Skip to main content
SpringerLink
Log in

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This paper presents a non-intrusive model order reduction (MOR) for the solution of parameterized electromagnetic scattering problems, which needs to prepare a database offline of full-order solution samples (snapshots) at some different parameter locations. The snapshot vectors are produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. Because the second dimension of snapshots matrix is large, a two-step or nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of POD-CSI method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Hesthaven, J.S., Rozza, G., Stamm, B., et al.: Certified reduced basis methods for parametrized partial differential equations. Springer, New York (2016)

    MATH  Google Scholar 

  2. Guo, M., Hesthaven, J.S.: Data-driven reduced order modeling for time-dependent problems. Comput Meth Appl Mech Eng 345, 75–99 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Georgaka, S., Stabile, G., Rozza, G., Bluck, M.J.: Parametric POD-Galerkin model order reduction for unsteady-state heat transfer problems. Commun Comput Phys 27(1), 1–32 (2020)

    MathSciNet  Google Scholar 

  4. Vidal-Codina, F., Nguyen, N.C., Peraire, J.: Computing parametrized solutions for plasmonic nanogap structures. J Comput Phys 366, 89–106 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Yee, K.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans Antennas Propag 14(3), 302–307 (1966)

    MATH  Google Scholar 

  6. Hesthaven, J.S., Warburton, T.: Nodal discontinuous, Galerkin methods algorithms, analysis, and applications. Springer, New York (2007)

    MATH  Google Scholar 

  7. Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev 57(4), 483–531 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Peherstorfer, B., Willcox, K., Gunzburger, M.: Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev 60(3), 550–591 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Haasdonk, B., Ohlberger, M.: Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math Comput Modell Dyn Syst 17(2), 145–161 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Pasetto, D., Putti, M., Yeh, W.W.-G.: A reduced-order model for groundwater flow equation with random hydraulic conductivity: application to monte carlo methods. Water Resour Res 49(6), 3215–3228 (2013)

    Google Scholar 

  11. Hesthaven, J.S., Stamm, B., Zhang, S.: Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods. ESAIM Math Modell Num Anal 48(1), 259–283 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Sirovich, L.: Turbulence and the dynamics of coherent structures. I. Coherent structures. Qr Appl Math 45(3), 561–571 (1987)

    MathSciNet  MATH  Google Scholar 

  13. Rathinam, M., Petzold, L.R.: A new look at proper orthogonal decomposition. SIAM J Num Anal 41(5), 1893–1925 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Pinnau, R.: Model reduction via proper orthogonal decomposition, in: model order reduction, theory: research aspects and applications, pp. 95–109. Springer, New York (2008)

    MATH  Google Scholar 

  15. Chaturantabut, S., Sorensen, D.C.: A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J Num Anal 50(1), 46–63 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Li, K., Huang, T.-Z., Li, L., Lanteri, S.: POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell’s equations. J Comput Phys 396, 106–128 (2019)

    MathSciNet  MATH  Google Scholar 

  17. Ballarin, F., Manzoni, A., Quarteroni, A., Rozza, G.: Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations. Int J Num Meth Eng 102(5), 1136–1161 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Wang, Q., Hesthaven, J.S., Ray, D.: Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem. J Comput Phys 384, 289–307 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Lass, O., Volkwein, S.: POD-Galerkin schemes for nonlinear elliptic-parabolic systems. SIAM J Sci Comput 35(3), A1271–A1298 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Ullmann, S., Rotkvic, M., Lang, J.: POD-Galerkin reduced-order modeling with adaptive finite element snapshots. J Comput Phys 325, 244–258 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Ballarin, F., Faggiano, E., Ippolito, S., Manzoni, A., Quarteroni, A., Rozza, G., Scrofani, R.: Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization. J Comput Phys 315, 609–628 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Strazzullo, M., Ballarin, F., Rozza, G.: POD-Galerkin model order reduction for parametrized nonlinear time dependent optimal flow control: an application to shallow water equations. J Sci Comput (2020). https://doi.org/10.1007/s10915-020-01232-x

    Article  MATH  Google Scholar 

  23. Baur, U., Benner, P., Feng, L.: Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch Comput Meth Eng 21(4), 331–358 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Carlberg, K., Barone, M., Antil, H.: Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction. J Comput Phys 330, 693–734 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Ly, H.V., Tran, H.T.: Modeling and control of physical processes using proper orthogonal decomposition. Math Comput Modell 33(1–3), 223–236 (2001)

    MATH  Google Scholar 

  26. Li, K., Huang, T.-Z., Li, L., Lanteri, S., Xu, L., Li, B.: A reduced-order discontinuous Galerkin method based on POD for electromagnetic simulation. IEEE Trans Antennas Prop 66(1), 242–254 (2017)

    Google Scholar 

  27. Fu, H., Wang, H., Wang, Z.: POD/DEIM reduced-order modeling of time-fractional partial differential equations with applications in parameter identification. J Sci Comput 74(1), 220–243 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Luo, Z., Ren, H.: A reduced-order extrapolated finite difference iterative method for the riemann-liouville tempered fractional derivative equation. Appl Num Math 157, 307–314 (2020)

    MathSciNet  MATH  Google Scholar 

  29. Shen, J., Singler, J.R., Zhang, Y.: HDG-POD reduced order model of the heat equation. J Comput Appl Math 362, 663–679 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Yıldız, S., Goyal, P., Benner, P., Karasozen, B.: Data-driven learning of reduced-order dynamics for a parametrized shallow water equation (2020). arxiv preprint arXiv:2007.14079

  31. Yu, J., Yan, C., Jiang, Z., Yuan, W., Chen, S.: Adaptive non-intrusive reduced order modeling for compressible flows. J Comput Phys 397, 108855 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Yu, J., Yan, C., Guo, M.: Non-intrusive reduced-order modeling for fluid problems: a brief review. Proceed Inst Mech Eng Part G J Aerospace Eng 233(16), 5896–5912 (2019)

    Google Scholar 

  33. Casenave, F., Ern, A., Leliévre, T.: A nonintrusive reduced basis method applied to aeroacoustic simulations. Adv Comput Math 41(5), 961–986 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Hesthaven, J.S., Ubbiali, S.: Non-intrusive reduced order modeling of nonlinear problems using neural networks. J Comput Phys 363, 55–78 (2018)

    MathSciNet  MATH  Google Scholar 

  35. San, O., Maulik, R., Ahmed, M.: An artificial neural network framework for reduced order modeling of transient flows. Commun Nonlinear Sci Numer Simul 77, 271–287 (2019)

    MathSciNet  MATH  Google Scholar 

  36. Guo, M., Hesthaven, J.S.: Reduced order modeling for nonlinear structural analysis using gaussian process regression. Comput Meth Appl Mech Eng 341, 807–826 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Audouze, C., De Vuyst, F., Nair, P.B.: Nonintrusive reduced-order modeling of parametrized time-dependent partial differential equations. Num Meth Partial Diff Equ 29(5), 1587–1628 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Xiao, D., Fang, F., Pain, C.C., Navon, I.M., Salinas, P., Muggeridge, A.: Non-intrusive reduced order modeling of multi-phase flow in porous media using the POD-RBF method. J Comput Phys 1, 1–25 (2015)

    Google Scholar 

  39. Dehghan, M., Abbaszadeh, M.: The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations. J Comput Phys 351, 478–510 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Constantine, P.G., Gleich, D.F., Hou, Y., Templeton, J.: Model reduction with mapreduce-enabled tall and skinny singular value decomposition. SIAM J Sc Comput 36(5), S166–S191 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Sun, X., Pan, X., Choi, J.-I.: A non-intrusive reduced-order modeling method using polynomial chaos expansion (2019). arxiv preprint arXiv:1903.10202

  42. Bui-Thanh, T., Damodaran, M., Willcox, K.: Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics, In Proceedings of the 21st Applied Aerodynamics AIAA Conference, Orlando, Florida (2003)

  43. Oulghelou, M., Allery, C.: Non intrusive method for parametric model order reduction using a bi-calibrated interpolation on the grassmann manifold. J Comput Phys 426, 109924 (2021)

    MathSciNet  Google Scholar 

  44. Schmidt, E.: On the theory of linear and nonlinear integral equations. I. development of arbitrary function according to systems prescribed. Math Ann 63, 433–476 (1907)

    MathSciNet  Google Scholar 

  45. Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936)

    MATH  Google Scholar 

  46. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J Num Anal 40(2), 492–515 (2002)

    MathSciNet  MATH  Google Scholar 

  47. Georgaka, S., Stabile, G., Star, K., Rozza, G., Bluck, M.J.: A hybrid reduced order method for modelling turbulent heat transfer problems. Comput Fluids 208, 104615 (2020)

  48. McKinley, S., Levine, M.: Cubic spline interpolation. Coll Redwoods 45(1), 1049–1060 (1998)

    Google Scholar 

  49. Behforooz, G.H.: A comparison of the E(3) and not-a-knot cubic splines. Appl Math Comput 72(2–3), 219–223 (1995)

    MathSciNet  MATH  Google Scholar 

  50. Hasan, M.S., Islam, S.K., Blalock, B.J.: Modeling of soi four-gate transistor (g4fet) using multidimensional spline interpolation method. Microelectron J 76, 33–42 (2018)

    Google Scholar 

  51. Hasan, M.S., Amer, S., Islam, S. K., Rose, G. S.: Multivariate cubic spline: a versatile DC modeling technique suitable for different deep submicron transistors, In: Proceedings of IEEE SoutheastCon, 1–8 (2019)

  52. Wang, R.-H.: Multivariate spline functions and their applications. Springer, New York (2013)

    Google Scholar 

  53. Avenda no-Valencia, L. D., Chatzi, E. N., Koo, K. Y., Brownjohn, J. M.: Gaussian process time-series models for structures under operational variability, Front Built Environ 3, 69 (2017)

  54. Trehan, S., Carlberg, K.T., Durlofsky, L.J.: Error modeling for surrogates of dynamical systems using machine learning. Int J Num Meth Eng 112(12), 1801–1827 (2017)

    MathSciNet  Google Scholar 

  55. El Bouajaji, M., Dolean, V., Gander, M.J., Lanteri, S., Perrussel, R.: Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell equations. Electron Trans Num Anal 44, 572–592 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful for the constructive advices from anonymous reviewers and the help of the editors.

Author information

Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Kun Li

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, People’s Republic of China

    Ting-Zhu Huang & Liang Li

  3. INRIA, 2004 Route des Lucioles, BP 93, 06902, Sophia Antipolis Cedex, France

    Stéphane Lanteri

Authors
  1. Kun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting-Zhu Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stéphane Lanteri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liang Li.

Ethics declarations

Fundings

This research was supported by NSFC (Grant No. 61772003) and Key Projects of Applied Basic Research in Sichuan Province (Grant No. 2020YJ0216).

Conflicts of interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Code availability

Not applicable

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, K., Huang, TZ., Li, L. et al. Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation. J Sci Comput 87, 52 (2021). https://doi.org/10.1007/s10915-021-01467-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01467-2

Keywords

Search

Navigation