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doi: 10.3934/mcrf.2022003
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Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates

1. 

Department of Radiology, School of Medicine, University of California - Davis, Sacramento, California 95817, USA

2. 

Center of Mathematics and Artificial Intelligence (CMAI) and, Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

3. 

Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

*Corresponding author: Quyen Tran

Received  June 2021 Revised  October 2021 Early access February 2022

Fund Project: The first author is partially supported by School of Medicine, University of California - Davis, USA and Institute for Numerical and Applied Mathematics, University of Goettingen, Germany.
The second author is partially supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-19-1-0036 and NSF grants DMS-2110263, DMS-1913004, DMS-2111315.

We consider an optimal control problem 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 for the control. We present a reduced basis method for the optimal control problem and establish the uniform convergence of the reduced order solutions to that of the original full-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.

Citation: Quyen Tran, Harbir Antil, Hugo Díaz. Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022003
References:
[1]

A. A. Ali and M. Hinze, Reduced basis methods–an application to variational discretization of parametrized elliptic optimal control problems, SIAM J. Sci. Comput., 42 (2020), A271–A291. doi: 10.1137/18M1227147.

[2]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.

[3]

H. AntilM. Heinkenschloss and D. C. Sorensen, Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems, Reduced Order Methods for Modeling and Computational Reduction, 9 (2014), 101-136.  doi: 10.1007/978-3-319-02090-7_4.

[4]

E. BaderM. KärcherM. A. Grepl and K. Veroy, Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints, SIAM J. Sci. Comput., 38 (2016), 3921-3946.  doi: 10.1137/16M1059898.

[5]

M. BarraultY. MadayN. C. Nguyen and A. T. Patera, An 'ampirical interpolation' method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339 (2004), 667-672.  doi: 10.1016/j.crma.2004.08.006.

[6]

P. Benner and M. Hess, Reduced basis approximations for Maxwell's equations in dispersive media, In Model Reduction of Parametrized Systems, Springer-Verlag, 17 (2017), 107–119.

[7]

V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261.  doi: 10.1051/m2an/2015041.

[8]

S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[9]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129-151. 

[10]

G. Caselli, Optimal control of an eddy current problem with a dipole source, J. Math. Anal. Appl., 489 (2020), 124152, 20 pp. doi: 10.1016/j.jmaa.2020.124152.

[11]

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, Springer-Verlag, 76 (2011), 141–152. doi: 10.1007/978-3-642-15337-2_11.

[12]

Y. ChenJ. HesthavenY. Maday and J. Rodríguez, Certified reduced basis methods and output bounds for the harmonic Maxwell's equations, SIAM J. Sci. Comput., 32 (2010), 970-996.  doi: 10.1137/09075250X.

[13]

P. CiarletH. 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.  doi: 10.1137/120899856.

[14]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[15]

L. Dedè, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems, SIAM J. Sci. Comput., 32 (2010), 997-1019.  doi: 10.1137/090760453.

[16]

J. L. EftangA. 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.  doi: 10.1137/090780122.

[17]

N. G. Gatica, A Simple Introduction to the Mixed Finite Element Method, Theory and applications. SpringerBriefs in Mathematics. Springer, Cham, 2014. doi: 10.1007/978-3-319-03695-3.

[18]

B. Haasdonk, Reduced basis methods for parametrized pdes – a tutorial introduction for stationary and instationary problems, Model Reduction and Approximation, 15 (2017), 65-136.  doi: 10.1137/1.9781611974829.ch2.

[19]

B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM: M2AN, 42 (2008), 277-302.  doi: 10.1051/m2an:2008001.

[20]

M. Hammerschmidt, S. Herrmann, J. Pomplun, L. Zschiedrich, S. Burger and F. Schmidt, Reduced basis method for Maxwell's equations with resonance phenomena, In SPIE Optical Systems Design., SPIE, Philadelphia, USA, 2015.

[21]

M. Hess and P. Benner, Fast evaluation of time-harmonic Maxwell's equations using the reduced basis method, IEEE Transactions on Microwave Theory and Techniques, 61 (2013), 2265-2274.  doi: 10.1109/TMTT.2013.2258167.

[22]

M. HessS. Grundel and P. Benner, Estimating the inf-sup constant in reduced basis methods for time-harmonic Maxwell's equations, IEEE Transactions on Microwave Theory and Techniques, 63 (2015), 3549-3557.  doi: 10.1109/TMTT.2015.2473157.

[23]

J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs in Mathematics, 2016. doi: 10.1007/978-3-319-22470-1.

[24]

M. Hinze, A variational discretization concept in control constrained optimization: The linear- quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[25]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23. Springer, New York, 2009.

[26]

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), 237-339.  doi: 10.1017/S0962492902000041.

[27]

K. Ito and S. S. Ravindran, A reduced-order method for simulation and control of fluid flows, J. Comput. Phys., 143 (1998), 403-425.  doi: 10.1006/jcph.1998.5943.

[28]

U. Kangro and R. Nicolaides, Divergence boundary conditions for vector Helmholtz equations with divergence constraints, ESAIM: M2AN, 3 (1999), 479-492.  doi: 10.1051/m2an:1999148.

[29]

M. Kärcher and M. A. Grepl, A certified reduced basis method for parametrized elliptic optimal control problems, ESAIM: COCV, 20 (2014), 416-441.  doi: 10.1051/cocv/2013069.

[30]

M. KärcherZ. TokoutsiM. A. Grepl and K. Veroy, Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls, J. Sci. Comput., 75 (2018), 276-307.  doi: 10.1007/s10915-017-0539-z.

[31]

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.  doi: 10.1137/110842533.

[32] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[33]

J. C. Nédélec, Mixed finite elements in $\mathbb{R}^3$, Numer. Math., 35 (1980), 315-341.  doi: 10.1007/BF01396415.

[34]

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. doi: 10.1137/120894737.

[35]

S. NicaiseS. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), 555-573.  doi: 10.1515/cmam-2014-0022.

[36]

S. Nicaise and F. Tröltzsch, Optimal control of some quasilinear Maxwell equations of parabolic type, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1375-1391.  doi: 10.3934/dcdss.2017073.

[37]

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, An introduction. Unitext, 92. La Matematica per il 3+2. Springer, Cham, 2016. doi: 10.1007/978-3-319-15431-2.

[38]

T. TonnK. 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.  doi: 10.1080/13873954.2011.547678.

[39]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Providence, American Mathematical Society, RI, 2010. doi: 10.1090/gsm/112.

[40]

F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors, Optimization, 65 (2016), 1651-1673.  doi: 10.1080/02331934.2016.1179301.

[41]

W. WeiH. M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Anal., 75 (2012), 2024-2036.  doi: 10.1016/j.na.2011.10.003.

[42]

I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581.  doi: 10.1007/s10589-011-9422-2.

[43]

I. Yousept, Optimal control of non-smooth hyperbolic evolution Maxwell equations in type-II superconductivity, SIAM J. Control Optim., 55 (2017), 2305-2332.  doi: 10.1137/16M1074229.

[44]

I. Yousept and J. Zou, Edge element method for optimal control of stationary Maxwell system with Gauss law, SIAM J. Numer. Anal., 55 (2017), 2787-2810.  doi: 10.1137/17M1117021.

show all references

References:
[1]

A. A. Ali and M. Hinze, Reduced basis methods–an application to variational discretization of parametrized elliptic optimal control problems, SIAM J. Sci. Comput., 42 (2020), A271–A291. doi: 10.1137/18M1227147.

[2]

C. AmroucheC. BernardiM. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.  doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.

[3]

H. AntilM. Heinkenschloss and D. C. Sorensen, Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems, Reduced Order Methods for Modeling and Computational Reduction, 9 (2014), 101-136.  doi: 10.1007/978-3-319-02090-7_4.

[4]

E. BaderM. KärcherM. A. Grepl and K. Veroy, Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints, SIAM J. Sci. Comput., 38 (2016), 3921-3946.  doi: 10.1137/16M1059898.

[5]

M. BarraultY. MadayN. C. Nguyen and A. T. Patera, An 'ampirical interpolation' method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339 (2004), 667-672.  doi: 10.1016/j.crma.2004.08.006.

[6]

P. Benner and M. Hess, Reduced basis approximations for Maxwell's equations in dispersive media, In Model Reduction of Parametrized Systems, Springer-Verlag, 17 (2017), 107–119.

[7]

V. Bommer and I. Yousept, Optimal control of the full time-dependent Maxwell equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 237-261.  doi: 10.1051/m2an/2015041.

[8]

S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3$^{rd}$ edition, Springer-Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.

[9]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129-151. 

[10]

G. Caselli, Optimal control of an eddy current problem with a dipole source, J. Math. Anal. Appl., 489 (2020), 124152, 20 pp. doi: 10.1016/j.jmaa.2020.124152.

[11]

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, Springer-Verlag, 76 (2011), 141–152. doi: 10.1007/978-3-642-15337-2_11.

[12]

Y. ChenJ. HesthavenY. Maday and J. Rodríguez, Certified reduced basis methods and output bounds for the harmonic Maxwell's equations, SIAM J. Sci. Comput., 32 (2010), 970-996.  doi: 10.1137/09075250X.

[13]

P. CiarletH. 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.  doi: 10.1137/120899856.

[14]

M. Costabel, A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains, Math. Methods Appl. Sci., 12 (1990), 365-368.  doi: 10.1002/mma.1670120406.

[15]

L. Dedè, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems, SIAM J. Sci. Comput., 32 (2010), 997-1019.  doi: 10.1137/090760453.

[16]

J. L. EftangA. 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.  doi: 10.1137/090780122.

[17]

N. G. Gatica, A Simple Introduction to the Mixed Finite Element Method, Theory and applications. SpringerBriefs in Mathematics. Springer, Cham, 2014. doi: 10.1007/978-3-319-03695-3.

[18]

B. Haasdonk, Reduced basis methods for parametrized pdes – a tutorial introduction for stationary and instationary problems, Model Reduction and Approximation, 15 (2017), 65-136.  doi: 10.1137/1.9781611974829.ch2.

[19]

B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM: M2AN, 42 (2008), 277-302.  doi: 10.1051/m2an:2008001.

[20]

M. Hammerschmidt, S. Herrmann, J. Pomplun, L. Zschiedrich, S. Burger and F. Schmidt, Reduced basis method for Maxwell's equations with resonance phenomena, In SPIE Optical Systems Design., SPIE, Philadelphia, USA, 2015.

[21]

M. Hess and P. Benner, Fast evaluation of time-harmonic Maxwell's equations using the reduced basis method, IEEE Transactions on Microwave Theory and Techniques, 61 (2013), 2265-2274.  doi: 10.1109/TMTT.2013.2258167.

[22]

M. HessS. Grundel and P. Benner, Estimating the inf-sup constant in reduced basis methods for time-harmonic Maxwell's equations, IEEE Transactions on Microwave Theory and Techniques, 63 (2015), 3549-3557.  doi: 10.1109/TMTT.2015.2473157.

[23]

J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs in Mathematics, 2016. doi: 10.1007/978-3-319-22470-1.

[24]

M. Hinze, A variational discretization concept in control constrained optimization: The linear- quadratic case, Comput. Optim. Appl., 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.

[25]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23. Springer, New York, 2009.

[26]

R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer., 11 (2002), 237-339.  doi: 10.1017/S0962492902000041.

[27]

K. Ito and S. S. Ravindran, A reduced-order method for simulation and control of fluid flows, J. Comput. Phys., 143 (1998), 403-425.  doi: 10.1006/jcph.1998.5943.

[28]

U. Kangro and R. Nicolaides, Divergence boundary conditions for vector Helmholtz equations with divergence constraints, ESAIM: M2AN, 3 (1999), 479-492.  doi: 10.1051/m2an:1999148.

[29]

M. Kärcher and M. A. Grepl, A certified reduced basis method for parametrized elliptic optimal control problems, ESAIM: COCV, 20 (2014), 416-441.  doi: 10.1051/cocv/2013069.

[30]

M. KärcherZ. TokoutsiM. A. Grepl and K. Veroy, Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls, J. Sci. Comput., 75 (2018), 276-307.  doi: 10.1007/s10915-017-0539-z.

[31]

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.  doi: 10.1137/110842533.

[32] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.
[33]

J. C. Nédélec, Mixed finite elements in $\mathbb{R}^3$, Numer. Math., 35 (1980), 315-341.  doi: 10.1007/BF01396415.

[34]

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. doi: 10.1137/120894737.

[35]

S. NicaiseS. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields, Comput. Methods Appl. Math., 14 (2014), 555-573.  doi: 10.1515/cmam-2014-0022.

[36]

S. Nicaise and F. Tröltzsch, Optimal control of some quasilinear Maxwell equations of parabolic type, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1375-1391.  doi: 10.3934/dcdss.2017073.

[37]

A. Quarteroni, A. Manzoni and F. Negri, Reduced Basis Methods for Partial Differential Equations, An introduction. Unitext, 92. La Matematica per il 3+2. Springer, Cham, 2016. doi: 10.1007/978-3-319-15431-2.

[38]

T. TonnK. 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.  doi: 10.1080/13873954.2011.547678.

[39]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Providence, American Mathematical Society, RI, 2010. doi: 10.1090/gsm/112.

[40]

F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors, Optimization, 65 (2016), 1651-1673.  doi: 10.1080/02331934.2016.1179301.

[41]

W. WeiH. M. Yin and J. Tang, An optimal control problem for microwave heating, Nonlinear Anal., 75 (2012), 2024-2036.  doi: 10.1016/j.na.2011.10.003.

[42]

I. Yousept, Optimal control of Maxwell's equations with regularized state constraints, Comput. Optim. Appl., 52 (2012), 559-581.  doi: 10.1007/s10589-011-9422-2.

[43]

I. Yousept, Optimal control of non-smooth hyperbolic evolution Maxwell equations in type-II superconductivity, SIAM J. Control Optim., 55 (2017), 2305-2332.  doi: 10.1137/16M1074229.

[44]

I. Yousept and J. Zou, Edge element method for optimal control of stationary Maxwell system with Gauss law, SIAM J. Numer. Anal., 55 (2017), 2787-2810.  doi: 10.1137/17M1117021.

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