Pith. sign in

REVIEW 4 major objections 5 minor 85 references

A modified Levenberg-Marquardt method for estimating the elastic material parameters of polymer waveguides using residuals between autocorrelated frequency responses

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A geometry-derived step-size rule and an autocorrelation-phase residual identify polymer-waveguide elastic parameters with fewer forward-model evaluations than a quasi-Newton baseline.

desk verdict A clean derivation of a step-size-adapted Levenberg-Marquardt method and an autocorrelation-phase objective, but the headline speed claim rests on unproven convexification and same-model virtual measurements. read the letter →

arxiv 2507.01706 v1 pith:V5UXTJAM submitted 2025-07-02 cs.CE

Dominik Itner , Dmitrij Dreiling , Hauke Gravenkamp , Bernd Henning , Carolin Birk This is my paper · ORCID
classification cs.CE
keywords materialparameterestimationinverseproblemLevenberg-Marquardtmethodautocorrelationresidualmodelmanifoldultrasonicwaveguidespolymersleastsquares
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

The paper tries to establish that two coupled changes make inverse material-parameter identification in ultrasonic waveguides faster and more reliable. The first is a parameter-free automatic step-size rule for the Levenberg-Marquardt method, derived by viewing the forward model's outputs as points on a curved model manifold and treating Gauss-Newton and gradient descent as dual projections. The second is a new objective function that compares the unwrapped phases of the autocorrelation of the squared signal envelopes in the frequency domain, instead of comparing raw time signals. The paper argues that this autocorrelation-phase transformation convexifies the objective over the relevant parameter ranges for isotropic polymers, so a local optimizer can find the true parameters from realistic starting points. On virtual measurements for PEEK, PA6 and polypropylene, the method reaches a relative parameter error below $10^{-6}$ with fewer total forward-model evaluations than the reference quasi-Newton method, and since the forward model is the computational bottleneck, fewer evaluations mean shorter identification times.

What carries the argument

The organizing object is the model manifold $\mathcal{M} = \{f(x) \in \mathbb{R}^n \mid x \in \mathbb{R}^m\}$, the curved surface swept out in signal space as the parameters vary. Gauss-Newton is the orthogonal projection of the residual $r = \hat{y} - f(x)$ onto the tangent space $T_{f(x)}\mathcal{M}$, while gradient descent projects onto the dual basis. The new machinery is the scalar factor $\lambda_k$ that rescales the pure descent direction to the metric length of the Gauss-Newton step, combined with the Levenberg-Marquardt damping $\eta = \bar{\eta}_k \lambda_k^{-1}$, and the objective residual $r = \mathrm{arg}_{\mathrm{stable}}(\hat{a}) - \mathrm{arg}_{\mathrm{stable}}(a)$ computed from the positive-frequency autocorrelation coefficients $E_k = \sum_{i=k}^{n_+-1} U_{i+1} \bar{U}_{i-k+1}$ of the squared envelope, stabilized by a damping factor $\gamma_k$. The autocorrelation-phase step is intended to "relax and untangle" the phases, producing a locally convex objective within the search space for isotropic symmetry, which is the property that makes a local optimizer reliable.

What would settle it

Sample the Hessian of the autocorrelation-phase least-squares objective on a fine grid of $(E,\nu)$ pairs inside the two-standard-deviation boxes of the three materials; if any sampled Hessian has a negative eigenvalue, the claimed convexification inside the search space is false. Equivalently, run the optimizer from many starting points on real measurement signals of PEEK, PA6 and PP and count failures to reach a $10^{-6}$ relative parameter error.

Watch

Extended reading notes

Core claim

The central claim is that the inverse problem of estimating frequency-dependent elastic parameters from ultrasonic waveguide responses can be made both more robust and cheaper by replacing the standard least-squares comparison of time signals with a phase residual between autocorrelations of the squared envelopes, and by equipping the Levenberg-Marquardt method with a self-contained geometric step-size selection. The step size is derived from the metric tensor $G=J^TJ$ of the model manifold: the proposed factor $\lambda_k = \sqrt{(\Delta x_k^*)^T G^{-1} \Delta x_k^* \big/ (\Delta x_k^*)^T G \Delta x_k^*}$ rescales the gradient-descent direction to match the scale of the Gauss-Newton step, and the Levenberg-Marquardt damping is set to $\bar{\eta}_k \lambda_k^{-1}$ with $\bar{\eta}_k = \|\Delta y_k\| / \|\Delta y_0\|$, requiring no hand-tuned hyperparameters. The paper demonstrates on virtual measurements for three isotropic polymers that this combination reaches relative parameter errors below $10^{-6}$ in fewer forward-model evaluations than the baseline quasi-Newton method, counting the line-search evaluations the baseline needs.

Load-bearing premise

The decisive premise is that the autocorrelation-phase residual makes the objective function convex over the whole relevant parameter space for isotropic materials, a property the paper explicitly states as a hypothesis rather than a proof; if the objective retains hidden local minima, the optimizer's speed and reliability claims would not generalize beyond the tested instances.

Editorial extensions

If this is right

  • For the isotropic case, the method reaches the $10^{-6}$ parameter-error cutoff in fewer total forward-model evaluations than the reference quasi-Newton method for PEEK, PA6 and PP; because the forward solve dominates compute time, this translates into shorter identification time.
  • The autocorrelation-phase objective makes the least-squares landscape convex over the tested parameter ranges, so first-order optimizers starting inside the material ranges should converge to the true parameters rather than to local minima.
  • The step-size rule is parameter-free, so it transfers to other least-squares inverse problems once Jacobians are available.
  • Counting line-search evaluations matters: the comparison shows that the baseline's implicit line search adds a significant number of forward solves, so fair speed comparisons should include them.

Reading between the lines

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

  • If the convexification property extends to transverse-isotropic symmetry and damping, which the authors flag as an open uncertainty, the same residual could stabilize a harder inversion problem that currently defeats first-order methods.
  • The phase-untangling mechanism is a hypothesis rather than a proof; a Hessian audit of the objective inside the search box would settle whether "convexify" is literal or merely empirical.
  • The geometric step-size derivation is independent of the autocorrelation objective, so either component could be swapped into other inverse problems; for instance, the same Levenberg-Marquardt scaling could accelerate any differentiable least-squares fit with expensive forward solves.
  • The gamma-distributed material priors give a ready-made test set for experimental validation: if real measurements on PEEK, PA6 or PP do not reproduce the virtual convergence statistics, the mismatch would point to model error rather than optimizer error.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes two methodological innovations for estimating elastic material parameters of polymer waveguides from ultrasonic transmission signals: (i) an automatic step-size rule for the Levenberg-Marquardt (LM) method, derived from a geometric interpretation of nonlinear least squares, and (ii) a new objective function based on the phase of the autocorrelation of the signal's frequency content. The method is tested on virtual measurements for three isotropic polymers (PEEK, PA6, PP), where the SBFEM forward model is used both to generate synthetic reference signals and to evaluate the objective during inversion. The central performance claim is that the proposed LM variant with the autocorrelation-phase residual reaches a relative parameter error below 1e-6 in fewer forward-model evaluations than BFGS with Hager-Zhang line search for all three materials considered.

Significance. If the claimed speedup and convexification hold beyond the presented test cases, the paper makes a useful contribution to ultrasonic material characterization by reducing the number of expensive wave-propagation solves in gradient-based inversion. The geometric derivation of the LM damping parameter is elegant and parameter-free, and the careful construction of marginal gamma distributions for PEEK, PA6, and PP from aggregated datasheet values is a reproducible and practical contribution. The comparison metric is fair in counting line-search evaluations of BFGS. However, the significance is tempered by two factors the manuscript itself acknowledges: the validation is entirely synthetic (references come from the same SBFEM forward model used in inversion), and the convexification property of the new residual is stated as a hypothesis, not a proven fact. As a methods paper with empirical support on a synthetic benchmark, the work is worthwhile, but the abstract and conclusions currently claim more than the evidence substantiates.

major comments (4)
  1. [Sec. 3.4, Fig. 8] The claim that the autocorrelation-phase residual 'completely convexifies the objective function within the search space' is load-bearing for the speedup claim, yet the same section states: 'we hypothesize that this mixture ... relaxes and untangles the phases, although we do not have a concrete proof.' The visual evidence in Fig. 8 shows only one reference (caption: 'for a particular reference') and only a two-standard-deviation box. Since the proposed LM method has no line search or global safeguard (Sec. 3.3), any unobserved local minima or flat regions outside this box could invalidate the convergence and the evaluation-count advantage. The authors should either provide a proof of unimodality on the search space (even for a simplified or reduced-order problem) or, failing that, run a systematic numerical scan over many randomly drawn references and initial points, reporting the basin of attraction and failure rates. Without this, the abstract's statement that the method 'shortens the time to identify the material parameters' is not fully supported.
  2. [Sec. 4.2] The performance comparison is self-referential and the initial conditions are not characterized. All 60 virtual measurements are produced by the same SBFEM forward model that is used in the inversion (the abstract acknowledges that the study 'primarily relies on simulation data'), so the test measures the optimizer's behavior on synthetic data with perfect model correspondence, not on real measured signals. More importantly, the initial estimates are 'generated using the methodology presented in [83]' and the distance of those starting points from the true parameters is never reported. If these starts lie inside the single visually convex basin of Fig. 8, the experiment measures local convergence speed only; it does not substantiate a claim about global robustness over the intended search region. Please report the distribution of initial parameter errors, success/failure counts as a function of initial distance, and ideally repeat the comparison with deliberately perturbed starting points spanning the full search space defined by the gamma distributions in Table 1.
  3. [Sec. 3.3, Eq. (34)] The proposed step-size rule λ_k squared equals (Δx*_k)ᵀG⁻¹Δx*_k / (Δx*_k)ᵀGΔx*_k, which requires the metric G = JᵀJ to be positive definite and the gradient descent step Δx*_k to be nonzero. Near a stationary point, in a flat direction, or when the Jacobian is rank-deficient (a common situation in parameter-identification problems with correlated parameters), λ_k and hence η = η_k λ_k⁻¹ may be undefined or unbounded. The manuscript states that η_k may exceed one for uphill steps and that 'This is not problematic in and of itself', but no safeguard or fallback is described. Since no line search is used, the authors should state explicitly under what conditions Eq. (38) yields a descent direction and describe what the implementation does when G is singular or when λ_k cannot be computed.
  4. [Sec. 3.4, Eq. (57)] The new residual introduces a new free hyperparameter, the damping coefficient C ∈ [1,10] in the stability weights γ_k. This is in tension with the paper's claim to avoid hyperparameters (Sec. 1), and the objective surface shown in Fig. 8 is computed with C=1 only. No sensitivity study with respect to C is presented, so it is unknown whether the claimed convexification and the subsequent speedup are robust to the choice of C. A small scan (e.g., C = 1, 2, 5, 10) for at least one material should be added, and the role of C as a user-set parameter should be acknowledged in the summary of the method.
minor comments (5)
  1. [Sec. 2, Eq. (5)] The text states 'We choose ¯f = 1 GHz' for the center frequency. This appears inconsistent with the time axis of Fig. 2 (in microseconds) and with typical ultrasonic polymer testing; if this is a typo for 1 MHz, please correct it in the text and in any related discussion.
  2. [Figure captions] Several figure captions contain unresolved placeholder text '( ??)', e.g., Fig. 1, Fig. 3, and Fig. 8. These internal cross-references need to be completed before the manuscript is sent to production.
  3. [Sec. 3.2, p. 7] The phrase 'has to equally fullfull' contains a typo and should read 'has to equally fulfill'.
  4. [Sec. 4.1] The procedure for fitting gamma distributions is described, but the actual data ranges from the cited sources are not reported (due to licensing). For reproducibility, the authors should at least provide the aggregated parameter intervals or quartiles that were used, so that readers can reproduce the marginal distributions without access to the original datasheets.
  5. [Sec. 4.2, reference [83]] The generation of initial estimates via 'the methodology presented in [83]' is not described in this paper. Since the initial-point distance is critical to the interpretation of the comparison, a one-sentence description of that methodology (or a citation to an accessible source) is needed for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimizer and objective transformation are derived analytically and the evaluation-count comparison is a genuine empirical benchmark, though the virtual-measurement validation is self-referential and the convexification claim is explicitly unproven.

full rationale

The central derivation is self-contained. The modified Levenberg-Marquardt step in Sec. 3.3 follows analytically from the geometric interpretation: the scaling factor lambda_k is defined by metric distances (Eqs. 33-34) and the interpolation factor eta_k by residual ratios (Eq. 37); no free parameter is fitted to force convergence. The autocorrelation-phase residual in Sec. 3.4 is derived from the Fourier-domain identity for the squared envelope (Eqs. 41-51), and the residual in Eq. (59) is then used directly as the objective. The claim that this residual 'completely convexifies the objective function within the search space' is explicitly hedged by the authors: 'we hypothesize that this mixture of Fourier coefficients in the autocorrelation relaxes and untangles the phases, although we do not have a concrete proof.' An unproven assumption is a correctness risk, not a circular definition. The validation in Sec. 4.2 uses 60 virtual references generated with the same SBFEM forward model used in the inversion, and initial estimates come from the self-cited methodology in [83]; these are validity concerns (inverse crime, unreported initial-condition distance), but they do not make the evaluation-count comparison tautological: the BFGS with Hager-Zhang line search could in principle have required fewer evaluations, so the reported speed advantage is an empirical outcome rather than a construction. No equation or fitted parameter in the paper reduces the claimed speedup to its inputs. Therefore no significant circularity is identified.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the SBFEM forward model as a faithful representation of the physical setup, on the unproven convexification of the autocorrelation-phase objective, and on a heuristic step-size scaling assumption. The only hand-picked numeric parameter in the method itself is the damping constant C, set to 1. The fictitious signal used for phase normalization is an invented mathematical construct with no independent evidence.

free parameters (2)
  • Damping coefficient C in Eq. (57) = 1 (chosen from the stated range [1,10])
    Controls the exponential damping of phase residuals for frequencies outside the excitation range. The convexified objective surface shown in Fig. 8 uses C=1; the choice affects the shape of the objective and is not varied in the study.
  • Gamma distribution shape and scale factors for rho, E, nu, G (Table 1) = Various values, see Table 1
    Fitted to manufacturer and literature data using Monte Carlo sampling; used to generate virtual measurement parameters. Not part of the proposed method but part of the test design, and the central claim depends on the test results.
assumptions (5)
  • domain assumption The SBFEM forward model, together with the Mason transducer model and the area-averaging assumption, accurately represents the physical waveguide measurement.
    The inverse problem treats simulated responses as equivalent to real measurements. Any model error would break the validation. Introduced in Sec. 2 (Eqs. 4-10).
  • ad hoc to paper The autocorrelation-based phase residual (Eq. 59) produces a locally convex objective function within the search space for the isotropic case.
    This property is asserted based on visual inspection of Fig. 8 and is explicitly unproven. It is essential for the local optimizer to converge to the global minimizer. Stated in Sec. 3.4.
  • ad hoc to paper The Gauss-Newton step scales the step optimally, so bounding the gradient descent step by the Gauss-Newton step via the metric distance (Eq. 33) is a valid heuristic.
    This assumption justifies the step-size adaptation formula; it is introduced without proof in Sec. 3.2.
  • domain assumption The residual ratio eta_k in Eq. (37) is bounded and the steps are descent directions, so the interpolation factor transitions from gradient descent to Gauss-Newton.
    The method relies on this for stability; the authors note in footnote 3 that it is not guaranteed without a line search. Sec. 3.3.
  • standard math Standard properties of the Hilbert transform, convolution, and the Fourier transform (Eqs. 41-51).
    Used in the analytic derivation of the autocorrelation objective. Standard results, no independent verification needed.
invented entities (1)
  • Fictitious signal with linear phase (Eq. 55)
    purpose: Normalizes the phases of the autocorrelated Fourier coefficients so that the phase residual in Eq. (59) is numerically robust.
    This is a mathematical construction introduced ad hoc to avoid phase discontinuities. It is not a physical entity and has no independent falsifiable handle.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A modified Levenberg-Marquardt method for estimating the elastic material parameters of polymer waveguides using residuals between autocorrelated frequency responses." pith.science (2026). https://pith.science/paper/V5UXTJAM

@misc{pith2026250701706,
  author       = {Pith},
  title        = {Pith review of: A modified Levenberg-Marquardt method for estimating the elastic material parameters of polymer waveguides using residuals between autocorrelated frequency responses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V5UXTJAM}},
  note         = {Machine review of arXiv:2507.01706}
}
read the original abstract

In this contribution, we address the estimation of the frequency-dependent elastic parameters of polymers in the ultrasound range, which is formulated as an inverse problem. This inverse problem is implemented as a nonlinear regression-type optimization problem, in which the simulation signals are fitted to the measurement signals. These signals consist of displacement responses in waveguides, focusing on hollow cylindrical geometries to enhance the simulation efficiency. To accelerate the optimization and reduce the number of model evaluations and wait times, we propose two novel methods. First, we introduce an adaptation of the Levenberg-Marquardt method derived from a geometrical interpretation of the least-squares optimization problem. Second, we introduce an improved objective function based on the autocorrelated envelopes of the measurement and simulation signals. Given that this study primarily relies on simulation data to quantify optimization convergence, we aggregate the expected ranges of realistic material parameters and derive their distributions to ensure the reproducibility of optimizations with proper measurements. We demonstrate the effectiveness of our objective function modification and step adaptation for various materials with isotropic material symmetry by comparing them with a state-of-the-art optimization method. In all cases, our method reduces the total number of model evaluations, thereby shortening the time to identify the material parameters.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 70 canonical work pages

  1. [83]

    Die Bestimmung viskoelastischer Materialparameter von Polymeren mittels eines Puls-Echo- Messverfahrens

    Dmitrij Dreiling et al. “Die Bestimmung viskoelastischer Materialparameter von Polymeren mittels eines Puls-Echo- Messverfahrens”. In:tm - Technisches Messen91.s1 (2024). doi:doi:10.1515/teme-2024-0045(cit. on p. 15)

    work page doi:10.1515/teme-2024-0045(cit 2024

  2. [1]

    Utilizing guided acoustic waves to measure dispersive material properties of polymers

    J. Rautenberg, F. Bause, and B. Henning. “Utilizing guided acoustic waves to measure dispersive material properties of polymers”. In: 2015.doi: 10.5162/sensor2015/A6.1 (cit. on p. 1)

    work page doi:10.5162/sensor2015/a6.1 2015

  3. [2]

    Transient modeling of ultrasonic guided waves in circular viscoelastic waveguides for inverse material characterization

    Fabian Bause et al. “Transient modeling of ultrasonic guided waves in circular viscoelastic waveguides for inverse material characterization”. In:Measurement Science and Technology 26.9 (2015), p. 095602.doi: 10.1088/0957-0233/26/9/095602 (cit. on p. 1)

    work page doi:10.1088/0957-0233/26/9/095602 2015

  4. [3]

    Ultrasonic transmission measurements in the characterization of viscoelasticity utilizing polymeric waveguides

    Fabian Bause et al. “Ultrasonic transmission measurements in the characterization of viscoelasticity utilizing polymeric waveguides”. In:Measurement Science and Technology27.10 (2016).doi: 10.1088/0957-0233/27/10/105601 (cit. on pp. 1, 2)

    work page doi:10.1088/0957-0233/27/10/105601 2016

  5. [4]

    Semi-analytical formulation for the guided waves-based recon- struction of elastic moduli

    Michele Sale, Piervincenzo Rizzo, and Alessandro Marzani. “Semi-analytical formulation for the guided waves-based recon- struction of elastic moduli”. In:Mechanical Systems and Signal Processing25.6 (2011).doi: 10.1016/j.ymssp.2011.02.004 (cit. on p. 1)

    work page doi:10.1016/j.ymssp.2011.02.004 2011

  6. [5]

    Lamb wave based approach to the determination of acoustic material parameters

    Sarah Johannesmann et al. “Lamb wave based approach to the determination of acoustic material parameters”. In:tm - Tech- nisches Messen89.7-8 (2022).doi: 10.1515/teme-2021-0134 (cit. on p. 1)

    work page doi:10.1515/teme-2021-0134 2022

  7. [6]

    Determination of isotropic elastic con- stants from dispersion images based on ultrasonic guided waves by using neural networks

    Mathias Held et al. “Determination of isotropic elastic con- stants from dispersion images based on ultrasonic guided waves by using neural networks”. In:Ultrasonics143 (2024).doi: 10.1016/j.ultras.2024.107403(cit. on p. 1)

    work page arXiv 2024

  8. [7]

    Increasing the sensitivity in the de- termination of material parameters by using arbitrary loads in ultrasonic transmission measurements

    Dmitrij Dreiling et al. “Increasing the sensitivity in the de- termination of material parameters by using arbitrary loads in ultrasonic transmission measurements”. In: 2020.doi: 10. 5162/SMSI2020/D1.3(cit. on p. 1)

    2020

Show all 85 references
  1. [8]

    Application and modelling of ultrasonic transducers using 1-3 piezoelectric composites with structured electrodes

    Dmitrij Dreiling et al. “Application and modelling of ultrasonic transducers using 1-3 piezoelectric composites with structured electrodes”. In: 2021 (cit. on pp. 1, 3)

    2021

  2. [9]

    Improved determination of viscoelastic material parameters using a pulse-echo measurement setup

    Dmitrij Dreiling et al. “Improved determination of viscoelastic material parameters using a pulse-echo measurement setup”. In: 2023.doi:10.5162/SMSI2023/P59(cit. on p. 1)

    2023 doi

  3. [10]

    A pulse-echo measurement setup to determine viscoelastic material parameters

    Dmitrij Dreiling et al. “A pulse-echo measurement setup to determine viscoelastic material parameters”. In:Journal of Physics: Conference Series2822.1 (2024).doi: 10.1088/1742- 6596/2822/1/012169(cit. on p. 1)

    2024 doi

  4. [11]

    Identification of elastic properties utilizing non-destructive vibrational evaluation methods with emphasis on definition of objective functions: a review

    Jun Hui Tam. “Identification of elastic properties utilizing non-destructive vibrational evaluation methods with emphasis on definition of objective functions: a review”. In:Structural and Multidisciplinary Optimization61.4 (2020).doi: 10.1007/ s00158-019-02433-1(cit. on p. 2)...

    2020

  5. [12]

    A method for the solution of certain non-linear problems in least squares

    Kenneth Levenberg. “A method for the solution of certain non-linear problems in least squares”. In:Quarterly of Applied Mathematics2 (1944).doi: 10.1090/QAM/10666 (cit. on pp. 2, 9)

    1944 doi

  6. [13]

    An Algorithm for Least-Squares Es- timation of Nonlinear Parameters

    Donald W. Marquardt. “An Algorithm for Least-Squares Es- timation of Nonlinear Parameters”. In:Journal of the Soci- ety for Industrial and Applied Mathematics11.2 (1963).doi: 10.1137/0111030(cit. on pp. 2, 9)

    1963 doi

  7. [14]

    Defect reconstruction in a two-dimensional semi-analytical waveguide model via derivative-based optimiza- tion

    J. Bulling et al. “Defect reconstruction in a two-dimensional semi-analytical waveguide model via derivative-based optimiza- tion”. In:The Journal of the Acoustical Society of America 152.2 (2022).doi:10.1121/10.0013574(cit. on pp. 2, 10)

    2022 doi

  8. [15]

    On the Rate of Convergence of the Levenberg-Marquardt Method

    N. Yamashita and M. Fukushima. “On the Rate of Convergence of the Levenberg-Marquardt Method”. In:Topics in Numerical Analysis. Springer Vienna, 2001.doi: 10.1007/978- 3- 7091- 6217-0_18(cit. on pp. 2, 9)

    2001 doi

  9. [16]

    On the Quadratic Conver- gence of the Levenberg-Marquardt Method without Nonsingu- larity Assumption

    Jin-yan Fan and Ya-xiang Yuan. “On the Quadratic Conver- gence of the Levenberg-Marquardt Method without Nonsingu- larity Assumption”. In:Computing74.1 (2005).doi: 10.1007/ s00607-004-0083-1(cit. on pp. 2, 9)

    2005

  10. [17]

    Some research on Leven- berg–Marquardt method for the nonlinear equations

    Changfeng Ma and Lihua Jiang. “Some research on Leven- berg–Marquardt method for the nonlinear equations”. In:Ap- plied Mathematics and Computation184.2 (2007).doi: 10 . 1016/j.amc.2006.07.004(cit. on pp. 2, 9)

    2007

  11. [18]

    A note on the Leven- berg–Marquardt parameter

    Jinyan Fan and Jianyu Pan. “A note on the Leven- berg–Marquardt parameter”. In:Applied Mathematics and Computation207.2 (2009).doi: 10.1016/j.amc.2008.10.056 (cit. on pp. 2, 9)

    2009 doi

  12. [19]

    The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence

    Jinyan Fan. “The modified Levenberg-Marquardt method for nonlinear equations with cubic convergence”. In:Mathematics of Computation81 (2012).doi: 10.1090/S0025- 5718- 2011- 02496-8(cit. on pp. 2, 9)

    2012 doi

  13. [20]

    An Adaptive Multi-step Levenberg–Marquardt Method

    Jinyan Fan, Jianchao Huang, and Jianyu Pan. “An Adaptive Multi-step Levenberg–Marquardt Method”. In:Journal of Sci- entific Computing78.1 (2019).doi: 10 . 1007 / s10915 - 018 - 0777-8(cit. on pp. 2, 9)

    2019

  14. [21]

    A New Adaptive Accelerated Levenberg–Marquardt Method for Solving Nonlin- ear Equations and Its Applications in Supply Chain Problems

    Rong Li, Mingyuan Cao, and Guoling Zhou. “A New Adaptive Accelerated Levenberg–Marquardt Method for Solving Nonlin- ear Equations and Its Applications in Supply Chain Problems”. In:Symmetry15.3 (2023).doi: 10.3390/sym15030588 (cit. on pp. 2, 9)

    2023 doi

  15. [22]

    Algebraic rules for computing the regularization parameter of the Levenberg–Marquardt method

    Sandra A.and Svaiter Benar F. Karas Elizabeth W.and Santos. “Algebraic rules for computing the regularization parameter of the Levenberg–Marquardt method”. In:Computational Opti- mization and Applications65.3 (2016).doi: 10.1007/s10589- 016-9845-x(cit. on pp. 2, 9)

    2016 doi

  16. [23]

    An efficient Levenberg–Marquardt method with a new LM pa- rameter for systems of nonlinear equations

    Keyvan Amini, Faramarz Rostami, and Giuseppe Caristi and. “An efficient Levenberg–Marquardt method with a new LM pa- rameter for systems of nonlinear equations”. In:Optimization 67.5 (2018).doi: 10.1080/02331934.2018.1435655 (cit. on pp. 2, 9)

    2018

  17. [24]

    Efficient Industrial Robot Calibration via a Novel Unscented Kalman Filter-Incorporated Variable Step-Size Levenberg–Marquardt Algorithm

    Zhibin Li, Shuai Li, and Xin Luo. “Efficient Industrial Robot Calibration via a Novel Unscented Kalman Filter-Incorporated Variable Step-Size Levenberg–Marquardt Algorithm”. In:IEEE Transactions on Instrumentation and Measurement72 (2023). doi:10.1109/TIM.2023.3265744(cit. on ...

    2023

  18. [25]

    A New Adaptive Leven- berg–Marquardt Method for Nonlinear Equations and Its Con- vergence Rate under the H¨ olderian Local Error Bound Con- dition

    Yang Han and Shaoping Rui. “A New Adaptive Leven- berg–Marquardt Method for Nonlinear Equations and Its Con- vergence Rate under the H¨ olderian Local Error Bound Con- dition”. In:Symmetry16.6 (2024).doi: 10.3390/sym16060674 (cit. on pp. 2, 9)

    2024 doi

  19. [26]

    Shun-ichi Amari.Information Geometry and Its Applications. 1st. Applied Mathematical Sciences. Springer Tokyo, 2016.doi: 10.1007/978-4-431-55978-8(cit. on pp. 2, 4)

    2016 doi

  20. [27]

    Modeling 1-3 composite piezo- electrics: thickness-mode oscillations

    W.A. Smith and B.A. Auld. “Modeling 1-3 composite piezo- electrics: thickness-mode oscillations”. In:IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control38.1 (1991).doi:10.1109/58.67833(cit. on p. 3)

    1991 doi

  21. [28]

    Sim- ulation of elastic guided waves interacting with defects in arbitrarily long structures using the Scaled Boundary Finite Element Method

    Hauke Gravenkamp, Carolin Birk, and Chongmin Song. “Sim- ulation of elastic guided waves interacting with defects in arbitrarily long structures using the Scaled Boundary Finite Element Method”. In:Journal of Computational Physics295 (2015).doi:10.1016/j.jcp.2015.04.032(cit. on p. 3)

    2015 doi

  22. [29]

    John Wiley & Sons, Ltd, 2018.doi: 10.1002/9781119388487 (cit

    Chongmin Song.The Scaled Boundary Finite Element Method. John Wiley & Sons, Ltd, 2018.doi: 10.1002/9781119388487 (cit. on p. 3)

    2018 doi

  23. [30]

    Efficient simulation of elastic guided waves interacting with notches, adhesive joints, delaminations and inclined edges in plate structures

    Hauke Gravenkamp. “Efficient simulation of elastic guided waves interacting with notches, adhesive joints, delaminations and inclined edges in plate structures”. In:Ultrasonics82 (2018).doi:10.1016/j.ultras.2017.07.019(cit. on p. 3)

    2018 doi

  24. [31]

    Efficient semi-analytical simulation of elastic guided waves in cylinders subject to arbitrary non- symmetric loads

    Dominik Itner et al. “Efficient semi-analytical simulation of elastic guided waves in cylinders subject to arbitrary non- symmetric loads”. In:Ultrasonics114 (2021).doi: 10.1016/j. ultras.2021.106389(cit. on pp. 3, 4)

    2021

  25. [32]

    Society for Industrial and Ap- plied Mathematics, 2005.doi:10.1137/1.9780898717921(cit

    Albert Tarantola.Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Ap- plied Mathematics, 2005.doi:10.1137/1.9780898717921(cit. on pp. 4, 7, 8)

    2005 doi

  26. [33]

    Tilo Strutz.Data Fitting and Uncertainty: A practical intro- duction to weighted least squares and beyond. First. Wiesbaden: Vieweg+Teubner, 2010 (cit. on pp. 4, 7, 9)

    2010

  27. [34]

    Shape sensitivity analysis of stress intensity factors by the scaled boundary finite element method

    Morsaleen Shehzad Chowdhury, Chongmin Song, and Wei Gao. “Shape sensitivity analysis of stress intensity factors by the scaled boundary finite element method”. In:Engineering Frac- ture Mechanics116 (2014).doi: 10.1016/j.engfracmech.2013. 12.007(cit. on p. 6)

    2014 doi

  28. [35]

    John Wiley & Sons, Ltd, 2016.doi: 10.1002/9781119136941 (cit

    Michel Kern.Numerical Methods for Inverse Problems. John Wiley & Sons, Ltd, 2016.doi: 10.1002/9781119136941 (cit. on p. 6)

    2016 doi

  29. [36]

    ˚Ake Bj¨ orck.Numerical Methods for Least Squares Problems. 2nd. Society for Industrial and Applied Mathematics, 2024. doi:10.1137/1.9781611977950(cit. on pp. 7, 8)

    2024 doi

  30. [37]

    Luenberger.Optimization by Vector Space Methods

    David G. Luenberger.Optimization by Vector Space Methods. John Wiley & Sons, Ltd, 1997 (cit. on p. 7)

    1997

  31. [38]

    Baltimore: Johns Hop- kins University Press, 2013 (cit

    Christian Hansen, V´ ıctor Pereyra, and Godela Scherer.Least Squares Data Fitting with Applications. Baltimore: Johns Hop- kins University Press, 2013 (cit. on p. 8)

    2013

  32. [39]

    Wright.Numerical Optimiza- tion

    Jorge Nocedal and Stephen J. Wright.Numerical Optimiza- tion. 2nd. Springer New York, 2006.doi: 10.1007/978-0-387- 40065-5(cit. on pp. 8, 9)

    2006 doi

  33. [40]

    Misfit Criteria for Quantitative Com- parison of Seismograms

    Miriam Kristekov´ a et al. “Misfit Criteria for Quantitative Com- parison of Seismograms”. In:Bulletin of the Seismological Society of America96.5 (2006).doi: 10.1785/0120060012 (cit. on p. 10)

    2006 doi

  34. [41]

    Time- frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals

    Miriam Kristekov´ a, Jozef Kristek, and Peter Moczo. “Time- frequency misfit and goodness-of-fit criteria for quantitative comparison of time signals”. In:Geophysical Journal Interna- tional178.2 (2009).doi: 10.1111/j.1365-246X.2009.04177.x (cit. on p. 10)

    2009 arXiv

  35. [42]

    On the forward simulation and cost functions for the ultra- sonic material characterization of polymers. Vol. 21. 1. 2021. doi:10.1002/pamm.202100128(cit. on p. 10)

    2021 doi

  36. [43]

    Prentice Hall PTR, 1995 (cit

    Leon Cohen.Time-frequency analysis. Prentice Hall PTR, 1995 (cit. on p. 10)

    1995

  37. [44]

    Alan V Oppenheim and Ronald W Schafer.Discrete-time sig- nal processing. 3rd. Harlow, [England]: Pearson, 2013 (cit. on p. 11)

    2013

  38. [45]

    2024.url: https : / / www

    Syensqo (Solvay Specialty Polymers).Data Sheet: KetaSpireR KT-880 Polyetheretherketone (PEEK). 2024.url: https : / / www . matweb . com / search / datasheet . aspx ? matguid = 5e21a17ebc5a43f0ac0fe4b7411ba1fb (visited on 07/10/2024) (cit. on p. 13)

    2024

  39. [46]

    2024.url: https : / / kunststoffedirekt

    Kunststoffedirekt.Data Sheet (PEEK). 2024.url: https : / / kunststoffedirekt . de/(visited on 07/18/2024) (cit. on p. 13)

    2024

  40. [47]

    2024.url: https: //www.loewen- kunststoffe.de/files/content/Downloads/ Datenblatt%20PEEK.pdf (visited on 07/18/2024) (cit

    Loewen Kunststoffe.Data Sheet (PEEK). 2024.url: https: //www.loewen- kunststoffe.de/files/content/Downloads/ Datenblatt%20PEEK.pdf (visited on 07/18/2024) (cit. on p. 13)

    2024

  41. [48]

    2024.url: https : / / www

    Ensinger.Data Sheet (PEEK, PA6). 2024.url: https : / / www . ensingerplastics . com / de - de / halbzeuge(visited on 07/10/2024) (cit. on p. 13)

    2024

  42. [49]

    HEUTE & COMP.Technische Information (PEEK). 2015. url: https://heutecomp.de/en/materials/peek/ (visited on 07/10/2024) (cit. on p. 13). Preprint 18

    2015

  43. [50]

    Liedtke Kunststoffetechnik.Data Sheet (PEEK, PA6.6). 2024. url: http://www.liedtke-kunststofftechnik.de/Liedtke3- Werkstoffdatenblaetter.html (visited on 07/10/2024) (cit. on p. 13)

    2024

  44. [51]

    2024.url: https : / / www

    TMK Zerspanungstechnik GmbH.Data Sheet (PEEK, PA6). 2024.url: https : / / www . tmk - zerspanungstechnik . de / material-technical-plastics.html (visited on 07/10/2024) (cit. on p. 13)

    2024

  45. [52]

    Mueller GmbH & Co.KG.Data Sheet (PEEK, PA6)

    Wilhelm Herm. Mueller GmbH & Co.KG.Data Sheet (PEEK, PA6). 2019.url: https://whm.net/en/company/ (visited on 12/19/2019) (cit. on p. 13)

    2019

  46. [53]

    2024.url: https : / / www

    chemie.de.Data Sheet (PEEK). 2024.url: https : / / www . chemie.de/produkte/labor/ (visited on 07/18/2024) (cit. on p. 13)

    2024

  47. [54]

    2016.url: https://kb-kunststoffdreherei.de/ (visited on 02/19/2025) (cit

    K&B Kunststoffdreherei GmbH.Data Sheet: (PEEK, PP, PA6- G, PA6-E). 2016.url: https://kb-kunststoffdreherei.de/ (visited on 02/19/2025) (cit. on p. 13)

    2016

  48. [55]

    2024.url: https : / / www

    Amsler & Frey AG.Data Sheet: (PEEK, PA6). 2024.url: https : / / www . amsler - frey . ch / downloads (visited on 02/19/2025) (cit. on p. 13)

    2024

  49. [56]

    Erwin Telle GmbH.Data Sheet: (PEEK, PP, PA6). 2024. url: https://telle.de/fertigung- produkte/kunststoffe/ werkstoffuebersicht(visited on 02/19/2025) (cit. on p. 13)

    2024

  50. [57]

    2024.url: https://www.sahlberg.de/ (visited on 02/19/2025) (cit

    SAHLBERG GmbH.Data Sheet: (PEEK, PA6). 2024.url: https://www.sahlberg.de/ (visited on 02/19/2025) (cit. on p. 13)

    2024

  51. [58]

    KG.Data Sheet (PEEK)

    Heck-Sevdic GmbH & Co. KG.Data Sheet (PEEK). 2024.url: https://www.heck-sevdic-gbr.de/index.php/kunststoffe- pa- pom- pvc- teflon/peek- polyetheretherketon (visited on 02/19/2025) (cit. on p. 13)

    2024

  52. [59]

    KG.Data Sheet (PEEK-Medical Grade)

    noltewerk GmbH & Co. KG.Data Sheet (PEEK-Medical Grade). 2015.url: https : / / noltewerk . de / en / downloads/ (visited on 02/19/2025) (cit. on p. 13)

    2015

  53. [60]

    2024.url: https: //designerdata.nl/materials/plastics/thermo- plastics/ (visited on 02/19/2025) (cit

    Designerdata.Data Sheet (PEEK, PP, PA6). 2024.url: https: //designerdata.nl/materials/plastics/thermo- plastics/ (visited on 02/19/2025) (cit. on p. 13)

    2024

  54. [61]

    2024.url: https://fietz

    Fietz Gruppe.Data Sheet (PEEK). 2024.url: https://fietz. com / produkte - leistungen / werkstoffe / hochtemperatur - thermoplaste/peek(visited on 02/19/2025) (cit. on p. 13)

    2024

  55. [62]

    2016.url: https : / / www

    Zell-Metall GmbH.Data Sheet (PEEK-ZELLAMID 1500). 2016.url: https : / / www . zellamid . com / en / Products / ZELLAMID % C2 % AE - HPM - High - Performance - Materials / ZELLAMID % C2 % AE - 1500 - X (visited on 02/19/2025) (cit. on p. 13)

    2016

  56. [63]

    2024.url: https: / / www

    Professional Plastics.Data Sheet (PEEK). 2024.url: https: / / www . professionalplastics . com(visited on 02/19/2025) (cit. on p. 13)

    2024

  57. [64]

    2015.url: https://www.polymehr

    Polymehr.Data Sheet (PP). 2015.url: https://www.polymehr. com/materialien/(visited on 07/18/2024) (cit. on p. 13)

    2015

  58. [65]

    2024.url: https : / / www

    Sonelastic.Data Sheet (PP). 2024.url: https : / / www . sonelastic . com / en / fundamentals / tables - of - materials - properties/polymers.html (visited on 07/10/2024) (cit. on p. 13)

    2024

  59. [66]

    The British Plastics Federation (BPF).Data Sheet (PP). 2015. url: https : / / www . bpf . co . uk / plastipedia / polymers / polymer-thermoplastics.aspx#polypropylenepp (visited on 07/10/2024) (cit. on p. 13)

    2015

  60. [67]

    INEOS Olefins and Polymers USA.Data Sheet (PP). 2024.url: https://www.ineos.com/products/?bu=INEOS+O+%26+P+USA& c=All+Countries&b=All+Brands&i=All+Industries&si=All+ Sub- Industries&m=All+Materials&sm=All+Sub- Materials (visited on 07/10/2024) (cit. on p. 13)

    2024

  61. [68]

    Plansee Composite Materials GmbH.Data Sheet (PP). 2024. url: https://www.plansee.com/de/unternehmen/kontakt/ deutschland.html(visited on 07/10/2024) (cit. on p. 13)

    2024

  62. [69]

    2024.url: https : / / www

    AZO Materials.Data Sheet (PP). 2024.url: https : / / www . azom . com / article . aspx ? ArticleID = 834 (visited on 07/10/2024) (cit. on p. 13)

    2024

  63. [70]

    2024.url: https://protoxyz

    ProtoXYZ.Data Sheet (PP). 2024.url: https://protoxyz. com / materials / Plastic / Polypropylene(PP) (visited on 02/19/2025) (cit. on p. 13)

    2024

  64. [71]

    K¨ onig GmbH Kunststoffprodukte.Data Sheet (PA6). 2016. url: https : / / www . koenig - kunststoffe . de / de / product - filter/(visited on 07/10/2024) (cit. on p. 13)

    2016

  65. [72]

    2014.url:https://www

    Pollen AM Inc.Data Sheet (PA6.6). 2014.url:https://www. pollen.am/performance_thermoplastics_pa_6_6/ (visited on 07/10/2024) (cit. on p. 13)

    2014

  66. [73]

    2024.url: https://www

    GEHRlastics.Data Sheet (PA6.6). 2024.url: https://www. gehrplastics . com / products / semifinishedproducts / gehr - pa/gehr-pa-6-6/(visited on 07/10/2024) (cit. on p. 13)

    2024

  67. [74]

    2015.url: https: / / www

    Direct Plastics Ltd.Data Sheet (PA6.6). 2015.url: https: / / www . directplastics . co . uk / data - sheets (visited on 07/10/2024) (cit. on p. 13)

    2015

  68. [75]

    Steelplast CC Engineering Plastics.Data Sheet (PA6). 2024. url: https://www.steelplastsa.com/downloads (visited on 07/10/2024) (cit. on p. 13)

    2024

  69. [76]

    2020.url: https:// www.broncesval.com/en/plastics-technicians/nylon-pa6- polyamide-6/(visited on 07/10/2024) (cit

    Broncesval Central.Data Sheet (PA6). 2020.url: https:// www.broncesval.com/en/plastics-technicians/nylon-pa6- polyamide-6/(visited on 07/10/2024) (cit. on p. 13)

    2020

  70. [77]

    Materials Sci- ence and Engineering: An Introduction

    William D. Callister and David G. Rethwisch. “Materials Sci- ence and Engineering: An Introduction”. In: 2018 (cit. on p. 13)

    2018

  71. [78]

    2011.url: https : / / www

    MatWeb LLC.MatWeb Material Property Data. 2011.url: https : / / www . matweb . com(visited on 07/10/2024) (cit. on p. 13)

    2011

  72. [79]

    Ansys LLC.Ansys Granta Materials Data.url: https : / / www . ansys . com / resource - center / brochure / granta - data - products-overview(visited on 07/17/2024) (cit. on p. 13)

    2024

  73. [80]

    Surrogate-based optimisation using adaptively scaled radial basis functions

    Magnus Urquhart, Emil Ljungskog, and Simone Sebben. “Surrogate-based optimisation using adaptively scaled radial basis functions”. In:Applied Soft Computing88 (2020).doi: 10.1016/j.asoc.2019.106050(cit. on p. 14)

    2020

  74. [81]

    Formu- lation of the Optimal Latin Hypercube Design of Experi- ments Using a Permutation Genetic Algorithm

    Stuart Bates, Johann Sienz, and Vassili Toropov. “Formu- lation of the Optimal Latin Hypercube Design of Experi- ments Using a Permutation Genetic Algorithm”. In:45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy- namics & Materials Conference.doi: 10.2514/6.2004-2011 (cit....

    doi

  75. [82]

    Optim: A mathematical optimization package for Julia

    Patrick Kofod Mogensen and Asbjørn Nilsen Riseth. “Optim: A mathematical optimization package for Julia”. In:Journal of Open Source Software3.24 (2018).doi: 10.21105/joss.00615 (cit. on p. 15)

    2018 doi

  76. [84]

    Version v1.40.13

    Tom Breloff.Plots.jl. Version v1.40.13. 2025.doi: 10.5281/ zenodo.15236224(cit. on p. 16)

    2025

  77. [85]

    Makie.jl: Flexible high- performance data visualization for Julia

    Simon Danisch and Julius Krumbiegel. “Makie.jl: Flexible high- performance data visualization for Julia”. In:Journal of Open Source Software6.65 (2021).doi: 10.21105/joss.03349 (cit. on p. 16)

    2021 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.