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Optimal Control of Parameterized Maxwell's System: Reduced Basis, Convergence Analysis, and A Posteriori Error Estimates

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Pith's one-line read The paper constructs a reduced-basis surrogate for control-constrained optimal control of stationary Maxwell's equations with Gauss's law, proves uniform convergence of the reduced optimal controls to the finite-element ones as snapshot…

desk verdict First RB treatment of constrained Maxwell optimal control; the convergence part is plausible, but the advertised a posteriori estimator rests on a false application of discrete coercivity and needs a divergence residual term. read the letter →

arxiv 1908.08846 v2 pith:FYUQ2E4Y submitted 2019-08-23 math.OC cs.NAmath.NA

Harbir Antil , Tran Nhan Tam Quyen This is my paper · ORCID
classification math.OCcs.NAmath.NA MSC 35Q6135Q9365M6065M1265K1049M25
keywords Maxwell'ssystemParameterizedpartialdifferentialequationOptimalcontrolReducedbasismethodModelorderreductionConvergenceanalysisAposteriorierrorestimates
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 builds a reduced-basis surrogate for optimal control problems governed by a parameter-dependent stationary Maxwell system: a control drives the electric field through curl and divergence equations with Gauss's law, while dielectric, magnetic permeability, and charge density depend on parameters from a compact set. The authors prove that if the snapshot parameter set is dense enough and the affine coefficients are Hölder continuous, the reduced optimal controls converge uniformly to the finite-element optimal controls at a rate controlled by the snapshot density. They further prove that the combined error in control, state, and adjoint is bounded above and below by computable norms of the state and adjoint residuals, yielding an absolute a posteriori estimator and, under a mild condition, a relative one. A sympathetic reader would care because fast repeated queries of such optimization problems are otherwise expensive, and the paper supplies the missing convergence and certification theory for a reduced-basis approach in this setting.

What carries the argument

The load-bearing objects are the reduced basis spaces E_N and V_N built by a greedy algorithm from snapshots of the optimal state E*_h(µ_i), the adjoint F*_h(µ_i), and their potential parts, together with parameter separability expansions $σ^{{-1}}$ = Σ Θ_q^σ $σ_q^{{-1}}$, ǫ = Σ Θ_q^ǫ ǫ_q, and analogous expansions for the desired state and control. The argument runs on three mechanisms: the discrete Helmholtz decomposition and the coercivity inequality on the reduced spaces, which makes the constrained saddle-point system well posed; Hölder continuity of the coefficient functions Θ, which turns parameter distance into control distance; and the residual functionals R_E and R_F whose dual norms bound the state and adjoint gaps above and below through the Riesz representation theorem. The residual-based bounds are what make the greedy sampling error estimator computable without solving the high-dimensional problem.

What would settle it

Run the proposed greedy scheme on a family of Maxwell optimal control problems with Hölder-continuous affine coefficients, and compare ||u*_h(µ) - u*_N(µ)|| to κ_N^γ over a fine parameter grid; if the ratio grows without bound as N increases, or if one exhibits a problem where the reduced solution does not reproduce the finite-element solution at a snapshot, the central convergence claim would be refuted. For the a posteriori bound, compute the residual norms and the true error on random parameters; an unbounded ratio between true error and the estimator would contradict Theorem 6.3.

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Extended reading notes

Core claim

The central claim is that the reduced-basis solution of the parameterized Maxwell optimal control problem is a faithful surrogate in a uniform sense: for every parameter µ in the compact parameter set, the L² error between the reduced optimal control u*_N(µ) and the finite-element target u*_h(µ) is bounded by C κ_N^γ, where κ_N is the largest distance from a parameter to the snapshot sample and γ is half the smallest Hölder exponent of the affine coefficients. The proof adds and subtracts the solution at the nearest snapshot, using Hölder continuity to convert parameter proximity into control proximity and using the snapshot consistency assumption to cancel the error at sample parameters. For the estimator part, the paper defines feasible state and adjoint fields from the reduced control, forms the residuals of the reduced state and adjoint equations, and proves that the true errors are trapped between two positive multiples of the residual norms; this gives an absolute a posteriori estimator for the control and an analogous bound for the cost functional.

Load-bearing premise

The proof assumes that the reduced basis solution agrees exactly with the finite-element solution at every snapshot parameter, a consistency condition cited to another paper rather than proved here; if it fails, the uniform convergence bound no longer follows.

Editorial extensions

If this is right

  • If the snapshot parameters are dense, the reduced optimal controls converge uniformly in the parameter set at a rate fixed by the smoothness of the parameter-to-coefficient maps.
  • The state and adjoint residuals provide both upper and lower bounds for the control, state, and adjoint errors, so an online computation can certify how far the reduced solution is from the full finite-element solution.
  • The cost-functional error is also controlled by the same residual norms with explicit constants, enabling certified evaluation of the objective without solving the full problem.
  • A relative error estimator holds whenever the absolute estimator is no larger than half the norm of the reduced control, giving a practical stopping criterion for the greedy algorithm.
  • The theory covers parameter-dependent dielectric, permeability, charge density, desired state, and desired control, so a single reduced basis can be reused across many parameter queries.

Reading between the lines

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

  • Inference: the same dense-snapshot plus Hölder argument should transfer to other constrained saddle-point optimal control problems, such as Stokes or mixed elasticity, whenever a discrete coercivity condition and snapshot consistency hold.
  • Inference: the residual-based estimator could be used online as a genuinely a posteriori stopping criterion, but the paper does not include numerical tests; a natural experiment is to monitor the ratio of true error to estimator as the greedy algorithm enriches the basis.
  • Inference: if the snapshot consistency assumption fails, the convergence rate would need an extra interpolation term; testing with a discretization known not to reproduce reduced solutions exactly would isolate that dependence.
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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

4 major / 6 minor

Summary. The paper develops a reduced basis method for control-constrained optimal control problems governed by a parameterized stationary Maxwell system with Gauss's law. The state is discretized with Nédélec finite elements, the control is treated by variational discretization, and the reduced spaces are built from state, adjoint, and Helmholtz gradient snapshots. The main theoretical results are a uniform convergence theorem for the reduced optimal control as the snapshot set becomes dense (Theorem 5.5) and residual-based absolute and relative a posteriori error estimators for the reduced state, adjoint, control, and cost functional (Theorems 6.3, 6.4, and 6.5). The convergence result is conditional on a snapshot consistency assumption, and the a posteriori results are derived from estimates involving the state and adjoint residuals.

Significance. If the advertised results were fully established, the paper would be a useful contribution to certified reduced-order methods for Maxwell-type optimal control: it identifies a parameter-separability mechanism, gives an explicit convergence rate in terms of the snapshot density and Hölder exponents, and proposes residual-based estimators that avoid solving the high-dimensional problem. The main theorems are stated with detailed proofs, and the convergence rate is concrete. However, the central a posteriori error estimates rest on a lemma whose proof is invalid, and the convergence argument relies on an unproved consistency property. The scope of the claimed contribution is therefore currently not met, though the framework appears repairable by adding a divergence residual term and by supplying the missing consistency proof.

major comments (4)
  1. [Section 6, Lemma 6.2] The proof of the upper bound in Lemma 6.2 applies the discrete coercivity inequality (3.9) to v = E_N^*(μ) - Ê_h(μ). However, (3.9) is stated only for v_h in D_h^{(ε)}, the discrete ε-divergence-free subspace of E_h, and E_N^* satisfies the divergence condition (4.5b) only against test functions in V_N, not against the full V_h. Consequently E_N^* - Ê_h is not generally in D_h^{(ε)}, and the first inequality in Lemma 6.2 is not justified. The issue is load-bearing: since R_E(∇φ_h) = (ε u_N^*, ∇φ_h) = 0 for all φ_h ∈ V_h, the residual R_E is blind to gradient components. In the configuration u_N^* = 0, E_N^* = 0, with ρ orthogonal to V_N but not to V_h, the reduced equations (4.4) hold, Ê_h is a nonzero curl-free solution of (6.1), and R_E = 0; the asserted upper bound would then imply ‖Ê_h‖_{H(curl)} = 0, which is false. Theorems 6.3, 6.4, and 6.5 all inherit this gap. The estimator must include a discrete divergence residual term such as D(φ_h) = (ε E_N^*, ∇φ_h) + (ρ, φ_h) on the full space V_h, and Lemma 6.2 must be reproved with that term.
  2. [Section 5, Theorem 5.5] The uniform convergence theorem is conditional on the snapshot consistency assumption u_h^*(μ) = u_N^*(μ) for all μ ∈ P_N, stated immediately before Theorem 5.5 and justified only by a reference to [1, p. A283]. This property is nontrivial in the present setting because the reduced state and adjoint satisfy the divergence equations in (4.5) only against V_N, not against the full V_h, and the greedy spaces include gradient components precisely to recover the full discrete Gauss law. If this consistency fails, the dense-snapshot argument in the proof of Theorem 5.5 collapses. The authors should either prove the consistency property for their construction of E_N and V_N or state and verify the precise result from [1] that implies it.
  3. [Section 5, Lemma 5.4] Lemma 5.4 supplies the parameter-Hölder continuity of the discrete and reduced optimal controls that is used in the proof of Theorem 5.5, but its proof is omitted with the remark that it follows exactly as in the continuous case. Since this lemma is load-bearing for the convergence theorem, the manuscript should include the proof or a detailed indication of how the continuous arguments in Lemma 5.3 transfer to the discrete and reduced settings, including the role of the discrete coercivity assumption (3.9) on (E_N, V_N).
  4. [Section 4, reduced problem setup] The existence of a unique solution to the reduced problem (P_N) is conditioned on the assumption that the coercivity inequality (3.9) holds on (E_N, V_N). No criterion or verification procedure is given for this assumption, and the greedy construction in Algorithm 4.1 does not include an inf-sup or coercivity certification step. Because the reduced spaces contain gradients of snapshots, this condition is not automatic, and the convergence and estimator results depend on it.
minor comments (6)
  1. [Section 2 and 3, notation] The constants in Lemma 5.3 and Theorem 6.4 involve quantities such as ‖u‖_{R^3}, although u is a function; the authors should define this as a uniform bound on the pointwise box constraints or replace it with max(|u|, |u|).
  2. [Section 5, Lemma 5.3 proof] In the estimate for J4 + J5 the expression contains a typo: the term should be ‖E_e^*(μ2)‖_{L^2(D)}, not E_e^*(μ2)‖_{L^2(D)} without a norm.
  3. [Section 5, Lemma 5.4 statement] In the second displayed inequality of Lemma 5.4 the norm notation appears as ‖·‖^2_{L(Ω)}; it should be ‖·‖_{L^2(Ω)}.
  4. [Section 6, Theorem 6.5] The relative estimator (6.12) divides by ‖u_N^*(μ)‖_{L^2(Ω)}; the case ‖u_N^*(μ)‖ = 0 should be treated separately or excluded explicitly.
  5. [Section 3.3, Theorem 3.1] The proof of Theorem 3.1 is omitted with the comment that it is standard; since the theorem is used to justify the high-dimensional discretization, a short proof sketch or a precise reference for the stated curl and divergence error estimates would improve the paper.
  6. [Introduction] There are several typographical issues, including 'exasperated' for 'exacerbated' and a missing footnote marker in the cost functional definition; these should be corrected in revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence theorem is explicitly conditional on a snapshot-consistency assumption, and the a posteriori bounds are residual-based rather than fitted.

full rationale

The derivation chain has no step in which a purported prediction is built from its own target. Theorem 5.5 is explicitly conditional: the text states 'we assume u∗h(µ)=u∗N(µ) for µ belonging to the parameter sample PN,' and then invokes Hölder continuity of the affine coefficient functions and density of P_N. The proof is a triangle-inequality/dense-sample argument; the consistency assumption is a hypothesis, not a hidden form of the conclusion, and it is justified by an external citation ([1]), not by a self-citation. The a posteriori results (Theorems 6.3, 6.4, 6.5) define residuals R_E and R_F solely through the reduced solution (u*_N, E*_N, F*_N) and the data, then bound true errors by residual norms using the discrete coercivity and Lipschitz constants; no parameter is fitted to the true error, and the residual is not defined as the error. The only in-house reference, [3], is a pointer for EIM-based separability and is not load-bearing. The paper's open reliance on an unproved consistency property, the assumed discrete coercivity on (E_N,V_N), and the potential gap in Lemma 6.2 (coercivity applied to E*_N − Ê_h, which may not lie in D_h^(ε)) are correctness or support concerns, not circularity: they assert missing justification, not equivalence by construction.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claims rest on standard Maxwell space embeddings plus several assumptions specific to the reduced-basis construction: parameter separability, Hölder continuity of expansion coefficients, coercivity on the reduced spaces, and exact snapshot reproduction. The latter two are not proved in the paper. No new entities are introduced and no parameters are fitted to data.

assumptions (9)
  • domain assumption P is compact and the coefficients obey 0<ǫ≤ǫ(x;µ)≤ǫ, 0<σ≤σ(x;µ)≤σ, and ρ≤ρ(x;µ)≤ρ a.e. (2.2).
    Used for well-posedness, coercivity, and uniform constants throughout the paper.
  • domain assumption The control and desired state satisfy Gauss-law compatibility: ∇·(ǫu)=0 for u∈Uad and ∇·(ǫE_d)=ρ in D (2.3 and Section 2).
    Ensures the curl-curl system with the divergence constraint is consistent.
  • standard math The embedding V↪L^2 is compact and the curl-free subspace of V is {0}, giving the Poincaré-Friedrichs inequality (3.4).
    Invoked to obtain coercivity (3.5); cited to references [13,38].
  • domain assumption Coefficients and desired data admit finite affine parameter expansions, Q_σ, Q_ǫ, Q_ud, Q_Ed finite (Definition 5.1).
    Parameter separability is the prerequisite for the continuity estimates in Lemma 5.2 through Lemma 5.4.
  • domain assumption The separability coefficient functions Θ are Hölder continuous with exponents γ_σ, γ_ǫ, γ_ud, γ_Ed (Theorem 5.5).
    This converts parameter distance into control distance in the convergence proof.
  • ad hoc to paper The discrete coercivity condition (3.9) holds on the reduced spaces (E_N,V_N).
    Stated in Section 4 to guarantee existence and uniqueness of the reduced problem (P_N); it is not certified.
  • ad hoc to paper Snapshot consistency: u*_h(µ)=u*_N(µ) for all µ in the parameter sample P_N.
    Load-bearing for the proof of Theorem 5.5; justified only by citation to [1, pp. A283].
  • standard math The triangulation is quasi-uniform and the Nédélec spaces satisfy ∇V_h⊂E_h and the discrete Poincaré-Friedrichs inequality (3.7).
    Used for well-posedness of the discrete problem and finite element approximation.
  • domain assumption For the a priori estimate in Theorem 3.1(ii), ǫ and σ^{-1} lie in W^{1,∞} and E lies in H^s(curl).
    Only needed for the h^s convergence rate, not for the main reduced-basis convergence and estimator theorems.

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Pith. "Pith review of Optimal Control of Parameterized Maxwell's System: Reduced Basis, Convergence Analysis, and A Posteriori Error Estimates." pith.science (2026). https://pith.science/paper/FYUQ2E4Y

@misc{pith2026190808846,
  author       = {Pith},
  title        = {Pith review of: Optimal Control of Parameterized Maxwell's System: Reduced Basis, Convergence Analysis, and A Posteriori Error Estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYUQ2E4Y}},
  note         = {Machine review of arXiv:1908.08846}
}
read the original abstract

We consider control constrained optimal control problems governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter set is assumed to be compact. We discretize the electric field by a finite element method and use variational discretization concept to discretize the control. We create a reduced basis method for the optimal control problem and establish uniform convergence of the reduced order solutions to that of the original high dimensional problem provided that the snapshot parameter sample is dense in the parameter set, with an appropriate parameter separability rule. Finally, we establish the absolute a posteriori error estimator for the reduced order solutions and the corresponding cost functions in terms of the state and adjoint residuals.

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Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Ali and M

    A.A. Ali and M. Hinze , Reduced Basis Methods–An Application to Variational Discr etiza- tion of Parametrized Elliptic Optimal Control Problems , SIAM J. Sci. Comput., 42(1), A271A291. 2020

    work page 2020

  2. [2]

    Amrouche, C

    C. Amrouche, C. Bernardi, M. Dauge and V. Girault , Vector potentials in three- dimensional non-smooth domains , Math. Methods Appl. Sci. 21(1998), 823–864

    work page 1998

  3. [3]

    Antil, M

    H. Antil, M. Heinkenschloss, and D.C. Sorensen , Application of the discrete empirical in- terpolation method to reduced order modeling of nonlinear a nd parametric systems , MS&A. Model. Simul. Appl., Springer, Cham, 9(2014), 101–136

    work page 2014

  4. [4]

    Bader, M

    E. Bader, M. K ¨archer, M. A. Grepl, and K. Veroy , Certified reduced basis methods for parametrized distributed elliptic optimal control proble ms with control constraints , SIAM J. Sci. Comput., 38(2016), 3921–3946

    work page 2016

  5. [5]

    Barrault, Y

    M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera , An empirical interpolationmethod: ap- plication to efficient reduced-basis discretization of part ial differential equations , Comptes Rendus Mathematique, 339(2004), 667–672

    work page 2004

  6. [6]

    Benner and M

    P. Benner and M. Hess , Reduced basis approximations for Maxwells equations in dis persive media, in Model Reduction of Parametrized Systems, 107–119, Spri nger: New York, 2017

    work page 2017

  7. [7]

    Bommer and I

    V. Bommer and I. Yousept , Optimal control of the full time-dependent Maxwell equatio ns, ESAIM Math. Model. Numer. Anal., 50(2016), 237261

    work page 2016

  8. [8]

    Brenner and R

    S. Brenner and R. Scott , The Mathematical Theory of Finite Element Methods , New York: Springer, 2008

    work page 2008

Show all 38 references
  1. [9]

    Brezzi , On the existence, uniqueness and approximation of saddle-p oint problems arising from Lagrangian multipliers, Rev

    F. Brezzi , On the existence, uniqueness and approximation of saddle-p oint problems arising from Lagrangian multipliers, Rev. Fran¸ caise Automat. Informat. Recherche Op´ erationnelle S´ er. Rouge 8(1974), 129–151

    1974

  2. [10]

    Y. Chen, J. Hesthaven and Y. Maday , A Seamless Reduced Basis Element Method for 2D Maxwell’s Problem: An Introduction , In Spectral and High Order Methods for Partial Differential Equations, 141–152: Springer, 2011

    2011

  3. [11]

    Y. Chen, J. Hesthaven, Y. Maday and J. Rodr ´ıguez, Certified Reduced Basis Methods and Output Bounds for the Harmonic Maxwell’s Equations , SIAM J. Sci. Comput. 32(2010), 970–996

    2010

  4. [12]

    P. Jr. Ciarlet, H. Wu and J. Zou , Edge element methods for Maxwell’s equations with strong convergence for Gauss’ laws , SIAM J. Numer. Anal. 52(2014), 779–807

    2014

  5. [13]

    Costabel , A Remark on the Regularity of Solutions of Maxwell’s Equatio ns on Lipschitz Domains, Math

    M. Costabel , A Remark on the Regularity of Solutions of Maxwell’s Equatio ns on Lipschitz Domains, Math. Methods Appl. Sci. 12(1990), 365–368

    1990

  6. [14]

    Ded `e, Reduced basis method and a posteriori error estimation for p arametrized linear- quadratic optimal control problems , SIAM J

    L. Ded `e, Reduced basis method and a posteriori error estimation for p arametrized linear- quadratic optimal control problems , SIAM J. Sci. Comput. 32(2010) 997–1019. 19

    2010

  7. [15]

    J. L. Eftang, A. T. Patera and E. M. Ronquist , An “hp” certified reduced basis method for parametrized elliptic partial differential equations , SIAM J. Sci. Comput. 32(2010), 3170–3200

    2010

  8. [16]

    N. G. Gatica , A Simple Introduction to the Mixed Finite Element Method , Heidelberg New York Dordrecht London: Springer, 2014

    2014

  9. [17]

    B. Haasdonk , Reduced basis methods for parametrized pdes – a tutorial int roduction for sta- tionary and instationary problems , Model Reduction and Approximation: Theory and Algorithms, SIAM, 2017

    2017

  10. [18]

    Haasdonk and M

    B. Haasdonk and M. Ohlberger , Reduced basis method for finite volume approximations of parametrized linear evolution equations , ESAIM: M2AN 42(2008), 277–302

    2008

  11. [19]

    Hammerschmidt, S

    M. Hammerschmidt, S. Herrmann, J. Pomplun, L. Zschiedrich, S. Burger and F. Schmidt, Reduced basis method for Maxwells equations with resonance phenomena, in SPIE Optical Systems Design., SPIE, Philadelphia, USA, 201 5

  12. [20]

    Hess and P

    M. Hess and P. Benner , Fast evaluation of time-harmonic Maxwell’s equations usin g the reduced basis method, IEEE Transactions on Microwave Theory and Techniques 61(2 013), 2265–2274

  13. [21]

    M. Hess, S. Grundel and P. Benner , Estimating the inf-sup constant in reduced basis meth- ods for time-harmonic Maxwell’s equations , IEEE Transactions on Microwave Theory and Techniques 63(2015): 3549–3557

    2015

  14. [22]

    Hesthaven, G

    J.S. Hesthaven, G. Rozza, and B. Stamm , Certified reduced basis methods for parametrized partial differential equations , SpringerBriefs in Mathematics (2016): xiii+131

    2016

  15. [23]

    Hinze , A variational discretization concept in control constrain ed optimization: the linear- quadratic case, Comput

    M. Hinze , A variational discretization concept in control constrain ed optimization: the linear- quadratic case, Comput. Optim. Appl. 30(2005), 45–61

    2005

  16. [24]

    Hinze, R

    M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich , Optimization with PDE constraints , volume 23. Springer Science & Business Media, 2008

    2008

  17. [25]

    Hiptmair , Finite elements in computational electromagnetism , Acta Numer

    R. Hiptmair , Finite elements in computational electromagnetism , Acta Numer. 11(2002), 237– 339

    2002

  18. [26]

    Ito and S

    K. Ito and S. S. Ravindran , A reduced-order method for simulation and control of fluid flo ws, J. Comput. Phys. 143(1998), 403–425

    1998

  19. [27]

    Kangro and R

    U. Kangro and R. Nicolaides , Divergence boundary conditions for vector Helmholtz equa- tions with divergence constraints , ESAIM: M2AN 3(1999), 479–492

    1999

  20. [28]

    K ¨archer and M

    M. K ¨archer and M. A. Grepl , A certified reduced basis method for parametrized elliptic optimal control problems , ESAIM: COCV 20(2014), 416–441

    2014

  21. [29]

    K ¨archer, Z

    M. K ¨archer, Z. Tokoutsi, M. A. Grepl and K. Veroy , Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distr ibuted Controls, J. Sci. Comput. 75(2018), 276–307

    2018

  22. [30]

    Kolmbauer and U

    M. Kolmbauer and U. Langer , A robust preconditioned MinRes solver for distributed time - periodic eddy current optimal control problems , SIAM J. Sci. Comput., 34(2012), 785–809

    2012

  23. [31]

    Monk , Finite Element Methods for Maxwell’s Equations , New York: Oxford University Press, 2003

    P. Monk , Finite Element Methods for Maxwell’s Equations , New York: Oxford University Press, 2003

    2003

  24. [32]

    Negri, G

    F. Negri, G. Rozza, A. Manzoni and A. Quarteroni , Reduced basis method for parametrized elliptic optimal control problems , SIAM J. Sci. Comput. 35(2013), A2316–A2340

    2013

  25. [33]

    J. C. N ´ed´elec, Mixed finite elements in R3, Numer. Math. 35(1980), 315–341

    1980

  26. [34]

    Nicaise, S

    S. Nicaise, S. Stingelin, and F. Tr ¨oltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14(2014), 555573

    2014

  27. [35]

    Quarteroni, A

    A. Quarteroni, A. Manzoni, and F. Negri , Reduced basis methods for partial differential equations, Unitext, Springer, Cham, 92(2016), xi+296

    2016

  28. [36]

    T. Tonn, K. Urban and S. Volkwein , Comparison of the reduced-basis and pod a posteriori error estimators for an elliptic linear-quadratic optimal control problem, Math. Comput. Model. Dyn. 17(2011), 355–369

    2011

  29. [37]

    Tr ¨oltzsch, Optimal Control of Partial Differential Equations , Providence, RI: American Mathematical Society, 2010

    F. Tr ¨oltzsch, Optimal Control of Partial Differential Equations , Providence, RI: American Mathematical Society, 2010

    2010

  30. [38]

    Yousept and J

    I. Yousept and J. Zou , Edge element method for optimal control of stationary Maxwe ll system with Gauss law , SIAM J. Numer. Anal. 55(2017), 2787–2810. 20

    2017

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