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Efficient evaluation of forward and inverse energy-based magnetic hysteresis operators

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that the regularized inverse magnetic hysteresis operator is the exact inverse of the regularized forward operator and can be evaluated in $O(K)$ time via a Schur complement.

desk verdict Worth reading and worth fixing: the regularization analysis is clean, but the Schur complement system in Eq. (15) is not the Newton system of Eq. (14). read the letter →

arxiv 2507.15289 v1 pith:5BLFAP7R submitted 2025-07-21 math.NA cs.NA

Herbert Egger , Felix Engertsberger , Andreas Schafelner This is my paper
classification math.NAcs.NA MSC 65K1065H10
keywords magnetichysteresisenergy-basedvectormodelinverseoperatorregularizationNewtonmethodSchurcomplementcomputationalcomplexitypotentialformulation
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

Energy-based magnetic hysteresis models describe a material's response through many small convex but non-smooth minimization problems, one per pinning force; solving them reliably at every material point is what makes these models usable in field solvers. This paper replaces the non-smooth pinning term with a smooth approximation $|x|_\varepsilon = \sqrt{|x|^2+\varepsilon}$, turning both the forward map $B = B(H;\{J_{k,p}\})$ and its inverse $H = H(B;\{J_{k,p}\})$ into strongly convex problems that can be solved by damped Newton iterations. The central results are that the regularization error can be controlled, that the regularized inverse operator is exactly the inverse of the regularized forward operator, and that a Schur complement reduces the otherwise dense $O(K^3)$ Newton systems of the inverse problem to $O(K)$ work. The payoff is practical: if these results hold, vector-potential finite element simulations can use a thermodynamically consistent hysteresis model without an expensive inversion step.

What carries the argument

The machinery has three parts. First, the regularized norm $|x|_\varepsilon = \sqrt{|x|^2+\varepsilon}$ smooths the non-differentiable pinning terms and bounds the error uniformly because the objectives are strongly convex. Second, a damped Newton method with Armijo backtracking solves each smooth minimization problem, with quadratic local convergence. Third, for the inverse problem the Newton system is block-structured: the blocks couple only through the shared term $\nu_0 \sum_\ell \delta J_\ell$, so introducing the total update $\delta J$ as an auxiliary variable makes the Hessian effectively block-diagonal plus a low-rank correction; the Schur complement system is only $d\times d$ and is set up in $O(K)$ operations.

What would settle it

Take an internal energy density that is strictly convex but not strongly convex, such as $U_k(J) = |J|^{3/2}$, and repeat the regularization experiments; if the Newton iterations stagnate or the computed error fails to vanish as $\varepsilon \to 0$, the load-bearing role of Assumption 1 is confirmed. Alternatively, time the Schur complement solver for increasing $K$ (e.g., $K=1000$); the measured complexity should grow linearly, not cubically.

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Extended reading notes

Core claim

Under Assumption 1, the paper claims the following. Let $|x|_\varepsilon = \sqrt{|x|^2+\varepsilon}$ be the regularized Euclidean norm. The minimizers of the regularized forward problem are within $O(\sqrt{\varepsilon})$ of the minimizers of the true non-smooth problem, and similarly for the inverse problem. By comparing the first-order optimality conditions, the paper proves that the inverse of the regularized forward operator is the regularized inverse operator: given $B = \mu_0 H + \sum_k J_k^\varepsilon$ with $J_k^\varepsilon$ solving the forward minimization, the same polarizations solve the coupled inverse minimization. Finally, the coupled Newton system can be reorganized with the total update $\delta J = \sum_\ell \delta J_\ell$ as an auxiliary variable, so each step only requires a block-diagonal Hessian solve plus a tiny $d\times d$ Schur complement; evaluation of the inverse operator therefore costs $O(K)$, matching the forward operator.

Load-bearing premise

Everything in the paper rests on Assumption 1, which gives each internal energy density a specific strongly convex logarithmic form with positive parameters; if a real material's energy density is less convex or the pinning is anisotropic, the proofs no longer apply.

Editorial extensions

If this is right

  • Evaluating the inverse hysteresis operator costs $O(K)$ operations, the same as the forward operator, removing the main obstacle to using energy-based hysteresis in vector-potential magnetic field formulations.
  • The regularization error can be made arbitrarily small by choosing $\varepsilon$; the proved bound is $O(\sqrt{\varepsilon})$ for the polarizations, and the numerical examples indicate the faster rate $O(\varepsilon)$ in the computed fluxes and fields.
  • Forward and inverse regularized models are exact inverses: a hysteresis loop computed with the forward operator and then read back with the inverse operator reproduces the original magnetic field.
  • The damped Newton iterations converge globally and locally quadratically, so the local material laws can be evaluated reliably with a fixed small number of iterations.
  • Wall-time measurements show the inverse operator implemented via the Schur complement runs at essentially the same speed as the forward operator for $K$ up to 100, while the standard $O(K^3)$ implementation becomes markedly slower as $K$ grows.

Reading between the lines

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

  • A natural extension the paper leaves implicit: the same Schur-complement trick should apply to the forward model with effective fields $H + \alpha J$, which the authors note leads to the same coupled minimization difficulty; if the analogy holds, that variant also becomes $O(K)$.
  • The numerical results suggest the true regularization rate is $O(\varepsilon)$ rather than the proved $O(\sqrt{\varepsilon})$; a sharper a-priori analysis is a testable open problem.
  • Because the paper shows how to compute the Jacobians of both operators by differentiating the optimality systems, full Newton-type solvers for the coupled field-and-material problem become possible; this is announced as future work.
  • In practice, linking $\varepsilon$ to the Newton tolerance or to the mesh size could yield an adaptive accuracy-cost trade-off when the inverse operator is embedded in a finite element solver.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper develops a regularized version of the energy-based vector hysteresis model of Francois-Lavet et al., in which the non-smooth pinning terms are replaced by smooth approximations |x|_epsilon = sqrt(|x|^2+epsilon). The authors prove strong-convexity-based error estimates for the regularized forward and inverse problems, show that the regularized inverse operator is the inverse of the regularized forward operator via first-order optimality conditions, and propose a damped Newton method for the coupled inverse minimization. The main algorithmic contribution is a Schur complement technique that reduces the per-step cost of evaluating the inverse operator from O(K^3) to O(K). The claims are supported by numerical experiments with uniform and rotational field excitations, which confirm the regularization error, the forward/inverse equivalence, and the expected wall-clock scaling.

Significance. If the main claims hold, the paper addresses a genuine bottleneck: the inverse energy-based hysteresis operator can then be used in vector-potential finite element formulations at essentially the same cost as the forward operator. The theoretical framework is appropriate and mostly clean: the strong-convexity arguments are the right tool, Assertion 6 gives an elegant direct proof of the forward/inverse correspondence, and the numerical results (Figures 1-3, Table I) support the central assertions, including the observed better-than-proven regularization rate. The authors are also transparent about the limitations of Assumption 1 and about the gap between the proven rate and the numerically observed rate. However, two load-bearing issues in the written analysis must be fixed before the paper can serve as a reliable recipe: the proof of Assertion 2 contains an algebraic error in deriving the O(sqrt(epsilon)) rate, and the block system (15) in Section III-D is not algebraically equivalent to the Newton system (14).

major comments (2)
  1. [Section II.A, Assertion 2 (and the related discussion in Section III.A)] The proof of Assertion 2 does not establish the stated rate |Jk - J_epsilon_k| = O(sqrt(epsilon)). From the strong-convexity inequality the authors obtain (gamma/2)|Jk - J_epsilon_k|^2 <= C sqrt(epsilon); this implies |Jk - J_epsilon_k| <= C epsilon^{1/4}, not O(sqrt(epsilon)). The displayed conclusion "|Jk - J_epsilon_k| <= 2 sqrt(epsilon)/gamma" does not follow from the preceding inequality. Since Assertion 5 is said to follow by the same argument and since the paper repeatedly refers to the O(sqrt(epsilon)) error control, this is not a cosmetic issue: either the proof must be repaired to yield the claimed rate, or the statement must be weakened to O(epsilon^{1/4}). The numerical results in Figure 1 suggest the intended O(sqrt(epsilon)) rate may be true, but the present proof does not deliver it.
  2. [Section III-D, Eq. (15) and the proof of Assertion 9] The block system (15) is not algebraically equivalent to the Newton system (14). Expanding (15) gives, for each k, nu_0 grad^2_{J_k} g_k delta J_k + delta J = RHS_k, whereas (14) requires grad^2_{J_k} g_k delta J_k + nu_0 delta J = RHS_k; the factor nu_0 is attached to the wrong term. In addition, the Schur complement derivation in the proof of Assertion 9 solves the first row as delta tilde J = H_K^{-1}(b - E_K^T delta J), which corresponds to a system with H_K (not nu_0 H_K) in the (1,1) block. The correct representation of (14) is (H_K, nu_0 E_K^T; E_K, -I), which gives the Schur complement I + nu_0 E_K H_K^{-1} E_K^T. As printed, Eq. (15) and the proof of Assertion 9 are mutually inconsistent, and a reader implementing the displayed system verbatim will compute directions that are not Newton directions. Because this block system is the entire basis for the O(K) inverse-evaluation claim, the equations and the proof must be aligned before the paper can be used as a recipe.
minor comments (5)
  1. [Section III-D, Eq. (14)] The right-hand side reads "nu_0(B - sum_l J_k)"; the subscript in the sum should be J_l, not J_k. This is a typo, but it occurs in a central equation.
  2. [Section III.A, proof of Assertion 5] The strong convexity parameter of G_epsilon is stated as gamma/2, but since the sum of the Hessians of U_k is bounded below by gamma I and the coupling term is positive semidefinite, the parameter is at least gamma. The factor is immaterial for the argument but should be made consistent.
  3. [Section III.D and Remark 8] The paper states that evaluating the inverse operator requires O(K) operations. The analysis actually shows O(K) operations per Newton step; the number of Newton iterations is only observed empirically to be O(1). The wording should distinguish per-step complexity from total complexity unless an iteration-count bound is proven.
  4. [Section IV.A] "with K the total number of pinning forces an Js = 1.545" contains a typo: "an" should be "and".
  5. [Table I caption] "Experiments where performed" should read "Experiments were performed".

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the forward/inverse equivalence is rederived from optimality conditions, and the O(K) Schur-complement claim is an algebraic reduction; only a minor non-load-bearing self-citation to [21] appears.

full rationale

The derivation chain is self-contained at every load-bearing point. Assertion 6 does not import the inverse property from the authors' prior work; it proves that the minimizers of the coupled inverse problem (9) coincide with the forward minimizers by verifying the stationarity system (10) at B = mu0 H + sum Jk^epsilon. This is a genuine optimality argument, not a definitional identity: the inverse operator is defined as a minimization problem and then shown to invert the forward map. The regularization error results (Assertions 2 and 5) rest on the assumed uniform strong convexity gamma from Assumption 1, with no fitted constants or data-dependent prediction. The O(K) complexity result is a Schur-complement elimination exploiting the block-diagonal structure of the Hessian; Table I confirms it against an independent dual-solver benchmark [11]. The only notable self-citation is [21], which supplies the inverse representation (4); because Assertion 6 rederives the regularized equivalence, this citation is not load-bearing. The block system (15) has a scaling/placement of nu0 that appears inconsistent with (14), but that is an algebraic correctness or typographical issue, not a circular reduction. No fitted input is renamed as a prediction, and no author-imported uniqueness theorem is used to force the construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central theoretical claims depend only on the strong-convexity structure of Assumption 1 and standard Newton convergence theory. The only hand-chosen numbers are the benchmark material parameters and the smoothing parameter; neither is fitted to data nor used as a substitute for a derivation. No new physical entities are introduced.

free parameters (2)
  • regularization parameter epsilon = not fitted; chosen e.g. 1e-8 with tol = 1e-8
    User-chosen smoothing scale that trades accuracy for differentiability. Its effect is controlled by the error estimates in Assertions 2 and 5, but the method's practical behavior depends on its value.
  • benchmark material parameters A_s,k, J_s,k = J_s/K, chi_k = A_s,k = 50, J_s = 1.545, chi_k = 140(k-1)/(K-1)
    Hand-chosen typical soft-magnetic parameters for the numerical tests. The paper explicitly notes these must be fitted to real measurements for applications; they are not fitted in this work and do not enter the central complexity or equivalence proofs.
assumptions (4)
  • domain assumption Assumption 1: internal energy densities have the form U_k(J) = -(2 A_s,k J_s,k / pi) log(cos(pi J / (2 J_s,k))) with A_s,k, J_s,k > 0 and pinning strengths chi_k >= 0.
    Invoked at the start of Section II; all strong convexity constants, error bounds, and Newton convergence statements depend on this assumption.
  • standard math Standard convex optimization results: damped Newton with Armijo backtracking converges globally and locally quadratically for smooth strongly convex objectives.
    Used in Assertions 3 and 7, citing the textbook by Nocedal and Wright [22].
  • domain assumption The energy-based hysteresis model of Francois-Lavet et al. [10] and the inverse representation from the authors' prior work [21] are accepted as valid material models.
    The paper builds on these models without re-deriving their physical validity; this is reasonable for a numerical methods contribution.
  • domain assumption Partial polarizations live in R^d with d <= 3, making each Hessian block d by d.
    Used in Remarks 4 and 8 to justify O(1) work per block and the small d x d Schur complement system; standard for magnetic field problems.

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Pith. "Pith review of Efficient evaluation of forward and inverse energy-based magnetic hysteresis operators." pith.science (2026). https://pith.science/paper/5BLFAP7R

@misc{pith2026250715289,
  author       = {Pith},
  title        = {Pith review of: Efficient evaluation of forward and inverse energy-based magnetic hysteresis operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5BLFAP7R}},
  note         = {Machine review of arXiv:2507.15289}
}
read the original abstract

The energy-based vector hysteresis model of Francois-Lavet et al. establishes an implicit relation between magnetic fields and fluxes via internal magnetic polarizations which are determined by convex but non-smooth minimization problems. The systematic solution of these problems for every material point is a key ingredient for the efficient implementation of the model into standard magnetic field solvers. We propose to approximate the non-smooth terms via regularization which allows to employ standard Newton methods for the evaluation of the local material models while being in control of the error in this approximation. We further derive the inverse of the regularized hysteresis operator which amounts to a regularized version of the inverse hysteresis model. The magnetic polarizations in this model are again determined by local minimization problems which here are coupled across the different pinning forces. An efficient algorithm for solving the Newton systems is proposed which allows evaluation of the inverse hysteresis operator at the same cost as the forward model. Numerical tests on standard benchmark problems are presented for illustration of our results.

Figures

Figures reproduced from arXiv: 2507.15289 by the authors.

Figure 1
Figure 1. Convergence plot of the relative error for the regular [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Hysteresis loop of the forward and inverse operator w [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Hysteresis loop of the forward and inverse operator w [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

  1. [1]

    Dlala, J

    E. Dlala, J. Saitz, and A. Arkkio. Inverted and forward pr eisach models for numerical analysis of electromagnetic field problems. Magnetics, IEEE Transactions on , 42:1963 – 1973, 2006

    work page 1963

  2. [2]

    M. M. Billah, F. Martin, and A. Belahcen. A computational ly effective method for iron loss estimation in a synchronous machine fro m a static field solution. In 2020 International Conference on Electrical Machines (ICEM), volume 1, pages 751–757, 2020

    work page 2020

  3. [3]

    D. C. Jiles and D. L. Atherton. Theory of ferromagnetic hy steresis. J. Magnetism. Magn. Materials , 61:48–60, 1986

    work page 1986

  4. [4]

    I. D. Mayergoyz and G. Friedman. Generalized Preisach mo del of hysteresis. IEEE Trans. Magn. , 24:212–217, 1988

    work page 1988

  5. [5]

    Serpico, M

    C. Serpico, M. d’Aquino, C. Visone, and D. Davino. A new cl ass of preisach-type isotropic vector model of hysteresis. Physica B: Condensed Matter, 343(1):117–120, 2004. Proceedings of the Fourth Intional Conference on Hysteresis and Micromagnetic Modeling

    work page 2004

  6. [6]

    Leite, N

    J.V . Leite, N. Sadowski, P . Kuo-Peng, and J.P .A. Bastos. A new anisotropic vector hysteresis model based on stop hysteron s. IEEE Transactions on Magnetics , 41(5):1500–1503, 2005

    work page 2005

  7. [7]

    Bergqvist

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

    work page 1997

  8. [8]

    Bergqvist, A

    A. Bergqvist, A. Lundgren, and G. Engdahl. Experimental testing of an anisotropic vector hysteresis model. IEEE Trans. Magn. , 33:4152 – 4154, 1997

    work page 1997

Show all 22 references
  1. [9]

    Henrotte, A

    F. Henrotte, A. Nicolet, and K. Hameyer. An energy-based vector hysteresis model for ferromagnetic materials. COMPEL, 25:71–80, 2006

    2006

  2. [10]

    Francois-Lavet, F

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

    2013

  3. [11]

    Prigozhin, V

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

    2016

  4. [12]

    Kaltenbacher, K

    M. Kaltenbacher, K. Roppert, L. D. Domenig, and H. Egger . Comparison of energy based hysteresis models. In COMPUMAG 2022 , pages 1–4, 2022

    2022

  5. [13]

    Sauseng, L

    A. Sauseng, L. Domenig, K. Roppert, and M. Kaltenbacher . Adaptions of the energy-based hysteresis model for correct rotational l osses. In 2022 IEEE 20th Biennial Conference on Electromagnetic Field Com putation (CEFC), pages 01–02, 2022

    2022

  6. [14]

    Roppert, M

    K. Roppert, M. Kaltenbacher, L. Domenig, and L. Daniel. Magneto- elastic vector hysteresis modelling for electromagnetic d evices: A combi- nation of a multiscale model with the energy-based hysteres is framework, 03 2025

    2025

  7. [15]

    Upadhaya, P

    B. Upadhaya, P . Rasilo, L. Perkki¨ o, P . Handgruber, A. B elahcen, and A. Arkkio. Comparison of anisotropic energy-based and jile s–atherton models of ferromagnetic hysteresis. IEEE Transactions on Magnetics , 56(4):1–7, 2020

    2020

  8. [16]

    Egger and F

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

    2025 arXiv

  9. [17]

    Jacques, R

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

    2015

  10. [18]

    Domenig, K

    L. Domenig, K. Roppert, and M. Kaltenbacher. Incorpora tion of a 3-d energy-based vector hysteresis model into the finite elemen t method using a reduced scalar potential formulation. IEEE Transactions on Magnetics , PP:1–1, 06 2024

    2024

  11. [19]

    Egger, F

    H. Egger, F. Engertsberger, L. Domenig, K. Roppert, and M. Kaltenbacher. On nonlinear magnetic field solvers using l ocal Quasi-Newton updates. Comp. Math. Appl. , 183:20–31, 2025

    2025

  12. [20]

    K. Jacques. Energy-Based Magnetic Hysteresis Models - Theoretical Development and Finite Element F ormulations . PhD thesis, Universit´ e de Li` ege, 2018

    2018

  13. [21]

    Egger, F

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

    2025

  14. [22]

    Nocedal and S

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

    2006

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