1...

    1987 doi

  17. [25]

    Taasan, One shot methods for optimal control of distributed paramet er systems 1: Finite dimensional control, tech

    S. Taasan, One shot methods for optimal control of distributed paramet er systems 1: Finite dimensional control, tech. report, NASA Langley Research Center, 1991

    1991

  18. [26]

    M. J. B. Theulings, R. Maas, L. No¨el, F. van Keulen, and M. Langelaar, Reducing time and memory requirements in topology optimization of transi ent problems , International Journal for Numerical Methods in Engineering, 125 (2024), p . e7461, https://doi.org/ https://doi.org/1...

    2024 doi

  19. [27]

    parareal

    S. Ulbrich , Preconditioners based on “parareal” time-domain decompos ition for time- dependent pde-constrained optimization , in Multiple Shooting and Time Domain Decom- position Methods, T. Carraro, M. Geiger, S. K¨ orkel, and R. R annacher, eds., Cham, 2015, Springer Intern...

    2015

  20. [28]

    F. Wang, B. S. Lazarov, and O. Sigmund , On projection methods, convergence and robust formulations in topology optimization , Structural and multidisciplinary optimization, 43 (2011), pp. 767–784

    2011

  21. [29]

    S. Wu, Y. Zhang, and S. Liu, Topology optimization for minimizing the maximum temperat ure of transient heat conduction structure , Structural and Multidisciplinary Optimization, 60 (2019), pp. 69–82

    2019

  22. [30]

    T. Zeng, H. Wang, M. Yang, and J. Alexandersen, Topology optimization of heat sinks for instantaneous chip cooling using a transient pseudo-3d the rmofluid model , International Journal of Heat and Mass Transfer, 154 (2020), p. 119681, https://doi.org/https://doi.org/ 10.1016/j...

    2020

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