• Previous Article
    Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications
  • IPI Home
  • This Issue
  • Next Article
    Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem
doi: 10.3934/ipi.2021012

A sampling type method in an electromagnetic waveguide

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Received  July 2020 Revised  November 2020 Published  January 2021

We propose a sampling type method to image scatterer in an electromagnetic waveguide. The waveguide terminates at one end and the measurements are on the other end and in the far field. The imaging function is based on integrating the measurements and a known function over the measurement surface directly. The design and analysis of such imaging function are based on a factorization of a data operator given by the measurements. We show by analysis that the imaging function peaks inside the scatterer, where the coercivity of the factorized operator and the design of the known function play a central role. Finally, numerical examples are provided to demonstrate the performance of the imaging method.

Citation: Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems & Imaging, doi: 10.3934/ipi.2021012
References:
[1]

T. ArensD. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM Journal on Applied Mathematics, 71 (2011), 753-772.  doi: 10.1137/100806333.  Google Scholar

[2]

A. B. BaggeroerW. A. Kuperman and P. N. Mikhalevsky, An overview of matched field methods in ocean acoustics, IEEE Journal of Oceanic Engineering, 18 (1993), 401-424.   Google Scholar

[3]

L. Borcea and S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Problems, 35 (2019), 124006, 33pp. doi: 10.1088/1361-6420/ab2c9b.  Google Scholar

[4]

L. Borcea and D. L. Nguyen, Imaging with electromagnetic waves in terminating waveguides, Inverse Problems and Imaging, 10 (2016), 915-941.  doi: 10.3934/ipi.2016027.  Google Scholar

[5]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 095004, 31pp. doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[6]

L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[7]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018, 20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[8]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19pp. doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[10]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, 188. Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[11]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, volume 88., SIAM, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[12]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, volume 80., SIAM, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[13]

J. Chen and G. Huang, A direct imaging method for inverse electromagnetic scattering problem in rectangular waveguide, Communications in Computational Physics, 23 (2018), 1415-1433.  doi: 10.4208/cicp.oa-2017-0048.  Google Scholar

[14]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, volume 93, Third edition. Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[16]

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005, 23pp. doi: 10.1088/0266-5611/27/8/085005.  Google Scholar

[17]

A. HaackJ. Schreyer and G. Jackel, State-of-the-art of non-destructive testing methods for determining the state of a tunnel lining, Tunnelling and Underground Space Technology Incorporating Trenchless Technology Research, 4 (1995), 413-431.   Google Scholar

[18]

A. Kirsch, The factorization method for maxwell's equations, Inverse Problems, 20 (2004), S117–S134. doi: 10.1088/0266-5611/20/6/S08.  Google Scholar

[19] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36., Oxford University Press, 2008.   Google Scholar
[20]

X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20pp. doi: 10.1088/1361-6420/aa777d.  Google Scholar

[21]

P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747.  doi: 10.3934/ipi.2012.6.709.  Google Scholar

[22]

P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain, Inverse Problems, 32 (2016), 055001, 26pp. doi: 10.1088/0266-5611/32/5/055001.  Google Scholar

[23]

P. Monk, V. Selgas and F. Yang, Near-field linear sampling method for an inverse problem in an electromagnetic waveguide, Inverse Problems, 35 (2019), 065001, 27pp. doi: 10.1088/1361-6420/ab0cdc.  Google Scholar

[24]

R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015. doi: 10.1088/0266-5611/26/7/074015.  Google Scholar

[25]

P. Rizzo, A. Marzani, J. Bruck, et al., Ultrasonic guided waves for nondestructive evaluation/structural health monitoring of trusses, Measurement Science and Technology, 21 (2010), 045701. Google Scholar

[26]

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, 1 (1997), 41-52.   Google Scholar

[27]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.  Google Scholar

[28]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM Journal on Imaging Sciences, 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.  Google Scholar

[29]

Y. XuC. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

show all references

References:
[1]

T. ArensD. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM Journal on Applied Mathematics, 71 (2011), 753-772.  doi: 10.1137/100806333.  Google Scholar

[2]

A. B. BaggeroerW. A. Kuperman and P. N. Mikhalevsky, An overview of matched field methods in ocean acoustics, IEEE Journal of Oceanic Engineering, 18 (1993), 401-424.   Google Scholar

[3]

L. Borcea and S. Meng, Factorization method versus migration imaging in a waveguide, Inverse Problems, 35 (2019), 124006, 33pp. doi: 10.1088/1361-6420/ab2c9b.  Google Scholar

[4]

L. Borcea and D. L. Nguyen, Imaging with electromagnetic waves in terminating waveguides, Inverse Problems and Imaging, 10 (2016), 915-941.  doi: 10.3934/ipi.2016027.  Google Scholar

[5]

L. Bourgeois and S. Fliss, On the identification of defects in a periodic waveguide from far field data, Inverse Problems, 30 (2014), 095004, 31pp. doi: 10.1088/0266-5611/30/9/095004.  Google Scholar

[6]

L. Bourgeois, F. Le Louër and E. Lunéville, On the use of Lamb modes in the linear sampling method for elastic waveguides, Inverse Problems, 27 (2011), 055001, 27pp. doi: 10.1088/0266-5611/27/5/055001.  Google Scholar

[7]

L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018, 20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar

[8]

L. Bourgeois and E. Lunéville, On the use of sampling methods to identify cracks in acoustic waveguides, Inverse Problems, 28 (2012), 105011, 18pp. doi: 10.1088/0266-5611/28/10/105011.  Google Scholar

[9]

L. Bourgeois and E. Lunéville, On the use of the linear sampling method to identify cracks in elastic waveguides, Inverse Problems, 29 (2013), 025017, 19pp. doi: 10.1088/0266-5611/29/2/025017.  Google Scholar

[10]

F. Cakoni and D. Colton, A Qualitative Approach to Inverse Scattering Theory, Applied Mathematical Sciences, 188. Springer, New York, 2014. doi: 10.1007/978-1-4614-8827-9.  Google Scholar

[11]

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, volume 88., SIAM, 2016. doi: 10.1137/1.9781611974461.ch1.  Google Scholar

[12]

F. Cakoni, D. Colton and P. Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering, volume 80., SIAM, 2011. doi: 10.1137/1.9780898719406.  Google Scholar

[13]

J. Chen and G. Huang, A direct imaging method for inverse electromagnetic scattering problem in rectangular waveguide, Communications in Computational Physics, 23 (2018), 1415-1433.  doi: 10.4208/cicp.oa-2017-0048.  Google Scholar

[14]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[15]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, volume 93, Third edition. Applied Mathematical Sciences, 93. Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3.  Google Scholar

[16]

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005, 23pp. doi: 10.1088/0266-5611/27/8/085005.  Google Scholar

[17]

A. HaackJ. Schreyer and G. Jackel, State-of-the-art of non-destructive testing methods for determining the state of a tunnel lining, Tunnelling and Underground Space Technology Incorporating Trenchless Technology Research, 4 (1995), 413-431.   Google Scholar

[18]

A. Kirsch, The factorization method for maxwell's equations, Inverse Problems, 20 (2004), S117–S134. doi: 10.1088/0266-5611/20/6/S08.  Google Scholar

[19] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, volume 36., Oxford University Press, 2008.   Google Scholar
[20]

X. Liu, A novel sampling method for multiple multiscale targets from scattering amplitudes at a fixed frequency, Inverse Problems, 33 (2017), 085011, 20pp. doi: 10.1088/1361-6420/aa777d.  Google Scholar

[21]

P. Monk and V. Selgas, Sampling type methods for an inverse waveguide problem, Inverse Problems and Imaging, 6 (2012), 709-747.  doi: 10.3934/ipi.2012.6.709.  Google Scholar

[22]

P. Monk and V. Selgas, An inverse acoustic waveguide problem in the time domain, Inverse Problems, 32 (2016), 055001, 26pp. doi: 10.1088/0266-5611/32/5/055001.  Google Scholar

[23]

P. Monk, V. Selgas and F. Yang, Near-field linear sampling method for an inverse problem in an electromagnetic waveguide, Inverse Problems, 35 (2019), 065001, 27pp. doi: 10.1088/1361-6420/ab0cdc.  Google Scholar

[24]

R. Potthast, A study on orthogonality sampling, Inverse Problems, 26 (2010), 074015. doi: 10.1088/0266-5611/26/7/074015.  Google Scholar

[25]

P. Rizzo, A. Marzani, J. Bruck, et al., Ultrasonic guided waves for nondestructive evaluation/structural health monitoring of trusses, Measurement Science and Technology, 21 (2010), 045701. Google Scholar

[26]

J. Schöberl, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Computing and Visualization in Science, 1 (1997), 41-52.   Google Scholar

[27]

J. Sun and C. Zheng, Reconstruction of obstacles embedded in waveguides, Contemporary Mathematics, 586 (2013), 341-350.  doi: 10.1090/conm/586/11652.  Google Scholar

[28]

C. TsogkaD. A. Mitsoudis and S. Papadimitropoulos, Imaging extended reflectors in a terminating waveguide, SIAM Journal on Imaging Sciences, 11 (2018), 1680-1716.  doi: 10.1137/17M1159051.  Google Scholar

[29]

Y. XuC. Mawata and W. Lin, Generalized dual space indicator method for underwater imaging, Inverse Problems, 16 (2000), 1761-1776.  doi: 10.1088/0266-5611/16/6/311.  Google Scholar

Figure 1.  Plot of $ |\mathcal{H} \Psi ({\boldsymbol{x}}_*;{\boldsymbol{z}}) {\boldsymbol{e}}_j|^2 $ as a function of the sampling point $ {\boldsymbol{z}} $ for a rectangular waveguide with cross section $ (0,10)\times(0,10) $ where $ {\boldsymbol{x}}_* = (5,5,-5) $ and $ k = 3 $. Top left: Iso-surface plot with iso-value $ 0.6 $. Top right: $ x_1x_2 $-cross section image. Bottom left: $ x_2x_3 $-cross section image. Bottom right: $ x_1x_3 $-cross section image
Figure 2.  Three dimensional view of the generated mesh
Figure 3.  Image of a cuboid. Top left: exact. Top middle: three dimensional image using iso-surface plot with iso-value $ 0.6 $. Top right: three dimensional image using iso-surface plot with iso-value $ 0.4 $. At the bottom, we plot the cross section images where the exact geometry is indicated by the dashed line. Bottom left: $ x_1x_2 $-cross section image. Bottom middle: $ x_2x_3 $-cross section image. Bottom right: $ x_1x_3 $-cross section image
Figure 4.  Image of a L-shape scatterer. Top left: exact. Top middle: three dimensional image using iso-surface plot with iso-value $ 0.5 $. Top right: three dimensional image using iso-surface plot with iso-value $ 0.4 $. At the bottom, we plot the cross section images where the exact geometry is indicated by the dashed line. Bottom left: $ x_1x_2 $-cross section image. Bottom middle: $ x_2x_3 $-cross section image. Bottom right: $ x_1x_3 $-cross section image
Figure 5.  Image of a cylindrical scatterer. Top left: exact. Top middle: $ k = 3 $; three dimensional image using iso-surface plot with iso-value $ 0.4 $. Top right: $ k = 5 $; three dimensional image using iso-surface plot with iso-value $ 0.3 $. In the middle ($ k = 3 $) and bottom ($ k = 5 $) row, we plot the cross section images where the exact geometry is indicated by the dashed line. Middle/bottom row left: $ x_1x_2 $-cross section image. Middle/bottom row middle: $ x_2x_3 $-cross section image. Middle/bottom row right: $ x_1x_3 $-cross section image
Figure 6.  Top: image of two balls. Bottom: image of three balls. Left: exact. Middle: three dimensional image using iso-surface plot (with iso-value 0.2). Right: $ x_1x_2 $-cross section image
Figure 7.  Image of a large ball. Left: exact. Middle: three dimensional image when $ k = 1 $. Right: three dimensional image when $ k = 3 $
Figure 8.  Iso-surface (with the same iso-value 0.4) image of the L-shape scatterer. Left: exact. Middle: $ 10\% $ noise. Right: $ 30\% $ noise
[1]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[2]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[3]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[4]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[5]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327

[6]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[7]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[8]

Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024

[9]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[10]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[11]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[12]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[13]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[14]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[15]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002

[16]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

[17]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149

[18]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[19]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[20]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

2019 Impact Factor: 1.373

Article outline

Figures and Tables

[Back to Top]