A Reduced Order Model Approach to Inverse Scattering in Lossy Layered Media

Authors

Liliana Borcea, Vladimir Druskin, Jörn Zimmerling

Abstract

We introduce a reduced order model (ROM) methodology for inverse electromagnetic wave scattering in layered lossy media, using data gathered by an antenna which generates a probing wave and measures the time resolved reflected wave. We recast the wave propagation problem as a passive infinite-dimensional dynamical system, whose transfer function is expressed in terms of the measurements at the antenna. The ROM is a low-dimensional dynamical system that approximates this transfer function. While there are many possible ROM realizations, we are interested in one that preserves passivity and in addition is: (1) data driven (i.e., is constructed only from the measurements) and (2) it consists of a matrix with special sparse algebraic structure, whose entries contain spatially localized information about the unknown dielectric permittivity and electrical conductivity of the layered medium. Localized means in the intervals of a special finite difference grid. The main result of the paper is to show with analysis and numerical simulations that these unknowns can be extracted efficiently from the ROM.

Keywords

Inverse scattering; Data driven reduced order model; Passive; Port-Hamiltonian dynamical system; 37N30; 65N21; 65L09; 86A22

Citation

• Journal: Journal of Scientific Computing
• Year: 2021
• Volume: 89
• Issue: 1
• Pages:
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s10915-021-01616-7

BibTeX

@article{Borcea_2021,
  title={{A Reduced Order Model Approach to Inverse Scattering in Lossy Layered Media}},
  volume={89},
  ISSN={1573-7691},
  DOI={10.1007/s10915-021-01616-7},
  number={1},
  journal={Journal of Scientific Computing},
  publisher={Springer Science and Business Media LLC},
  author={Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn},
  year={2021}
}

Download the bib file

References

• Beattie, C. A., Mehrmann, V. & Van Dooren, P. Robust port-Hamiltonian representations of passive systems. Automatica vol. 100 182–186 (2019) – 10.1016/j.automatica.2018.11.013
• Benner, P., Goyal, P. & Van Dooren, P. Identification of port-Hamiltonian systems from frequency response data. Systems & Control Letters vol. 143 104741 (2020) – 10.1016/j.sysconle.2020.104741
• Borcea, L., Druskin, V. & Knizhnerman, L. On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids. Communications on Pure and Applied Mathematics vol. 58 1231–1279 (2005) – 10.1002/cpa.20073
• Borcea, L., Druskin, V., Mamonov, A. V. & Zaslavsky, M. A model reduction approach to numerical inversion for a parabolic partial differential equation. Inverse Problems vol. 30 125011 (2014) – 10.1088/0266-5611/30/12/125011
• Borcea, L., Druskin, V., Mamonov, A. V. & Zaslavsky, M. Robust nonlinear processing of active array data in inverse scattering via truncated reduced order models. Journal of Computational Physics vol. 381 1–26 (2019) – 10.1016/j.jcp.2018.12.021
• Borcea, L., Druskin, V., Mamonov, A. V., Zaslavsky, M. & Zimmerling, J. Reduced Order Model Approach to Inverse Scattering. SIAM Journal on Imaging Sciences vol. 13 685–723 (2020) – 10.1137/19m1296355
• Borcea, L., Druskin, V., Mamonov, A. V. & Zaslavsky, M. Untangling the nonlinearity in inverse scattering with data-driven reduced order models. Inverse Problems vol. 34 065008 (2018) – 10.1088/1361-6420/aabb16
• Bruckstein, A. M., Levy, B. C. & Kailath, T. Differential Methods in Inverse Scattering. SIAM Journal on Applied Mathematics vol. 45 312–335 (1985) – 10.1137/0145017
• Buterin, S. A. & Yurko, V. A. Inverse problems for second-order differential pencils with Dirichlet boundary conditions. jiip vol. 20 855–881 (2012) – 10.1515/jip-2012-0062
• Chu, M. & Golub, G. Inverse Eigenvalue Problems. (2005) doi:10.1093/acprof:oso/9780198566649.001.0001 – 10.1093/acprof:oso/9780198566649.001.0001
• E Coddington. Coddington, E., Levinson, N.: Theory of Ordinary Differentail Equations. Differential Equations, pp. 16–1022. McGraw-Hill, New York (1955) (1955)
• Druskin, V., Mamonov, A. V., Thaler, A. E. & Zaslavsky, M. Direct, Nonlinear Inversion Algorithm for Hyperbolic Problems via Projection-Based Model Reduction. SIAM Journal on Imaging Sciences vol. 9 684–747 (2016) – 10.1137/15m1039432
• G Freiling. Freiling, G., Yurko, V.A.: Inverse Sturm-Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001) (2001)
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• Gustavsen, B. & Semlyen, A. Rational approximation of frequency domain responses by vector fitting. IEEE Transactions on Power Delivery vol. 14 1052–1061 (1999) – 10.1109/61.772353
• Gustavsen, B. & Semlyen, A. Enforcing passivity for admittance matrices approximated by rational functions. IEEE Transactions on Power Systems vol. 16 97–104 (2001) – 10.1109/59.910786
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Jaulent, M. The inverse scattering problem for G transmission lines. Journal of Mathematical Physics vol. 23 2286–2290 (1982) – 10.1063/1.525307
• Joubert, W. Lanczos Methods for the Solution of Nonsymmetric Systems of Linear Equations. SIAM Journal on Matrix Analysis and Applications vol. 13 926–943 (1992) – 10.1137/0613056
• T Kato. Kato, T.: Perturbation Theory for Linear Operators, vol. 132. Springer, Berlin (2013) (2013)
• Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal of Research of the National Bureau of Standards vol. 45 255 (1950) – 10.6028/jres.045.026
• Markus, A. Introduction to the Spectral Theory o Polynomial Operator Pencils. Translations of Mathematica Monographs (2012) doi:10.1090/mmono/071 – 10.1090/mmono/071
• Marshall, T. Synthesis of RLC Ladder Networks by Matrix Tridiagonalization. IEEE Transactions on Circuit Theory vol. 16 39–46 (1969) – 10.1109/tct.1969.1082885
• Morgan, M. A., Groves, W. M. & Boyd, T. A. Reflectionless Filter Topologies Supporting Arbitrary Low-Pass Ladder Prototypes. IEEE Transactions on Circuits and Systems I: Regular Papers vol. 66 594–604 (2019) – 10.1109/tcsi.2018.2872424
• Pronska, N.: Spectral Properties of Sturm-liouville Equations with Singular Energy-dependent Potentials. arXiv preprint arXiv:1212.6671 (2012)
• Pronska, N. Reconstruction of Energy-Dependent Sturm–Liouville Equations from two Spectra. Integral Equations and Operator Theory vol. 76 403–419 (2013) – 10.1007/s00020-013-2035-7
• Saad, Y. The Lanczos Biorthogonalization Algorithm and Other Oblique Projection Methods for Solving Large Unsymmetric Systems. SIAM Journal on Numerical Analysis vol. 19 485–506 (1982) – 10.1137/0719031
• Sorensen, D. C. Passivity preserving model reduction via interpolation of spectral zeros. Systems & Control Letters vol. 54 347–360 (2005) – 10.1016/j.sysconle.2004.07.006
• Schaft, A. J. Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems. Advanced Dynamics and Control of Structures and Machines 127–167 (2004) doi:10.1007/978-3-7091-2774-2_9 – 10.1007/978-3-7091-2774-2_9
• Willems, J. C. Dissipative Dynamical Systems. European Journal of Control vol. 13 134–151 (2007) – 10.3166/ejc.13.134-151
• Yagle, A. E. One-dimensional inverse scattering problems: an asymmetric two-component wave system framework. Inverse Problems vol. 5 641–646 (1989) – 10.1088/0266-5611/5/4/014