Optimization of Faber Polynomial-Based Propagators via Proper Orthogonal Decomposition

Abstract

The focus of this research lies in the extrapolation of the time-dependent Faber polynomial-based propagator applied to Maxwell’s equations with the Proper Orthogonal Decomposition (POD) approach. In particular, when investigating a system in the application area of THz-technology or nanophotonics, it is common to have geometries which require an extraordinary fine spatial resolution. Unfortunately, this is accompanied by the temporal resolution, which is restricted accordingly. For this reason, the explicit Faber approach is chosen to approximate the computationally intensive matrix exponential to enable large time step sizes. To achieve a further acceleration of the computation time, the POD approach is used to enormously reduce the complexity of the evaluation. In this context, a hybrid model is examined to establish a balance between the calculation time by means of a Reduced-Order Model (ROM) and the error accuracy via a Full-Order Model (FOM).

Keywords

Finite difference time domain (FDTD) / Matrix exponential / Faber polynomials / Proper orthogonal decomposition (POD) / Galerkin projection / THz-technology / Nanophotonics / Full-order model (FOM) / Reduced-order model (ROM) / Singular value decomposition (SVD) / Incremental singular value decomposition (ISVD) / 65D15

References

[1] Bai, F., Wang, Y.: DEIM reduced order model constructed by hybrid snapshot simulation. SN Appl. Sci. 2(12), 2165 (2020)

[2] Bai F, Wang Y. Reduced-order modeling based on hybrid snapshot simulation. Int. J. Comput. Methods, 2021, 18(01): 2050029.

[3] Bérenger, J.-P.: Perfectly matched layer (PML) for computational electromagnetics. Springer, Cham (2022)

[4] Berry MW. Large-scale sparse singular value computations. Int. J. Supercomput. Appl., 1992, 6(1): 13-49

[5] Borisov, A.G., Shabanov, S.V.: Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media. J. Comput. Phys. 216(1), 391–402 (2006)

[6] Brand, M.: Incremental singular value decomposition of uncertain data with missing values. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds) Proceedings European Conference on Computer Vision, pp. 707–720. Springer (2002)

[7] Brand, O., Fedder, G.K., Hierold, C., Tabata, O.: Micro and Nano Scale NMR: Technologies and Systems. John Wiley & Sons, Inc (2018)

[8] Brunton, S.L., Kutz, J.N.: Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, Cambridge (2022)

[9] Busch, K., Niegemann, J., Pototschnig, M., Tkeshelashvili, L.: A Krylov-subspace based solver for the linear and nonlinear Maxwell equations. Physica Status Solidi (b) 244(10), 3479–3496 (2007)

[10] Chatterjee, A.: An introduction to the proper orthogonal decomposition. Current Science 78(7), 808–817 (2000)

[11] Courant R, Friedrichs K, Lewy H. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 1928, 100(1): 32-74.

[12] Curtiss JH. Faber polynomials and the Faber series. Am. Math. Mon., 1971, 78(6): 577-596.

[13] Davidson, D.B.: Computational Electromagnetics for RF and Microwave Engineering. Cambridge University Press, Cambridge (2010)

[14] De Raedt, H., Michielsen, K., Kole, J.S., Figge, M.T.: Solving the Maxwell equations by the Chebyshev method: a one-step finite-difference time-domain algorithm. IEEE Trans. Antennas Propag. 51(11), 3155–3160 (2003)

[15] Ebeling, K.J.: Integrated Optoelectronics: Waveguide Optics, Photonics, Semiconductors. Springer, New York (2012)

[16] Everson, R., Sirovich, L.: Karhunen-Lo‌ève procedure for gappy data. J. Opt. Soc. Am. A 12(8), 1657–1664 (1995)

[17] Faber G. Über polynomische Entwicklungen. Math. Ann., 1903, 57(3): 389-408.

[18] Feingold, D.G., Varga, R.S.: Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem. Paci. J. Math. 4, 1241–1250 (1962)

[19] Fu, W., Tan, E.L.: Development of split-step FDTD method with higher-order spatial accuracy. Electron. Lett. 40(20), 1 (2004)

[20] Gaffar, Md., Jiao, D.: An explicit and unconditionally stable FDTD method for electromagnetic analysis. IEEE Trans. Microw. Theory Tech. 62(11), 2538–2550 (2014)

[21] Gaffar, Md., Jiao, D.: Alternative method for making explicit FDTD unconditionally stable. IEEE Trans. Microw. Theory Tech. 63(12), 4215–4224 (2015)

[22] Gavish M, Donoho DL. The optimal hard threshold for singular values is 4/$\sqrt{3}$. IEEE Trans. Inf. Theory, 2014, 60(8): 5040-5053.

[23] Greathouse, J.L., Daga, M.: Efficient sparse matrix-vector multiplication on GPUs using the CSR storage format. In SC’14: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 769–780. IEEE (2014)

[24] Han S, Feeny B. Application of proper orthogonal decomposition to structural vibration analysis. Mech. Syst. Signal Process., 2003, 17(5): 989-1001.

[25] Holmes, P.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (2012)

[26] Huisinga, W., Pesce, L., Kosloff, R., Saalfrank, P.: Faber and Newton polynomial integrators for open-system density matrix propagation. J. Chem. Phys. 110(12), 5538–5547 (1999)

[27] IEEE. IEEE standard for floating-point arithmetic. In: IEEE Std 754-2019 (Revision of IEEE 754-2008), pp. 1–148 (2019)

[28] Kleene, H., Schulz, D.: Time domain solution of Maxwell’s equations using Faber polynomials. IEEE Trans. Antennas Propag. 66(11), 6202–6208 (2018)

[29] Luo, Z., Gao, J.: A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium. J. Math. Anal. Appl. 444(1), 433–451 (2016)

[30] Mathworks. Sparse (2025). Access on November 1st, 2025. https://de.mathworks.com/help/matlab/ref/sparse.html

[31] Namiki, T.: 3-D ADI-FDTD method-unconditionally stable time-domain algorithm for solving full vector Maxwell’s equations. IEEE Trans. Microw. Theory Tech. 48(10), 1743–1748 (2000)

[32] Plotnikov, W., Inci, B.L., Schulz, D.: Investigation of conformal mappings for the approximation of Faber polynomial based propagators. In: 2022 IEEE MTT-S International Conference on Numerical Electromagnetic and Multiphysics Modeling and Optimization (NEMO), pp. 1–3. IEEE (2022)

[33] Plotnikov, W., Schulz, D.: Optimization of the Faber polynomial based propagator through parallelization. In: 2024 49th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz), Perth, Australia, 2024, pp. 1–2 (2024)

[34] Plotnikov, W., Schulz, D.: Performance analysis of Faber polynomial based local propagators for photonics. Opt. Quant. Electron. 57(3), 170 (2025)

[35] Rapún, M.-L., Vega, J.M.: Reduced order models based on local POD plus Galerkin projection. J. Comput. Phys. 229(8), 3046–3063 (2010)

[36] Steindl, A., Troger, H.: Methods for dimension reduction and their application in nonlinear dynamics. Int. J. Solids Struct. 38(10/11/12/13), 2131–2147 (2001)

[37] Strang, G.: Linear Algebra and Learning from Data. Wellesley-Cambridge Press, Wellesley (2019)

[38] Sun, G., Trueman, C.W.: Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method. IEEE Trans. Microw. Theory Tech. 54(5), 2275–2284 (2006)

[39] Taflove, A., Hagness, S.C., Piket-May, M.: Computational electromagnetics: the finite-difference time-domain method. In: Chen, W.-K. (ed) The Electrical Engineering Handbook, pp. 629–670. Academic Press (2005)

[40] Tan, E.L.: Unconditionally stable LOD-FDTD method for 3-D Maxwell’s equations. IEEE Microwave Wirel. Compon. Lett. 17(2), 85–87 (2007)

[41] Turk M, Pentland A. Eigenfaces for recognition. J. Cogn. Neurosci., 1991, 3(1): 71-86.

[42] Wang, K., Dong, X., Köhler, M.H., Kienle, P., Bian, Q., Jakobi, M., Koch, A.W.: Advances in optical fiber sensors based on multimode interference (MMI): a review. IEEE Sens. J. 21(1), 132–142 (2020)

[43] Xie, G., Hou, G., Feng, N., Song, K., Fang, M., Li, Y., Wu, X., Huang, Z.: Auxiliary differential equation (ADE) method based complying-divergence implicit FDTD method for simulating the general dispersive anisotropic material. Opt. Express 31(11), 18468–18486 (2023)

[44] Xie, G., Hou, G., Xu, K., Niu, K., Feng, N., Li, Y., Huang, Z., Elsherbeni, A.Z.: Extension of FLOD-FDTD method for multiterm modified Lorentz model. IEEE Trans. Antennas Propag. 71(11), 8830–8840 (2023)

[45] Yan, J., Jiao, D.: Fast explicit and unconditionally stable FDTD method for electromagnetic analysis. IEEE Trans. Microw. Theory Tech. 65(8), 2698–2710 (2017)

[46] 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)

[47] Zhang, Y.: An answer to an open question in the incremental SVD. arXiv:2204.05398 (2022)

Funding

German Research Funding Association DFG(SCHU 1016/6-2)

RIGHTS & PERMISSIONS

The Author(s)