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arxiv: 2605.21158 · v1 · pith:GAPNGAI4new · submitted 2026-05-20 · 🧮 math.AP · cs.NA· math.NA

Experimental detection of inclusions for the time-harmonic elastic wave equation

Sarah Eberle-Blick , Jochen Moll This is my paper

Pith reviewed 2026-05-21 03:09 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA
keywords inclusion reconstructiontime-harmonic elastic wavesmonotonicity methodinverse problemsnoisy measurementslaboratory experimentelastic body
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The pith

Reconstructing inclusions in elastic bodies from lab measurements works better with the time-harmonic wave equation than the stationary version.

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

The paper solves the inverse problem of detecting inclusions inside elastic bodies by using actual laboratory measurements of the time-harmonic elastic wave equation. It demonstrates that this time-harmonic formulation produces clearer reconstructions than the earlier stationary-wave approach on the same experimental setup. To cope with noise present in real data, the authors develop a modified linearized monotonicity method and apply it numerically to recover the inclusions.

Core claim

The investigation of the harmonic problem leads to a better reconstruction compared to the stationary one. By adapting the linearized monotonicity method to noisy data and applying it to laboratory measurements of the time-harmonic elastic wave equation, the inclusions are reconstructed numerically.

What carries the argument

Modified linearized monotonicity method for noisy data from the time-harmonic elastic wave equation.

Load-bearing premise

The modified linearized monotonicity method remains stable and accurate when applied to real noisy measurements of the time-harmonic elastic wave equation in a laboratory setting.

What would settle it

If numerical reconstructions obtained from the laboratory time-harmonic data are not visibly better or are less stable than those obtained from the corresponding stationary-wave data, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.21158 by Jochen Moll, Sarah Eberle-Blick.

Figure 1
Figure 1. Figure 1: Setup overview: 1: shaker, 2: optical sensor, 3: connection shaker and Makrolon plate, 4: two mounting points, where the Makrolon plate is fixed at the frame, 5: Makrolon plate Finally, we take a look at the schematic setting of the Makrolon plate [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic setting: Makrolon plate with positions of the mount￾ing (circles): Dirichlet conditions in blue and Neumann conditions in white. 4. Linearized monotonicity test In this section we will introduce the linearized monotonicity tests as considered in [10]. First of all, we give the required background: Our notations are to a large extent the 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sine sweep applied to each Neumann boundary The displacements on the Neumann boundary were then measured and the applied bound￾ary forces were recorded by the shaker. Both data sets were then made consistent by e.g. cutting of data recorded before and after the sweep. At last, both the applied force and the measured displacements were Fourier transformed into the frequency domain for a fixed frequency, whe… view at source ↗
Figure 4
Figure 4. Figure 4: Detected inclusions of the experiment with a 12cm central in￾clusion. The inclusion is marked in red. ω M δ 21rad s 6 9.775038 · 10−7 41rad s 6 9.7283458500 · 10−7 55.4rad s 6 7.929299 · 10−7 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Detected inclusions of the experiment wit a two separate 10cm decentralized inclusion. The reconstructed inclusion is marked in red. Pa￾rameters: ω = 20.2 rad s , M = 6, δ = 1.53598375 · 10−6 Finally, we want to remark that compared with the results for the stationary case (see [5]), we obtain better results especially for the two decentralized inclusions. Acknowledgement The first author thanks the German… view at source ↗
read the original abstract

We are concerned with the reconstruction of inclusions in elastic bodies based on measurements from a laboratory experiment. In doing so, we solve the inverse problem of the time-harmonic elastic wave equation, in contrast to the stationary wave equation and the corresponding lab experiment proposed earlier in Eberle and Moll (2021). The investigation of the harmonic problem leads to a better reconstruction compared to the stationary one. Since we deal with real measurement data, we have to take into account, that those measurements always include measurement errors, so that we have to handle noisy data. Thus, we consider the linearized monotonicity method for noisy data and introduce a modified version of this method. Based on this, we reconstruct the inclusions numerically.

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

3 major / 2 minor

Summary. The paper reconstructs inclusions in elastic bodies from real laboratory measurements of the time-harmonic elastic wave equation. It contrasts this approach with the stationary wave equation experiment in Eberle and Moll (2021), asserts that the harmonic formulation yields better reconstructions, and introduces a modified linearized monotonicity method to accommodate noisy data before presenting numerical results.

Significance. A controlled demonstration that time-harmonic data improve inclusion recovery over stationary data on identical laboratory measurements would strengthen the case for frequency-domain methods in practical elastic inverse problems. The use of actual experimental data rather than synthetics is a positive feature.

major comments (3)
  1. [Abstract] The central claim that the harmonic problem produces better reconstructions rests on a comparison to the separate 2021 stationary experiment. Because the manuscript does not re-process the identical current measurements with the stationary model and unmodified monotonicity method, differences in sensor placement, excitation frequencies, damping, or noise statistics cannot be ruled out as the source of any observed improvement (see Abstract and the discussion of the 2021 reference).
  2. [Abstract] No quantitative error metrics, baseline comparisons, or reconstruction-error tables are supplied to support the assertion of improved performance; the abstract states only that the harmonic approach 'leads to a better reconstruction' without reporting, for example, Hausdorff distances or L2 errors relative to ground truth.
  3. [Method section (noisy-data variant)] The modification of the linearized monotonicity method for noisy data is introduced without a statement of how the original proof is altered or what stability guarantees remain; this modification is load-bearing for the claim that the method works on real laboratory data.
minor comments (2)
  1. [Experimental setup] Clarify the precise experimental parameters (frequencies, sensor count, material properties) used in the present harmonic measurements versus those in the 2021 stationary study so that readers can assess comparability.
  2. [Numerical results / figures] Add explicit captions or legends that juxtapose the harmonic and stationary reconstructions side-by-side on the same figure panels.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments provided. We respond to each major comment in turn.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the harmonic problem produces better reconstructions rests on a comparison to the separate 2021 stationary experiment. Because the manuscript does not re-process the identical current measurements with the stationary model and unmodified monotonicity method, differences in sensor placement, excitation frequencies, damping, or noise statistics cannot be ruled out as the source of any observed improvement (see Abstract and the discussion of the 2021 reference).

    Authors: We acknowledge that our comparison is to the stationary wave experiment reported in Eberle and Moll (2021) rather than re-processing the current measurements using the stationary model. The laboratory setups are closely related, involving the same physical specimen and similar measurement configurations, but we recognize that differences in excitation and data collection could play a role. Nevertheless, the harmonic approach allows for frequency-specific information that enhances the reconstruction. In the revision, we will expand the discussion section to include a side-by-side comparison of key experimental parameters from both works and provide additional justification for attributing the observed improvement primarily to the time-harmonic formulation. We are open to further suggestions on this point. revision: partial

  2. Referee: [Abstract] No quantitative error metrics, baseline comparisons, or reconstruction-error tables are supplied to support the assertion of improved performance; the abstract states only that the harmonic approach 'leads to a better reconstruction' without reporting, for example, Hausdorff distances or L2 errors relative to ground truth.

    Authors: We agree with the referee that the abstract would benefit from quantitative support. We will revise the abstract and add a new subsection or table in the numerical results section that reports quantitative error metrics, including Hausdorff distances and relative L2 errors for the reconstructed inclusions compared to the known ground truth. This will provide a more rigorous basis for the claim of improved performance. revision: yes

  3. Referee: [Method section (noisy-data variant)] The modification of the linearized monotonicity method for noisy data is introduced without a statement of how the original proof is altered or what stability guarantees remain; this modification is load-bearing for the claim that the method works on real laboratory data.

    Authors: The modification to the linearized monotonicity method for handling noisy data is presented in the method section. We will clarify in the revised manuscript the specific alterations made to the original proof, such as the incorporation of noise bounds into the monotonicity test, and state the remaining stability guarantees under the assumption of bounded measurement noise. This will better support the applicability to real laboratory data. revision: yes

Circularity Check

0 steps flagged

No significant circularity; experimental results independent of prior self-citation.

full rationale

The manuscript presents laboratory measurements of the time-harmonic elastic wave equation and applies a modified linearized monotonicity method to reconstruct inclusions from real noisy data. The comparison to the stationary case references the authors' 2021 experiment on a separate dataset, but this citation supports only a qualitative contrast and does not enter any derivation or reconstruction step. No equation reduces a claimed prediction to a fitted parameter, no ansatz is smuggled via self-citation, and no uniqueness theorem is invoked to force the method. The work is grounded in external laboratory measurements rather than synthetic data generated from the model itself, rendering the central reconstruction self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling assumption that the time-harmonic elastic wave equation accurately describes the laboratory experiment and that the monotonicity method can be linearized and modified without losing its theoretical guarantees under realistic noise levels; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The time-harmonic elastic wave equation governs the laboratory measurements of wave propagation in the elastic body.
    Invoked when the authors state they solve the inverse problem of the time-harmonic elastic wave equation using real data.
  • ad hoc to paper The linearized monotonicity method remains valid after modification for noisy data.
    The abstract introduces a modified version of the method specifically to handle measurement errors.

pith-pipeline@v0.9.0 · 5647 in / 1428 out tokens · 47188 ms · 2026-05-21T03:09:59.171197+00:00 · methodology

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Reference graph

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