Abstract

We show that iterative solution of Maxwell’s equations using the finite-difference frequency-domain method can be significantly accelerated by using a Schur complement domain decomposition method. We account for the improvement by analyzing the spectral properties of the linear systems resulting from the use of the domain decomposition method.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Accelerating simulation of ensembles of locally differing optical structures via a Schur complement domain decomposition

Sacha Verweij, Victor Liu, and Shanhui Fan
Opt. Lett. 39(22) 6458-6461 (2014)

References

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  1. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nature Photonics 5(9), 523–530 (2011).
    [Crossref]
  2. N. Meinzer, W.L. Barnes, and I.R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nature Photonics 8(8), 889–898 (2014).
    [Crossref]
  3. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
    [Crossref] [PubMed]
  4. N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
    [Crossref]
  5. W. Cai and V. Shalaev, Optical Metamaterials(Springer, 2010).
    [Crossref]
  6. S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
    [Crossref]
  7. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials 13, 139–150 (2014).
    [Crossref] [PubMed]
  8. B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
    [Crossref] [PubMed]
  9. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: putting a new twist on light(Cambridge University Press, 2008)
  10. A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
    [Crossref] [PubMed]
  11. S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998).
    [Crossref]
  12. S. Verweij, V. Liu, and S. Fan, “Accelerating simulation of ensembles of locally differing optical structures via a Schur complement domain decomposition,” Opt. Lett. 39(22), 6458–6461 (2014).
    [Crossref] [PubMed]
  13. Y. Li and J.M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3D large-scale electromagnetic problems,” IEEE Transactions on Antennas and Propagation 54(10), 704–723 (2006).
    [Crossref]
  14. Peng Zhen, Lim Kheng-Hwee, and Lee Jin-Fa, “Non-conformal Domain Decomposition Methods for Solving Large Multi-scale Electromagnetic Scattering Problems,” Proceedings of the IEEE,  101(2), 298–319 (2013).
    [Crossref]
  15. Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
    [Crossref]
  16. K. Zhang and J.M. Jin, “Parallel FETI-DP algorithm for efficient simulation of large-scale electrmagnetic problems,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29(5), 897–914 (2016).
    [Crossref]
  17. P. Jaysaval, D. Datta, M. Sen, and A. Arnulf, “A Schur complement based fast 2D finite-difference multimodel modeling of acoustic wavefield in the frequency domain,” in).SEG Technical Program Expanded Abstracts, (Society of Exploration Geophysicists, 2017).
  18. J. Maryska, M. Rozloznik, and M. Tuma, “Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem,” SIAM Journal on Scientific Computing 22(2), 704–723 (2000).
    [Crossref]
  19. B. Alavikia, N. Soltani, and O. M. Ramahi, “Efficient 2-D Finite-Difference Frequency-Domain Method for Switching Noise Analysis in Multilayer Boards,” IEEE Transactions on Components, Packaging and Manufacturing Technology 3(5), 841–848 (2013).
    [Crossref]
  20. O. G. Ernst and M.J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I.G. Graham, T.Y. Hou, O. Lakkis, and R. Scheichl, eds. (SpringerBerlin Heidelberg, 2012).
    [Crossref]
  21. W. Shin, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” Journal of Computational Physics 231(8), 3406–3431 (2012).
    [Crossref]
  22. K. S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC-Press, 1993. Section 3.2
  23. V. Simoncini and D.B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numerical Linear Algebra with Applications 14(1), 1–59 (2007).
    [Crossref]
  24. W. Shin and S. Fan. “Simulation of phenomena characterized by partial differential equations,” US Patent Application13/744, 999
  25. J. Liesen and P. Tichy, “Convergence analysis of Krylov subspace methods,” GAMM-Mitteilungen 27(2), 153–173 (2004).
    [Crossref]
  26. W. Shin and S. Fan, “Accelerated solution of the frequency-domain Maxwell’s equations by engineering the eigenvalue distribution of the operator,” Opt. Express 21(19), 22578–22595 (2013).
    [Crossref] [PubMed]
  27. J. Mandel, “On block diagonal and Schur complement preconditioning,” Numerische Mathematik 58(1), 79–93 (1990).
    [Crossref]
  28. A. R. Horn and F. Zhang, The Schur complement and its applications(Springer, 2005), Chap. 1.
  29. A. Pyzara, B. Bylina, and J. Bylina, “The influence of a matrix condition number on iterative methods convergence,” in Federated Conference on Computer Science and Information Systems(FedCSIS, 2011), pp. 459–464.
  30. M. Arioli and F. Romani, “Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices,” Numerische Mathematik 46(1), 31–42 (1985).
    [Crossref]
  31. M. Neytcheva, “On element-by-element Schur complement approximations,” Linear Algebra and its Applications 434(11), 2308–2324 (2011).
    [Crossref]
  32. M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
    [Crossref]
  33. L. Kulas and M. Mrozowski, “Low-reflection subgridding,” IEEE Transactions on Microwave Theory and Techniques 53(5), 1587–1592 (2005).
    [Crossref]
  34. T. Wu and Z. Chen, “A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics 267, 82–95 (2014).
    [Crossref]
  35. L. Giraud, A. Haidar, and Y. Saad, “Sparse approximations of the Schur complement for parallel algebraic hybrid linear solvers in 3D,” [Research Report] RR-7237, INRIA. 2010, pp.18.
  36. Y. Saad and B. Suchomel, “ARMS: an algebraic recursive multilevel solver for general sparse linear systems,” Numerical Linear Algebra with Applications 9(5), 359–378 (2002).
    [Crossref]

2017 (1)

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

2016 (2)

Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
[Crossref]

K. Zhang and J.M. Jin, “Parallel FETI-DP algorithm for efficient simulation of large-scale electrmagnetic problems,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29(5), 897–914 (2016).
[Crossref]

2015 (2)

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
[Crossref]

2014 (4)

N. Meinzer, W.L. Barnes, and I.R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nature Photonics 8(8), 889–898 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials 13, 139–150 (2014).
[Crossref] [PubMed]

T. Wu and Z. Chen, “A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics 267, 82–95 (2014).
[Crossref]

S. Verweij, V. Liu, and S. Fan, “Accelerating simulation of ensembles of locally differing optical structures via a Schur complement domain decomposition,” Opt. Lett. 39(22), 6458–6461 (2014).
[Crossref] [PubMed]

2013 (3)

Peng Zhen, Lim Kheng-Hwee, and Lee Jin-Fa, “Non-conformal Domain Decomposition Methods for Solving Large Multi-scale Electromagnetic Scattering Problems,” Proceedings of the IEEE,  101(2), 298–319 (2013).
[Crossref]

W. Shin and S. Fan, “Accelerated solution of the frequency-domain Maxwell’s equations by engineering the eigenvalue distribution of the operator,” Opt. Express 21(19), 22578–22595 (2013).
[Crossref] [PubMed]

B. Alavikia, N. Soltani, and O. M. Ramahi, “Efficient 2-D Finite-Difference Frequency-Domain Method for Switching Noise Analysis in Multilayer Boards,” IEEE Transactions on Components, Packaging and Manufacturing Technology 3(5), 841–848 (2013).
[Crossref]

2012 (1)

W. Shin, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” Journal of Computational Physics 231(8), 3406–3431 (2012).
[Crossref]

2011 (2)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nature Photonics 5(9), 523–530 (2011).
[Crossref]

M. Neytcheva, “On element-by-element Schur complement approximations,” Linear Algebra and its Applications 434(11), 2308–2324 (2011).
[Crossref]

2008 (1)

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
[Crossref] [PubMed]

2007 (1)

V. Simoncini and D.B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numerical Linear Algebra with Applications 14(1), 1–59 (2007).
[Crossref]

2006 (2)

Y. Li and J.M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3D large-scale electromagnetic problems,” IEEE Transactions on Antennas and Propagation 54(10), 704–723 (2006).
[Crossref]

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

2005 (1)

L. Kulas and M. Mrozowski, “Low-reflection subgridding,” IEEE Transactions on Microwave Theory and Techniques 53(5), 1587–1592 (2005).
[Crossref]

2004 (1)

J. Liesen and P. Tichy, “Convergence analysis of Krylov subspace methods,” GAMM-Mitteilungen 27(2), 153–173 (2004).
[Crossref]

2002 (1)

Y. Saad and B. Suchomel, “ARMS: an algebraic recursive multilevel solver for general sparse linear systems,” Numerical Linear Algebra with Applications 9(5), 359–378 (2002).
[Crossref]

2000 (1)

J. Maryska, M. Rozloznik, and M. Tuma, “Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem,” SIAM Journal on Scientific Computing 22(2), 704–723 (2000).
[Crossref]

1998 (1)

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998).
[Crossref]

1996 (1)

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

1990 (1)

J. Mandel, “On block diagonal and Schur complement preconditioning,” Numerische Mathematik 58(1), 79–93 (1990).
[Crossref]

1985 (1)

M. Arioli and F. Romani, “Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices,” Numerische Mathematik 46(1), 31–42 (1985).
[Crossref]

Aieta, F.

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

Alavikia, B.

B. Alavikia, N. Soltani, and O. M. Ramahi, “Efficient 2-D Finite-Difference Frequency-Domain Method for Switching Noise Analysis in Multilayer Boards,” IEEE Transactions on Components, Packaging and Manufacturing Technology 3(5), 841–848 (2013).
[Crossref]

Arioli, M.

M. Arioli and F. Romani, “Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices,” Numerische Mathematik 46(1), 31–42 (1985).
[Crossref]

Arnulf, A.

P. Jaysaval, D. Datta, M. Sen, and A. Arnulf, “A Schur complement based fast 2D finite-difference multimodel modeling of acoustic wavefield in the frequency domain,” in).SEG Technical Program Expanded Abstracts, (Society of Exploration Geophysicists, 2017).

Barnes, W.L.

N. Meinzer, W.L. Barnes, and I.R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nature Photonics 8(8), 889–898 (2014).
[Crossref]

Bohn, B. J.

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

Brian, MacKie-Mason

Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
[Crossref]

Bylina, B.

A. Pyzara, B. Bylina, and J. Bylina, “The influence of a matrix condition number on iterative methods convergence,” in Federated Conference on Computer Science and Information Systems(FedCSIS, 2011), pp. 459–464.

Bylina, J.

A. Pyzara, B. Bylina, and J. Bylina, “The influence of a matrix condition number on iterative methods convergence,” in Federated Conference on Computer Science and Information Systems(FedCSIS, 2011), pp. 459–464.

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials(Springer, 2010).
[Crossref]

Capasso, F.

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials 13, 139–150 (2014).
[Crossref] [PubMed]

Chen, J.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Chen, J.C.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Chen, Z.

T. Wu and Z. Chen, “A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics 267, 82–95 (2014).
[Crossref]

Chu, C.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Dalcin, L.

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

Datta, D.

P. Jaysaval, D. Datta, M. Sen, and A. Arnulf, “A Schur complement based fast 2D finite-difference multimodel modeling of acoustic wavefield in the frequency domain,” in).SEG Technical Program Expanded Abstracts, (Society of Exploration Geophysicists, 2017).

Ellenbogen, T.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
[Crossref]

Ernst, O. G.

O. G. Ernst and M.J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I.G. Graham, T.Y. Hou, O. Lakkis, and R. Scheichl, eds. (SpringerBerlin Heidelberg, 2012).
[Crossref]

Fan, S.

S. Verweij, V. Liu, and S. Fan, “Accelerating simulation of ensembles of locally differing optical structures via a Schur complement domain decomposition,” Opt. Lett. 39(22), 6458–6461 (2014).
[Crossref] [PubMed]

W. Shin and S. Fan, “Accelerated solution of the frequency-domain Maxwell’s equations by engineering the eigenvalue distribution of the operator,” Opt. Express 21(19), 22578–22595 (2013).
[Crossref] [PubMed]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
[Crossref] [PubMed]

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998).
[Crossref]

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

W. Shin and S. Fan. “Simulation of phenomena characterized by partial differential equations,” US Patent Application13/744, 999

Gander, M.J.

O. G. Ernst and M.J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I.G. Graham, T.Y. Hou, O. Lakkis, and R. Scheichl, eds. (SpringerBerlin Heidelberg, 2012).
[Crossref]

Giraud, L.

L. Giraud, A. Haidar, and Y. Saad, “Sparse approximations of the Schur complement for parallel algebraic hybrid linear solvers in 3D,” [Research Report] RR-7237, INRIA. 2010, pp.18.

Haidar, A.

L. Giraud, A. Haidar, and Y. Saad, “Sparse approximations of the Schur complement for parallel algebraic hybrid linear solvers in 3D,” [Research Report] RR-7237, INRIA. 2010, pp.18.

Hendler, N.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
[Crossref]

Hillenbrand, R.

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

Hooper, I.R.

N. Meinzer, W.L. Barnes, and I.R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nature Photonics 8(8), 889–898 (2014).
[Crossref]

Horn, A. R.

A. R. Horn and F. Zhang, The Schur complement and its applications(Springer, 2005), Chap. 1.

Jaysaval, P.

P. Jaysaval, D. Datta, M. Sen, and A. Arnulf, “A Schur complement based fast 2D finite-difference multimodel modeling of acoustic wavefield in the frequency domain,” in).SEG Technical Program Expanded Abstracts, (Society of Exploration Geophysicists, 2017).

Jin, J.M.

K. Zhang and J.M. Jin, “Parallel FETI-DP algorithm for efficient simulation of large-scale electrmagnetic problems,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29(5), 897–914 (2016).
[Crossref]

Y. Li and J.M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3D large-scale electromagnetic problems,” IEEE Transactions on Antennas and Propagation 54(10), 704–723 (2006).
[Crossref]

Jin-Fa, Lee

Peng Zhen, Lim Kheng-Hwee, and Lee Jin-Fa, “Non-conformal Domain Decomposition Methods for Solving Large Multi-scale Electromagnetic Scattering Problems,” Proceedings of the IEEE,  101(2), 298–319 (2013).
[Crossref]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: putting a new twist on light(Cambridge University Press, 2008)

Joannopoulos, J.D.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998).
[Crossref]

Johannopoulos, J.D.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: putting a new twist on light(Cambridge University Press, 2008)

Kats, M. A.

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

Keren-Zur, S.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
[Crossref]

Kheng-Hwee, Lim

Peng Zhen, Lim Kheng-Hwee, and Lee Jin-Fa, “Non-conformal Domain Decomposition Methods for Solving Large Multi-scale Electromagnetic Scattering Problems,” Proceedings of the IEEE,  101(2), 298–319 (2013).
[Crossref]

Kuan, C.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Kulas, L.

L. Kulas and M. Mrozowski, “Low-reflection subgridding,” IEEE Transactions on Microwave Theory and Techniques 53(5), 1587–1592 (2005).
[Crossref]

Kunz, K. S.

K. S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC-Press, 1993. Section 3.2

Kurland, I.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Lai, Y.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Li, T.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Li, Y.

Y. Li and J.M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3D large-scale electromagnetic problems,” IEEE Transactions on Antennas and Propagation 54(10), 704–723 (2006).
[Crossref]

Liesen, J.

J. Liesen and P. Tichy, “Convergence analysis of Krylov subspace methods,” GAMM-Mitteilungen 27(2), 153–173 (2004).
[Crossref]

Liu, V.

Lu, S.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Luebbers, R.J.

K. S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC-Press, 1993. Section 3.2

Mandel, J.

J. Mandel, “On block diagonal and Schur complement preconditioning,” Numerische Mathematik 58(1), 79–93 (1990).
[Crossref]

Maryska, J.

J. Maryska, M. Rozloznik, and M. Tuma, “Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem,” SIAM Journal on Scientific Computing 22(2), 704–723 (2000).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: putting a new twist on light(Cambridge University Press, 2008)

Meinzer, N.

N. Meinzer, W.L. Barnes, and I.R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nature Photonics 8(8), 889–898 (2014).
[Crossref]

Mekis, A.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Mrozowski, M.

L. Kulas and M. Mrozowski, “Low-reflection subgridding,” IEEE Transactions on Microwave Theory and Techniques 53(5), 1587–1592 (2005).
[Crossref]

Neytcheva, M.

M. Neytcheva, “On element-by-element Schur complement approximations,” Linear Algebra and its Applications 434(11), 2308–2324 (2011).
[Crossref]

Nigro, N.

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

Paz, R.

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

Pyzara, A.

A. Pyzara, B. Bylina, and J. Bylina, “The influence of a matrix condition number on iterative methods convergence,” in Federated Conference on Computer Science and Information Systems(FedCSIS, 2011), pp. 459–464.

Ralf, Hiptmair

Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
[Crossref]

Ramahi, O. M.

B. Alavikia, N. Soltani, and O. M. Ramahi, “Efficient 2-D Finite-Difference Frequency-Domain Method for Switching Noise Analysis in Multilayer Boards,” IEEE Transactions on Components, Packaging and Manufacturing Technology 3(5), 841–848 (2013).
[Crossref]

Romani, F.

M. Arioli and F. Romani, “Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices,” Numerische Mathematik 46(1), 31–42 (1985).
[Crossref]

Rozloznik, M.

J. Maryska, M. Rozloznik, and M. Tuma, “Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem,” SIAM Journal on Scientific Computing 22(2), 704–723 (2000).
[Crossref]

Saad, Y.

Y. Saad and B. Suchomel, “ARMS: an algebraic recursive multilevel solver for general sparse linear systems,” Numerical Linear Algebra with Applications 9(5), 359–378 (2002).
[Crossref]

L. Giraud, A. Haidar, and Y. Saad, “Sparse approximations of the Schur complement for parallel algebraic hybrid linear solvers in 3D,” [Research Report] RR-7237, INRIA. 2010, pp.18.

Schnell, M.

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

Segal, N.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
[Crossref]

Sen, M.

P. Jaysaval, D. Datta, M. Sen, and A. Arnulf, “A Schur complement based fast 2D finite-difference multimodel modeling of acoustic wavefield in the frequency domain,” in).SEG Technical Program Expanded Abstracts, (Society of Exploration Geophysicists, 2017).

Shalaev, V.

W. Cai and V. Shalaev, Optical Metamaterials(Springer, 2010).
[Crossref]

Shin, W.

W. Shin and S. Fan, “Accelerated solution of the frequency-domain Maxwell’s equations by engineering the eigenvalue distribution of the operator,” Opt. Express 21(19), 22578–22595 (2013).
[Crossref] [PubMed]

W. Shin, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” Journal of Computational Physics 231(8), 3406–3431 (2012).
[Crossref]

W. Shin and S. Fan. “Simulation of phenomena characterized by partial differential equations,” US Patent Application13/744, 999

Simoncini, V.

V. Simoncini and D.B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numerical Linear Algebra with Applications 14(1), 1–59 (2007).
[Crossref]

Soltani, N.

B. Alavikia, N. Soltani, and O. M. Ramahi, “Efficient 2-D Finite-Difference Frequency-Domain Method for Switching Noise Analysis in Multilayer Boards,” IEEE Transactions on Components, Packaging and Manufacturing Technology 3(5), 841–848 (2013).
[Crossref]

Sonzogni, V.

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nature Photonics 5(9), 523–530 (2011).
[Crossref]

Storti, M.

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

Su, V.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Suchomel, B.

Y. Saad and B. Suchomel, “ARMS: an algebraic recursive multilevel solver for general sparse linear systems,” Numerical Linear Algebra with Applications 9(5), 359–378 (2002).
[Crossref]

Szyld, D.B.

V. Simoncini and D.B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numerical Linear Algebra with Applications 14(1), 1–59 (2007).
[Crossref]

Tichy, P.

J. Liesen and P. Tichy, “Convergence analysis of Krylov subspace methods,” GAMM-Mitteilungen 27(2), 153–173 (2004).
[Crossref]

Tsai, D.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Tuma, M.

J. Maryska, M. Rozloznik, and M. Tuma, “Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem,” SIAM Journal on Scientific Computing 22(2), 704–723 (2000).
[Crossref]

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
[Crossref] [PubMed]

Verweij, S.

Villeneuve, P.R.

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998).
[Crossref]

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

Wang, S.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Wang, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
[Crossref] [PubMed]

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nature Photonics 5(9), 523–530 (2011).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: putting a new twist on light(Cambridge University Press, 2008)

Wu, P.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Wu, T.

T. Wu and Z. Chen, “A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics 267, 82–95 (2014).
[Crossref]

Xu, B.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Yang, Shao

Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
[Crossref]

Yommi, A.

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials 13, 139–150 (2014).
[Crossref] [PubMed]

Yu, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
[Crossref] [PubMed]

Zhang, F.

A. R. Horn and F. Zhang, The Schur complement and its applications(Springer, 2005), Chap. 1.

Zhang, K.

K. Zhang and J.M. Jin, “Parallel FETI-DP algorithm for efficient simulation of large-scale electrmagnetic problems,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29(5), 897–914 (2016).
[Crossref]

Zhen, Peng

Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
[Crossref]

Peng Zhen, Lim Kheng-Hwee, and Lee Jin-Fa, “Non-conformal Domain Decomposition Methods for Solving Large Multi-scale Electromagnetic Scattering Problems,” Proceedings of the IEEE,  101(2), 298–319 (2013).
[Crossref]

Zhu, S.

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Advances in Engineering Software (1)

M. Storti, L. Dalcin, R. Paz, A. Yommi, V. Sonzogni, and N. Nigro, “A preconditioner for the Schur complement matrix,” Advances in Engineering Software 37(11), 754–762 (2006).
[Crossref]

GAMM-Mitteilungen (1)

J. Liesen and P. Tichy, “Convergence analysis of Krylov subspace methods,” GAMM-Mitteilungen 27(2), 153–173 (2004).
[Crossref]

IEEE Transactions on Antennas and Propagation (2)

Y. Li and J.M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3D large-scale electromagnetic problems,” IEEE Transactions on Antennas and Propagation 54(10), 704–723 (2006).
[Crossref]

Peng Zhen, Hiptmair Ralf, Shao Yang, and MacKie-Mason Brian, “Domain decomposition preconditioning for surface integral equations in solving challenging electromagnetic scattering problems,” IEEE Transactions on Antennas and Propagation 64(1), 210–223 (2016).
[Crossref]

IEEE Transactions on Components, Packaging and Manufacturing Technology (1)

B. Alavikia, N. Soltani, and O. M. Ramahi, “Efficient 2-D Finite-Difference Frequency-Domain Method for Switching Noise Analysis in Multilayer Boards,” IEEE Transactions on Components, Packaging and Manufacturing Technology 3(5), 841–848 (2013).
[Crossref]

IEEE Transactions on Microwave Theory and Techniques (1)

L. Kulas and M. Mrozowski, “Low-reflection subgridding,” IEEE Transactions on Microwave Theory and Techniques 53(5), 1587–1592 (2005).
[Crossref]

International Journal of Numerical Modelling: Electronic Networks, Devices and Fields (1)

K. Zhang and J.M. Jin, “Parallel FETI-DP algorithm for efficient simulation of large-scale electrmagnetic problems,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29(5), 897–914 (2016).
[Crossref]

Journal of Computational and Applied Mathematics (1)

T. Wu and Z. Chen, “A dispersion minimizing subgridding finite difference scheme for the Helmholtz equation with PML,” Journal of Computational and Applied Mathematics 267, 82–95 (2014).
[Crossref]

Journal of Computational Physics (1)

W. Shin, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” Journal of Computational Physics 231(8), 3406–3431 (2012).
[Crossref]

Linear Algebra and its Applications (1)

M. Neytcheva, “On element-by-element Schur complement approximations,” Linear Algebra and its Applications 434(11), 2308–2324 (2011).
[Crossref]

Nano Letters (1)

B. J. Bohn, M. Schnell, M. A. Kats, F. Aieta, R. Hillenbrand, and F. Capasso, “Near-Field Imaging of Phased Array Metasurfaces”, Nano Letters 15(6), 3851–3858 (2015).
[Crossref] [PubMed]

Nature Communications (1)

S. Wang, P. Wu, V. Su, Y. Lai, C. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. Tsai, “Broadbrand achromatic optical metasurface devices,” Nature Communications 8, 187 (2017).
[Crossref]

Nature Materials (1)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials 13, 139–150 (2014).
[Crossref] [PubMed]

Nature Photonics (3)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nature Photonics 5(9), 523–530 (2011).
[Crossref]

N. Meinzer, W.L. Barnes, and I.R. Hooper, “Plasmonic meta-atoms and metasurfaces,” Nature Photonics 8(8), 889–898 (2014).
[Crossref]

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nature Photonics 8(9), 180–184 (2015).
[Crossref]

Numerical Linear Algebra with Applications (2)

V. Simoncini and D.B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numerical Linear Algebra with Applications 14(1), 1–59 (2007).
[Crossref]

Y. Saad and B. Suchomel, “ARMS: an algebraic recursive multilevel solver for general sparse linear systems,” Numerical Linear Algebra with Applications 9(5), 359–378 (2002).
[Crossref]

Numerische Mathematik (2)

M. Arioli and F. Romani, “Relations between condition numbers and the convergence of the Jacobi method for real positive definite matrices,” Numerische Mathematik 46(1), 31–42 (1985).
[Crossref]

J. Mandel, “On block diagonal and Schur complement preconditioning,” Numerische Mathematik 58(1), 79–93 (1990).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Johannopoulos, “High Transmission through Sharp Bends in Photonic Crystal Waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref] [PubMed]

S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, “Channel Drop Tunneling through Localized States,” Phys. Rev. Lett. 80(5), 960–963 (1998).
[Crossref]

Physical Review Letters (1)

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Physical Review Letters,  100, 0239022008.
[Crossref] [PubMed]

Proceedings of the IEEE (1)

Peng Zhen, Lim Kheng-Hwee, and Lee Jin-Fa, “Non-conformal Domain Decomposition Methods for Solving Large Multi-scale Electromagnetic Scattering Problems,” Proceedings of the IEEE,  101(2), 298–319 (2013).
[Crossref]

SIAM Journal on Scientific Computing (1)

J. Maryska, M. Rozloznik, and M. Tuma, “Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem,” SIAM Journal on Scientific Computing 22(2), 704–723 (2000).
[Crossref]

Other (9)

P. Jaysaval, D. Datta, M. Sen, and A. Arnulf, “A Schur complement based fast 2D finite-difference multimodel modeling of acoustic wavefield in the frequency domain,” in).SEG Technical Program Expanded Abstracts, (Society of Exploration Geophysicists, 2017).

W. Cai and V. Shalaev, Optical Metamaterials(Springer, 2010).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: putting a new twist on light(Cambridge University Press, 2008)

W. Shin and S. Fan. “Simulation of phenomena characterized by partial differential equations,” US Patent Application13/744, 999

K. S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC-Press, 1993. Section 3.2

O. G. Ernst and M.J. Gander, “Why it is difficult to solve Helmholtz problems with classical iterative methods,” in Numerical Analysis of Multiscale Problems, I.G. Graham, T.Y. Hou, O. Lakkis, and R. Scheichl, eds. (SpringerBerlin Heidelberg, 2012).
[Crossref]

A. R. Horn and F. Zhang, The Schur complement and its applications(Springer, 2005), Chap. 1.

A. Pyzara, B. Bylina, and J. Bylina, “The influence of a matrix condition number on iterative methods convergence,” in Federated Conference on Computer Science and Information Systems(FedCSIS, 2011), pp. 459–464.

L. Giraud, A. Haidar, and Y. Saad, “Sparse approximations of the Schur complement for parallel algebraic hybrid linear solvers in 3D,” [Research Report] RR-7237, INRIA. 2010, pp.18.

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Figures (8)

Fig. 1
Fig. 1 Two-dimensional simulation domain consisting of an 8 × 8 grid of square meta-atoms. Each meta-atom contains a dielectric inclusion shown as a gray shape. A point source is placed in the center. The dotted lines denote the interfaces between meta-atoms. The lightly shaded region on the borders of the domain represents the perfectly matched layer boundary condition.
Fig. 2
Fig. 2 The first column shows the fields (real for dielectric, absolute value for metallic) from the iterative solution (using QMR) of Eq. (4) for a) the dielectric case (ϵ = 12) and c) the metallic case (ϵ = −3 − 0.3i). The second column shows the equivalents obtained with a direct solver for b) the dielectric case and d) the metallic case. Since the fields are complex in the metallic case, we show the absolute value of the fields. The source wavelength in all cases is 1.5a
Fig. 3
Fig. 3 Convergence rates for the unreduced and reduced systems on the domain containing an 8 × 8 lattice of meta-atoms from Fig. 1 with a periodic boundary condition for a) the dielectric case and b) the metallic case. The source wavelength in both cases is 1.5a.
Fig. 4
Fig. 4 Comparison of convergence rates of the unreduced and reduced systems on the domain containing an 8 × 8 lattice of meta-atoms from Fig. 1 with a perfectly matched layer 15 grid points deep in a) the dielectric case and b) the metallic case. The source wavelength in both cases is 1.5a
Fig. 5
Fig. 5 Number of iterations to convergence for the unreduced and reduced systems, as a function of increasing domain size. Here, the horizontal axis shows the number of meta-atoms per side in an N × N lattice of meta-atoms. We consider the dielectric case with a) perfectly matched layer boundary condition and b) periodic boundary condition. The source wavelength is 1.5a.
Fig. 6
Fig. 6 A lattice of cubic meta-atoms containing dielectric or metallic cubes. The volume between the dielectric or metallic cubes is air.
Fig. 7
Fig. 7 Comparison of convergence rates for the unreduced and reduced systems terminated by a periodic boundary condition. a) corresponds to the dielectric case and b) the metallic case. In both cases, the reduced system converges in fewer iterations than the unreduced system. The source wavelength is 2.5a.
Fig. 8
Fig. 8 Condition number of the unreduced matrix (A), the reduced matrix (S), and the interface sub-block of A (A00), as a function of domain size, for the physical system described in Fig. 5 where the dielectric inclusions have ϵ = 12 with (a) periodic and (b) perfectly matched layer boundary conditions.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

[ ( x ϵ 1 ( x , y ) ) x + ( y ϵ 1 ( x , y ) ) y ] H z + ω 2 μ 0 H z = i ω M z
[ A 00 B 01 B 02 B 0 n C 10 D 11 0 0 C 20 0 D 22 0 0 C n 0 0 0 0 D n n ] [ x 0 x 1 x 2 x n ] = [ b 0 b 1 b 2 b n ]
x i = D i i 1 ( b i C i 0 C i 0 x 0 )
S x 0 = ( A 00 i = 1 N B 0 i D i i 1 C i 0 ) x 0 = b 0 i = 1 N B i 0 D i i 1 b i
× × E [ 1 ϵ ( x , y , z ) ( ϵ ( x , y , z ) E ) ] ω 2 μ 0 ϵ ( x , y , z ) E = i ω μ 0 J i ω [ 1 ϵ ( x , y , z ) J ]
cond ( A ) = σ max ( A ) σ min ( A )
cond ( S ) cond ( A 00 ) 1 γ B 2
γ B 2 = ρ ( A 00 1 B D 1 C )
σ i ( A ) = | ω i , A 2 + ω 2 | μ 0
σ max , A = | ω max , A 2 + ω 2 | μ 0 | ( c A π Δ ) 2 | μ 0
σ min ( A ) = min i | ω i , A 2 + ω 2 | μ 0 | ω min , A 2 + ω 2 | μ 0 ω 2 μ 0
ω 2 μ 0 ~ ( c A π / a ) 2 μ 0
σ min ( A 00 ) = min i | ω i , A 00 2 + ω 2 | μ 0 | ω min , A 00 2 + ω 2 | μ 0 ( c A 00 π t ) 2 μ 0 .
σ min ( A 00 ) ( c A 00 π / t ) 2 μ 0 ( c A π / a ) 2 μ 0 σ min ( A ) .
T A Σ ( d ) C ( d ) [ N P t ] d = { 5 [ N P t ] 2 , d = 2 39 [ N P t ] 3 , d = 3
T S N d [ d C ( d ) P e d 1 ] 2 = { N 2 [ 2 P e ] 2 , d = 2 N 3 [ 9 P e 2 ] 2 , d = 3
T A T S Σ ( d ) C ( d ) P t d ( d C ( d ) P e ( d 1 ) ) 2 = { 5 P t 2 4 P e 2 , d = 2 13 P t 3 27 P e 4 , d = 3

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