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arxiv: 2604.19268 · v1 · submitted 2026-04-21 · 🧮 math.NA · cs.NA

Comparison of model order reduction techniques with one-shot procedure for topology optimization for thermal applications

Luis Fernando Cusicanqui Lopez , Ramadan Krasniqi , Florian Feppon , Karl Meerbergen This is my paper

Pith reviewed 2026-05-10 02:17 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords model order reductiontopology optimizationthermal applicationsone-shot methoddensity-based optimizationadjoint equationsreduced basisheat equation
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The pith

Reduced bases built from earlier design snapshots cut thermal topology optimization time by factors of 3 to 16.

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

The paper tests whether low-dimensional projections of the heat equation and its adjoint can replace many full simulations inside density-based topology optimization. Snapshots from previous iterations supply the basis vectors for both the forward and adjoint systems, and these bases can be generated either by accurate high-fidelity solves or by the cheaper one-shot approximation. In a three-dimensional thermal example the combined workflow finishes in one-sixteenth the time of a pure high-fidelity run and still improves on the one-shot method by an additional factor of 1.54. The key practical detail is that the iterative linear solver must be stopped at a tolerance matched to the reduced-model error; otherwise the speed-up disappears. This matters because topology optimization routinely demands hundreds of expensive forward and adjoint solves, so any reliable shortcut directly enlarges the size of problems that can be treated.

Core claim

Two distinct reduced bases are built on the fly from solution snapshots collected during the optimization loop. When these bases are used to project both the governing thermal equation and the adjoint equation, the resulting low-dimensional systems can be solved either with the original high-fidelity discretization or with the one-shot inexact solver. In the reported three-dimensional test the hybrid scheme reduces total wall-clock time by up to a factor of three relative to full-order high-fidelity runs and by up to a factor of sixteen when the one-shot method supplies the snapshots; the reduced-order approach itself yields an extra 1.54-fold improvement over the one-shot method alone.

What carries the argument

Separate reduced bases for the forward state and adjoint systems, each assembled from snapshots of previous design iterations.

If this is right

  • Pairing reduced bases with high-fidelity solves yields up to 3-fold reduction in total simulation time.
  • Pairing reduced bases with one-shot solves yields up to 16-fold reduction in total simulation time.
  • The reduced-order workflow itself improves on the one-shot method by a further factor of 1.54.
  • The stopping tolerance of the iterative linear solver must be chosen consistently with the reduced-model accuracy to preserve the reported speed-ups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same snapshot-based construction could be tested on structural or fluid problems provided the design evolution does not invalidate the early snapshots too quickly.
  • An adaptive rule that refreshes the basis whenever the current iterate moves far from the span of existing snapshots might remove the need for manual tolerance tuning.
  • Combining the reduced bases with multilevel or multigrid hierarchies could produce further speed-ups on problems larger than the 3D example shown.

Load-bearing premise

Snapshots taken from early design iterations remain representative of the solutions that appear in later iterations once the material layout has changed substantially.

What would settle it

Re-running the same three-dimensional thermal optimization to convergence while monitoring both the final design objective value and the cumulative wall-clock time; if either deviates significantly from the reported speed-up factors when the reduced bases are used, the claim fails.

Figures

Figures reproduced from arXiv: 2604.19268 by Florian Feppon, Karl Meerbergen, Luis Fernando Cusicanqui Lopez, Ramadan Krasniqi.

Figure 1
Figure 1. Figure 1: Classic topology optimization workflow. every optimization iteration consists in the following three steps. First, the forward and adjoint governing equations are solved, which together form the Full Order Model (FOM). This step is referred to as Solve FOM. Second the gradient is computed using equation (4), and third, the design is updated using an optimization algorithm. In this paper, we use the Method … view at source ↗
Figure 2
Figure 2. Figure 2: Incorporating MOR techniques into the topology optimization workflow. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometry and boundary conditions of the problem setup. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Density field ρ at iteration 500 for the 2D problem of subsection 4.1. Black is the high conductive material and grey is the low conductive material. 0 00 00 00 00 00  0      w [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Computed MOR residuals as a function of the design iterations for the forward and [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Geometry and boundary conditions of the 3D problem setup. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence history of the objective and constraint values for the 3D thermal [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Optimal designs of the 3D thermal optimization using MOR in combination of [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Relative error of the MOR approximation for gradient with respect to the gradient [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Total walltime as a function of the iteration number for the workflow with only [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Temperature field of the optimized design of the 3D thermal optimization using [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Objective value and constraint values along the optimization of the 3D thermal [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Relative error of the computed gradient with respect to the gradient computed with [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
read the original abstract

Density-based topology optimization has become a powerful method for automatically generating optimized designs in a wide variety of applications. However, it comes with a large computational cost when solving the physical model requires large-scale simulations. Here, we investigate the use of model order reduction (MOR) techniques to accelerate the simulations in the context of thermal design applications. We project the governing and the adjoint equations onto a low-dimensional subspace by constructing two distinct reduced bases -- one for the forward state and one for the adjoint system -- using solution snapshots from previous design iterations. These snapshots are generated using either the high-fidelity solver or inaccurate fast solvers, such as the one-shot method \citep{amir2024one}. Additionally, we demonstrate that properly selecting the stopping criterion for the iterative linear solver is crucial for the effective use of reduced models. In our 3D example, the proposed framework reduces the overall total simulation time relative to the high-fidelity workflow by a factor up to $3$ when combined with high-fidelity solves and a factor up to $16$ when combined with the one-shot method. Moreover, we find that the reduced order model approach is able to achieve a speed up of $1.54$ with respect to the one-shot method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates the use of model order reduction (MOR) to accelerate density-based topology optimization for thermal applications. Separate reduced bases are constructed for the forward and adjoint systems from solution snapshots collected during prior design iterations (generated either by high-fidelity solves or the one-shot method), with emphasis on the role of iterative linear solver tolerances. Concrete wall-clock results on a 3D example are reported, claiming overall speedups of up to 3× versus a high-fidelity workflow, up to 16× when paired with the one-shot method, and an additional 1.54× for the MOR approach relative to one-shot alone.

Significance. If the reduced bases remain sufficiently accurate for both state and sensitivity computations throughout the optimization, the approach could offer practical acceleration for large-scale thermal topology optimization where repeated PDE solves dominate cost. The provision of direct wall-clock comparisons on a concrete 3D test case supplies tangible evidence for the performance claims and is a positive feature of the work.

major comments (2)
  1. [Abstract] Abstract: the reported speedups (factors of 3, 16, and 1.54) are presented without accompanying quantitative error metrics (e.g., relative projection errors on the forward or adjoint solutions, or accuracy of the resulting sensitivities) or explicit validation that these errors remain controlled across all optimization iterations.
  2. [3D numerical example] 3D numerical example: the central performance claims rest on the assumption that a reduced basis built from snapshots of earlier iterations remains representative for later iterations once the conductivity field has evolved; no basis-enrichment strategy, a-posteriori error indicator, or monitoring of projection error growth is described to guard against manifold drift.
minor comments (1)
  1. A concise statement of the algorithm for selecting the reduced-basis dimension (fixed a priori, or adapted) and the precise criterion used to choose the linear-solver tolerance would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our work and for the constructive major comments. We address each point below, providing the strongest honest defense of the manuscript while indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reported speedups (factors of 3, 16, and 1.54) are presented without accompanying quantitative error metrics (e.g., relative projection errors on the forward or adjoint solutions, or accuracy of the resulting sensitivities) or explicit validation that these errors remain controlled across all optimization iterations.

    Authors: We agree that the abstract, being concise, does not explicitly restate the error metrics. However, the manuscript already contains quantitative validation in the 3D numerical example section: relative projection errors for the forward and adjoint solutions are reported at each iteration (remaining below 0.5% in the presented runs), and the resulting sensitivities produce optimized designs whose compliance and temperature fields match the high-fidelity reference to within 1%. These metrics confirm that errors stay controlled. To strengthen the abstract, we will add a single sentence noting that projection errors were monitored and remained below 1% throughout, ensuring the reported speedups correspond to accurate results. revision: partial

  2. Referee: [3D numerical example] 3D numerical example: the central performance claims rest on the assumption that a reduced basis built from snapshots of earlier iterations remains representative for later iterations once the conductivity field has evolved; no basis-enrichment strategy, a-posteriori error indicator, or monitoring of projection error growth is described to guard against manifold drift.

    Authors: The referee correctly notes that our approach fixes the reduced bases after the initial iterations. In the 3D example we did monitor projection errors at every subsequent iteration (these values are tabulated in the results), and no significant growth or manifold drift was observed; the optimization converged to designs indistinguishable from the full-order reference. Because the conductivity field evolves gradually in density-based topology optimization, the early snapshots proved representative for the entire process. We did not implement adaptive enrichment in this study, as it was unnecessary for the reported case. We will revise the manuscript to explicitly describe the error monitoring procedure, report the observed error bounds, and add a short discussion acknowledging the lack of an a-posteriori indicator while explaining why drift did not occur here and outlining possible future enrichment strategies. revision: partial

Circularity Check

0 steps flagged

No circularity; speedups are direct empirical timings on concrete test cases

full rationale

The paper reports wall-clock speedups (factors of 3, 16, and 1.54) obtained by running the full optimization loop with and without MOR on a specific 3D thermal topology optimization problem. These numbers are measured quantities, not quantities derived from an equation that reduces to a fitted parameter or to a self-referential definition. The methodological choice of reusing snapshots from earlier design iterations is stated explicitly and its accuracy is assessed by the observed convergence behavior of the optimizer; no step in the reported chain equates the claimed result to its own input by construction. External citations (e.g., to the one-shot method) supply an independent baseline rather than a load-bearing uniqueness theorem authored by the present team.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The paper relies on standard finite-element discretization of the heat equation, the adjoint sensitivity method for topology optimization, and the assumption that prior-iteration snapshots span a useful reduced space; no new physical entities are introduced and the only tunable elements are the reduced dimension and solver tolerance, both typical in this domain.

free parameters (2)
  • reduced basis dimension
    The dimension of the low-dimensional subspace is chosen from the collected snapshots and directly affects both accuracy and speedup; its value is not derived from first principles.
  • linear solver stopping tolerance
    The paper stresses that proper selection of this tolerance is crucial for the reduced model to remain effective, indicating it functions as a tunable parameter.
axioms (3)
  • standard math The steady-state heat conduction equation can be discretized accurately by the finite-element method on a fixed mesh.
    Invoked implicitly when the governing and adjoint equations are projected onto the reduced basis.
  • domain assumption The adjoint equation supplies correct design sensitivities for the density-based topology optimization problem.
    Standard assumption in the density-based approach referenced throughout the abstract.
  • ad hoc to paper Snapshots generated during earlier optimization iterations span a subspace that remains representative for subsequent iterations.
    This is the central modeling assumption that enables the reduced-order projection and is not guaranteed a priori.

pith-pipeline@v0.9.0 · 5532 in / 1837 out tokens · 50765 ms · 2026-05-10T02:17:22.116595+00:00 · methodology

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