A semi-smooth Newton method for efficient evaluation of the inverse hysteresis operator

Felix Engertsberger; Herbert Egger

REVIEW 4 major objections 4 minor 17 references

A semi-smooth Newton method for efficient evaluation of the inverse hysteresis operator

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves that inverse hysteresis evaluation can be reduced to a convex minimization solved by a globally convergent, mesh-independent semismooth Newton method.

desk verdict This is a competent, useful paper on inverse hysteresis evaluation; the main caveat is a load-bearing semismooth lemma quoted from the authors' companion paper, and a benchmark comparison that fits the same data it then compares against. read the letter →

arxiv 2608.03191 v1 pith:ODE46N3H submitted 2026-08-04 math.NA cs.NA

Herbert Egger , Felix Engertsberger This is my paper

classification math.NAcs.NA MSC 65K1065N30

keywords magnetichysteresisinverseoperatorsemismoothNewtonmethodArmijolinesearchvectorpotentialformulationfiniteelementmesh-independentconvergenceconvexduality

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 aims to show that evaluating the inverse of an energy-based vector hysteresis operator—the map that returns magnetic field $\mathbf{H}$ from magnetic flux $\mathbf{B}$ and a memory state $J_{\mathrm{prev}}$—can be posed as a convex minimization problem and solved robustly by a semismooth Newton iteration with Armijo line search. The same construction is embedded in a vector potential finite element formulation of magnetostatics, giving nonsmooth nonlinear systems at each load step. The authors prove global linear convergence with a contraction factor independent of the mesh size and illustrate the behavior on a transformer benchmark. If the claims hold, hysteresis effects enter electromagnetic device simulation through a deterministic, parameter-free solve rather than through ad-hoc damping or mesh-dependent tuning.

What carries the argument

The load-bearing identity is convex duality: the inverse hysteresis map is the gradient of the conjugate energy density $w(\mathbf{B};J_{\mathrm{prev}})=\sup_{\mathbf{H}}\{\langle\mathbf{B},\mathbf{H}\rangle-w^*(\mathbf{H};J_{\mathrm{prev}})\}$, and its generalized Jacobian is the matrix inverse of the forward generalized Jacobian, $S_{\mathbf{B}}^{\mathbf{H}}=(S_{\mathbf{B}}^{\mathbf{B}})^{-1}$. This turns inversion of a nonsmooth hysteresis law into minimization of a strongly convex function, which the Armijo backtracking semismooth Newton iteration (14)–(16) solves with guaranteed decrease. The vector potential formulation (18) uses the same energy density $w(\operatorname{curl} A;J_{\mat

What would settle it

Drive a single material point with the Table 1 pinning data along a flux loop that repeatedly crosses the magnetization switching threshold and record the Armijo step sizes and residuals of iteration (17). If at any load step the accepted step size collapses toward zero or the residual stalls above the $10^{-8}$ tolerance, the semismooth derivative coverage used in the proof is not adequate near switching points. Independently, compute $S_{\mathbf{B}}^{\mathbf{H}}$ by numerical differencing of the inverse map at those same points and test whether it equals $(S_{\mathbf{B}}^{\mathbf{B}})^{-1}$

Watch

Extended reading notes

Core claim

Under the log-cosine internal energy densities of Assumption 1, the forward hysteresis map $\mathbf{B}(\mathbf{H};J_{\mathrm{prev}})=\nabla_{\mathbf{H}} w^*(\mathbf{H};J_{\mathrm{prev}})$ is strongly monotone, Lipschitz, and semismooth. Convex duality then gives the inverse map as the gradient of the conjugate energy density $w(\mathbf{B};J_{\mathrm{prev}})=\sup_{\mathbf{H}}\{\langle\mathbf{B},\mathbf{H}\rangle-w^*(\mathbf{H};J_{\mathrm{prev}})\}$, so evaluating the inverse means minimizing $j(\mathbf{H})=w^*(\mathbf{H};J_{\mathrm{prev}})-\langle\mathbf{B},\mathbf{H}\rangle$. The proposed iteration is an Armijo line-search semismooth Newton method whose update can be written $\mathbf{H}_{k+1

Load-bearing premise

The proof relies on the quoted semismooth calculus fact that the generalized Jacobian of the inverse hysteresis map is exactly the inverse of the forward map's generalized Jacobian, and that this matrix has the bounded, symmetric positive definite structure required by the line-search theory; this fact is asserted from the companion paper rather than reproved here.

Editorial extensions

If this is right

Evaluating the inverse hysteresis operator at a material point becomes a globally convergent iteration needing only a few steps; in the paper's tests about 3–4 iterations per load step suffice.
The vector potential finite element solve can start from any initial potential $A_0$ and still converge linearly, so no specialized initialization or damping strategy is needed.
The contraction factor in (22) is independent of the mesh size, so refinement does not degrade the nonlinear solver; computed iteration counts stay near 5–6 across five mesh resolutions.
The same constitutive model can be used in both forward and inverse modes, since both are gradients of conjugate energy densities, which provides a thermodynamically consistent path for computing iron losses.

Reading between the lines

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

Editorial inference: because only strong convexity, Lipschitz continuity, and semismoothness of the energy are used, the method should transfer to other energy-based hysteresis laws whose internal energy densities satisfy the same structural assumptions, such as different pinning distributions or anisotropic variants.
Editorial inference: the avoided failure mode of the earlier fixed-point scheme [11] can be stress-tested directly by benchmarking the Armijo iteration against that scheme on flux trajectories that repeatedly touch switching thresholds.
Editorial inference: the proof is carried out on the continuous level before discretization, so higher-order or hp-adaptive finite element discretizations should retain the same mesh-independent contraction factor, though that would require an additional discrete stability argument.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Hi [name],

Quick take on arXiv:2608.03191. This paper does two useful things: it gives a robust way to evaluate the inverse of the energy-based vector hysteresis operator via a semismooth Newton method with Armijo line search, and it embeds that operator in a vector-potential finite element problem with a global linear convergence theorem whose contraction factor is independent of mesh size. The algorithm directly addresses a known failure mode in Jacques et al. by adding the line search and exploiting the convex-conjugate structure of the model. The numerical results support the claims: single-point inversion converges in 3–4 iterations, and the transformer benchmark shows average iteration counts barely moving as the mesh is refined.

The main soft spot is exactly what the stress-test flags: Lemma 3(iii), which asserts that the inverse operator is semismooth and that an element of its generalized Jacobian is the inverse of the forward Jacobian, is not proved in this manuscript. The authors say it follows from standard convex analysis and cite their companion paper [9] for the detailed proof. That is a legitimate division of labor, but it makes the central convergence theorem conditional on an unverified property. If [9] does not contain a full proof, or if the property fails near switching points, the theory is incomplete. I would want the editor to ask for a self-contained proof or a clear public statement of where the lemma is proved.

The second soft spot is the validation circularity. The 11 pinning forces in Table 1 are fitted to the TEAM 32 measurement data [17], and the comparison in Figure 2 is against that same data. So the benchmark match is a demonstration of the model's calibration, not a predictive test. That is worth saying but it is not fatal: the convergence experiments and the mesh-independence table do not depend on the fitted parameters, and the algorithm itself is being tested, not the material model.

The proofs of Theorems 1 and 2 are brief sketches, but they cite standard globalization theory and the authors' own work. For a numerical analysis readership this is acceptable, though a referee might want a bit more self-containment.

Overall, this deserves a serious peer review. I would cite it if I worked on inverse hysteresis or magnetic field simulation. The core idea—using the convex dual to get a globally convergent damped Newton method—is sound, and the paper is honest about what it relies on. My recommendation: send it to review, but with a request to make Lemma 3(iii) verifiable.

Best, [You]

Referee Report

4 major / 4 minor

Summary. The paper studies the efficient numerical evaluation of an energy-based vector hysteresis model, specifically the inverse hysteresis operator H(B; J_prev) = ∇_B w(B; J_prev). The authors characterize the forward and inverse operators via convex duality, propose an Armijo line-search semismooth Newton iteration (14)--(16) for evaluating the inverse at a material point, and then embed this in a vector potential finite element formulation (18), solved by the damped Newton scheme (19)--(21). The main theoretical claims are global linear convergence with a contraction factor depending only on model constants (Theorem 1) and, at the FEM level, mesh-independent linear convergence (Theorem 2, Eq. (22)). Numerical experiments on a single material point and on the TEAM 32 transformer benchmark illustrate iteration counts and mesh behavior.

Significance. If the results are correct, the paper provides a theoretically justified and practical solver for a difficult non-smooth inversion problem, and the mesh-independent convergence is a valuable property for large-scale magnetic field simulations. The algorithmic idea is simple and the experiments are consistent with the claimed linear and mesh-independent behavior. However, the paper's main convergence theorems are not self-contained: they rely on Lemma 3(iii), whose proof is delegated to the companion paper [9], and on results taken from [16]. The numerical validation also contains a circular component because the material parameters in Table 1 are fitted to the same TEAM 32 measurement data used for the comparison in Figure 2. These issues do not necessarily invalidate the central claim, but they need to be addressed before the paper can be considered complete.

major comments (4)

[Section 3, Lemma 3(iii)] Lemma 3(iii) is the load-bearing step for the global and superlinear convergence claims: it asserts that the inverse hysteresis operator is semismooth and that an element of its generalized Jacobian is S_H = (S_B)^{-1}, with S_H symmetric positive definite and uniformly bounded. The proof is not given; the text only says 'The assertions follow from standard properties about conjugate functions; see [7, 14]. A detailed proof can be found in [9].' This is not sufficient for a journal submission. The identity S_H = (S_B)^{-1} is nontrivial: inverting one element of the generalized Jacobian of the forward operator does not automatically produce an element of the generalized Jacobian of the inverse without additional semismoothness and nonsingularity conditions. In particular, the behavior at the switching points J_i = J_{i,prev}, where (7) sets S_i^J = 0 and hence S_B = μ0 I, must be justifi
[Section 3, Theorem 1] The proof of Theorem 1 is only a citation to [13, Ch. 3] and [10]. While using standard results is acceptable, the manuscript does not verify that the required assumptions are satisfied, especially that the chosen scaling matrices S_k are uniformly symmetric positive definite with eigenvalues bounded away from zero and infinity, and that the semismoothness of the composed operator holds in the sense needed by [10]. There is also a referencing error: the text says 'For the choice S_k = S_k^H, see Lemma 1(iii)', but Lemma 1(iii) concerns the generalized Jacobian of the forward operator, not of the inverse. The correct reference appears to be Lemma 3(iii). Please correct this and provide a proof sketch or a precise statement of the standard theorem being invoked, including the verification of the assumptions.
[Section 4, Theorem 2] The mesh-independent linear convergence result of Theorem 2 is asserted to follow 'immediately' from the analysis in [16]. Since this is one of the main contributions, the dependency on [16] should be made explicit and the hypotheses of the relevant result must be checked against the present setting. In particular, the paper does not show that the energy functional (18) satisfies the conditions used in [16], nor does it derive the contraction factor's independence from the mesh size within this paper. If [16] is a companion paper, please include the exact theorem and its verification, or provide a self-contained proof.
[Section 5, Table 1 and Figure 2] The validation is partially circular. The material parameters in Table 1 are fitted to the measurement data reported in [17], and the same data source is used for the C6 comparison in Figure 2. Thus the agreement with measurements in Figure 2 is a consistency check rather than an independent validation of the hysteresis model. This does not undermine the convergence claims, which are the main focus, but the manuscript should state this limitation explicitly or use a separate validation data set.

minor comments (4)

[Section 3, Theorem 1] The reference to Lemma 1(iii) for S_k^H should be Lemma 3(iii) as noted above.
[Throughout] Some notation is garbled in the typeset text (e.g., 'u1D464.alt*' for the co-energy density and related symbols). Please ensure all mathematical symbols are correctly rendered.
[Section 5, Table 2] The column 'Time' would benefit from specifying whether it is wall-clock time on the described hardware and whether the iteration counts are averaged over all load steps or only over nonlinear iterations; the caption says 'Iterations' but the text says 'average iteration counts'.
[Section 2, Remark 1] The sentence 'If Y_i = ∇J φ_i(J_i,prev) fulfills the constraint, then J_i = J_i,prev' should be checked for consistency with (8); this is a detail but important for the algorithmic description.

Circularity Check

2 steps flagged · score 4.0 of 10

Empirical validation is calibrated on the same benchmark data used for comparison; central convergence theorems are delegated to companion papers by the same authors.

fitted input called prediction [Section 5, Numerical validation (Table 1 and Figure 2)]
"For the experiment, we use 11 pinning forces fitted to the measurement data reported in [17]. ... The magnetic induction at the pick-up coil C6 is depicted in the right plot of Fig. 2 and compared to measurements and an anhysteretic simulation."

The 11 pinning forces (and reference polarizations) in Table 1 are fitted to the TEAM 32 measurement data of [17], and the same TEAM 32 dataset supplies the C6 measurement against which the simulation is compared in Fig. 2. The agreement at C6 is therefore a consistency check of the parameter fit, not an independent prediction of the material model. This circularity affects the empirical validation only; it does not enter the convergence proofs.
self citation load bearing [Section 3 (Lemma 3(iii)) and Section 4 (Theorem 2)]
"The assertions follow from standard properties about conjugate functions; see [7, 14]. A detailed proof can be found in [9]. ... From the analysis developed in [16], we can immediately deduce the following convergence results."

The load-bearing semismooth inverse Jacobian property S_H=(S_B)^{-1} (Lemma 3(iii)) and the global mesh-independent linear convergence (Theorem 2, Eq. (22)) are not proved in this manuscript; they are delegated to [9] and [16], both authored by the same research group. The derivation chain therefore cannot be audited from the present text alone. Because [9] invokes external standard results [7,14] and [16] is a published theorem, this is self-citation load-bearing rather than a definitional equivalence; the algorithmic contribution would survive if the cited theorems are sound.

full rationale

The central mathematical derivation—defining the inverse hysteresis operator as the gradient of the convex conjugate energy and solving the inversion by an Armijo line-search semismooth Newton method—is a legitimate convex-analytic argument with independent content. The convergence theorems are not statistical predictions and do not reduce to the fitted material parameters. However, the paper's main convergence claims rest on Lemma 3(iii) and Theorem 2, whose proofs are outsourced to companion papers [9] and [16] by the same authors; while these are supported by external references, they are load-bearing self-citations. In addition, the numerical validation calibrates the 11 pinning forces to the same TEAM 32 measurement data that is then used as the benchmark comparison, so the empirical agreement at C6 is not an out-of-sample prediction. Neither issue makes the core derivation definitionally circular, but together they justify a moderate circularity score of 4.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The mathematical machinery is standard convex analysis and semismooth Newton theory; no new physical entities are introduced. The material parameters in Table 1 are fitted to the benchmark data, which weakens the independent-evidence value of the numerical comparison. Several load-bearing proof steps are delegated to the authors' own prior papers.

free parameters (2)

pinning strengths kappa_i (i=1..11) = 0, 43, 73, 128, 164, 202, 241, 280, 320, 360, 400
Fitted to TEAM 32 measurement data [17] and used in the numerical benchmark in Figure 2 and Table 2.
reference polarizations J_0,i (i=1..11) = 0.08, 0.55, 0.49, 0.05, 0.04, 0.07, 0.02, 0.02, 0.02, 0.02, 0.03
Used as material data in Table 1; part of the same fitted set used for the TEAM 32 validation.

assumptions (5)

standard math Convex conjugate duality and Legendre-Fenchel transform properties
Used to define the inverse operator via the supremum in (9) and to prove strong monotonicity of the resulting gradient; see [7,14] and Lemma 3.
standard math Semismooth Newton convergence theory (Ulbrich)
Basis for local superlinear convergence in Theorem 1 and Remark 3, following [10].
domain assumption Energy-based hysteresis model with Assumption 1 (log-cos energy density)
The specific form of the energy density in Assumption 1 provides strong convexity and semismoothness; all convergence results depend on this model structure from [3,5].
domain assumption Global convergence and mesh-independence analysis from prior work [16]
Theorem 2 and the mesh-independent contraction factor are deduced from the analysis in [16], a companion paper by the same authors, rather than proved in this text.
standard math Closed subspace V of H(curl) with Friedrichs-type inequality ||v||_L2 <= c5 ||curl v||_L2
Assumed in Theorem 2 and needed for the variational formulation (18) to be coercive and for the convergence bound (22).

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Cite this review

Pith. "Pith review of A semi-smooth Newton method for efficient evaluation of the inverse hysteresis operator." pith.science (2026). https://pith.science/paper/ODE46N3H

@misc{pith2026260803191,
  author       = {Pith},
  title        = {Pith review of: A semi-smooth Newton method for efficient evaluation of the inverse hysteresis operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ODE46N3H}},
  note         = {Machine review of arXiv:2608.03191}
}

read the original abstract

We study the numerical evaluation of an energy-based vector hysteresis model and its incorporation into finite element simulations based on a vector potential formulation. The inherent non-smoothness of the hysteresis model poses challenges for both numerical analysis and implementation. Using tools from convex analysis, we characterize the forward and inverse hysteresis operators in terms of energy densities. This characterization yields well-posedness of the resulting models and leads to robust algorithms for their evaluation based on generalized semi-smooth Newton methods. It furthermore enables a seamless integration into magnetic field simulations, leading to nonlinear and non-smooth optimization problems at every load step. We discuss the finite element discretization, present a semi-smooth Newton method for the iterative solution, and establish global linear convergence with mesh-independent convergence rates. The theoretical results are illustrated by numerical experiments.

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[9]

A semi-smooth Newton method for magnetic field problems with hysteresis

H. Egger and F. Engertsberger. A semi-smooth Newton metho d for magnetic ﬁeld problems with hysteresis. arXiv:2506.01499, 2025

work page Pith review arXiv 2025
[17]

Bottauscio, M

O. Bottauscio, M. Chiampi, C. Ragusa, L. Rege, and M. Repe tto. A test case for validation of magnetic ﬁeld analysis with vector hysteresis. IEEE Trans. Magn., 38:893–896, 2002

work page 2002
[16]

Egger, F

H. Egger, F. Engertsberger, and B. Radu. On the convergen ce of higher-order ﬁnite element methods for nonlinear magnetostatics. SIAM J. Numer. Anal., 64:55–75, 2026

work page 2026
[10]

M. Ulbrich. Semismooth Newton Methods for Variational Inequalities and Constrained Optimiza- tion Problems in Function Spaces. SIAM, Philadelphia, PA, 2011

work page 2011
[1]

Sadowski, N

N. Sadowski, N. J. Batistela, J. P . A. Bastos, and M. Lajoie -Mazenc. An inverse Jiles-Atherton model to take into account hysteresis in time-stepping ﬁnit e-element calculations. IEEE Trans. Magn., 38:797–800, 2002

work page 2002
[2]

Upadhaya, P

B. Upadhaya, P . Rasilo, L. Perkki ¨o, P . Handgruber, A. Belahcen, and A. Arkkio. Comparison of anisotropic energy-based and Jiles–Atherton models of f erromagnetic hysteresis. IEEE Trans. Magn., 56:1–7, 2020

work page 2020
[3]

Francois-Lavet, F

V . Francois-Lavet, F. Henrotte, L. Stainier, L. Noels, and C. Geuzaine. An energy-based variational model of ferromagnetic hysteresis for ﬁnite element computations. J. Comp. Appl. Math., 246:243– 250, 2013

work page 2013
[4]

Bergqvist

A. Bergqvist. Magnetic vector hysteresis model with dry f riction-like pinning. Physica B: Condensed Matter, 233:342–347, 1997

work page 1997

Show all 17 references

[5]

Prigozhin, V

L. Prigozhin, V . Sokolovsky, J. W. Barrett, and S. E. Zirka. On the energy-based variational model for vector magnetic hysteresis. IEEE Trans. Magn., 52:1–11, 2016

2016
[6]

Domenig, K

L. Domenig, K. Roppert, and M. Kaltenbacher. Incorporati on of a 3-d energy-based vector hysteresis model into the ﬁnite element method using a reduc ed scalar potential formulation. IEEE Trans. Magn., 60:1–8, 2024

2024
[7]

A. Beck. First-Order Methods in Optimization. SIAM, Philadelphia, PA, 2017

2017
[8]

Egger, F

H. Egger, F. Engertsberger, L. Domenig, K. Roppert, and M. Kaltenbacher. On forward and inverse magnetic vector hysteresis operators. IEEE Trans. Magn., 61:7300207, 2025

2025
[11]

Jacques, R

K. Jacques, R. Sabariego, C. Geuzaine, and J. Gyselinck. Inclusion of a direct and inverse energy- consistent hysteresis model in dual magnetostatic ﬁnite element formulations. IEEE Trans. Magn., 52:7300304, 2015

2015
[12]

Mielke and T

A. Mielke and T. Roub ´ıˇcek. Rate-Independent Systems. Theory and Applications. Springer, 2015

2015
[13]

Nocedal and S

J. Nocedal and S. Wright. Numerical Optimization. 2nd ed. . Springer, 2006

2006
[14]

M. S. Gowda. Inverse and implicit function theorems for /u1D43B-diﬀerentiable and semismooth functions. Optim. Methods Softw., 19:443–461, 2004

2004
[15]

G. Meunier. The Finite Element Method for Electromagnetic Modeling . Wiley, 2008

2008

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [9]

A semi-smooth Newton method for magnetic field problems with hysteresis

H. Egger and F. Engertsberger. A semi-smooth Newton metho d for magnetic ﬁeld problems with hysteresis. arXiv:2506.01499, 2025

work page Pith review arXiv 2025

[2] [17]

Bottauscio, M

O. Bottauscio, M. Chiampi, C. Ragusa, L. Rege, and M. Repe tto. A test case for validation of magnetic ﬁeld analysis with vector hysteresis. IEEE Trans. Magn., 38:893–896, 2002

work page 2002

[3] [16]

Egger, F

H. Egger, F. Engertsberger, and B. Radu. On the convergen ce of higher-order ﬁnite element methods for nonlinear magnetostatics. SIAM J. Numer. Anal., 64:55–75, 2026

work page 2026

[4] [10]

M. Ulbrich. Semismooth Newton Methods for Variational Inequalities and Constrained Optimiza- tion Problems in Function Spaces. SIAM, Philadelphia, PA, 2011

work page 2011

[5] [1]

Sadowski, N

N. Sadowski, N. J. Batistela, J. P . A. Bastos, and M. Lajoie -Mazenc. An inverse Jiles-Atherton model to take into account hysteresis in time-stepping ﬁnit e-element calculations. IEEE Trans. Magn., 38:797–800, 2002

work page 2002

[6] [2]

Upadhaya, P

B. Upadhaya, P . Rasilo, L. Perkki ¨o, P . Handgruber, A. Belahcen, and A. Arkkio. Comparison of anisotropic energy-based and Jiles–Atherton models of f erromagnetic hysteresis. IEEE Trans. Magn., 56:1–7, 2020

work page 2020

[7] [3]

Francois-Lavet, F

V . Francois-Lavet, F. Henrotte, L. Stainier, L. Noels, and C. Geuzaine. An energy-based variational model of ferromagnetic hysteresis for ﬁnite element computations. J. Comp. Appl. Math., 246:243– 250, 2013

work page 2013

[8] [4]

Bergqvist

A. Bergqvist. Magnetic vector hysteresis model with dry f riction-like pinning. Physica B: Condensed Matter, 233:342–347, 1997

work page 1997