Reconstruction of a defect in an open waveguide. (English) Zbl 1304.35760
Summary: The paper is concerned with the reconstruction of a defect in the core of a two-dimensional open waveguide from the scattering data. Since only a finite numbers of modes can propagate without attenuation inside the core, the problem is similar to the one-dimensional inverse medium problem. In particular, the inverse problem suffers from a lack of uniqueness and is known to be severely ill-posed. To overcome these difficulties, we consider multi-frequency scattering data. The uniqueness of the solution to the inverse problem is established from the far field scattering information over an interval of low frequencies.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P25 | Scattering theory for PDEs |
References:
| [1] | Ammari H, Iakovleva E, Kang H. Reconstruction of a small inclusion in a two-dimensional open waveguide. SIAM J Appl Math, 2005, 65: 2107-2127 · Zbl 1081.78010 · doi:10.1137/040615389 |
| [2] | Ammari H, Triki F. Resonances for microstrip transmission lines. SIAM J Appl Math, 2004, 64: 601-636 · Zbl 1126.78303 · doi:10.1137/S0036139902418390 |
| [3] | Bao G, Li P. Inverse medium scattering problems for electromagnetic waves. Inverse Probl, 2005, 21: 1621-1641 · Zbl 1086.35120 · doi:10.1088/0266-5611/21/5/007 |
| [4] | Bao G, Li P. Inverse medium scattering problems for electromagnetic waves. SIAM J Appl Math, 2005, 65: 2049-2066 · Zbl 1114.78006 · doi:10.1137/040607435 |
| [5] | Bao G, Triki F. Error estimates for recursive linearization of inverse medium problems. J Comput Math, 2010, 28: 725-744 · Zbl 1240.35574 |
| [6] | Bao G, Lin J, Triki F. A multi-frequency inverse source problem. J Differential Equations, 2010, 249: 3443-3465 · Zbl 1205.35336 · doi:10.1016/j.jde.2010.08.013 |
| [7] | Bao G, Lin J, Triki F. An inverse source problem with multiple frequency data. Comptes rendus Mathematiques, 2011, 349: 855-859 · Zbl 1231.35296 · doi:10.1016/j.crma.2011.07.009 |
| [8] | Bartashevskii E L, Drobakhin O O, Slavin I V. Interpretation of results of multifrequency measurements in cheking multilayer dielectric structures. Measurement Techniques, 1984, 27: 511-514 · doi:10.1007/BF00838753 |
| [9] | Bonnet-Ben Dhia A S, Chorfi L, Dakhia G, et al. Diffraction by a defect in an open waveguide: a mathematical analysis based on a modal radiation condition. SIAM J Appl Math, 2009, 70: 677-693 · Zbl 1193.35018 · doi:10.1137/080740155 |
| [10] | Chen Y, Rokhlin V. On the inverse scattering problem for the Helmholtz equation in one dimension. Inverse Probl, 1992, 8: 365-391 · Zbl 0760.34017 · doi:10.1088/0266-5611/8/3/002 |
| [11] | Ciraolo G, Magnanini R. A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides. Math Methods Appl Sci, 2009, 32: 1183-1206 · Zbl 1168.78306 · doi:10.1002/mma.1084 |
| [12] | Coddington, E. A.; Levinson, N., Theory of Differential Equations (1955) · Zbl 0064.33002 |
| [13] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (1992), Berlin · Zbl 0760.35053 |
| [14] | Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. New York: Springer-Verlag, 1977 · Zbl 0361.35003 · doi:10.1007/978-3-642-96379-7 |
| [15] | Magnanini R, Santosa F. Wave propagation in a 2-D optical waveguide. SIAM J Appl Math, 2000, 61: 1237-1252 · Zbl 0973.78018 |
| [16] | Snyder A W, Love D. Optical Waveguide Theory. London: Chapman & Hall, 1974 |
| [17] | Wilcox C. Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water. Arch Rat Mech Anal, 1975, 60: 259-300 · Zbl 0346.76061 · doi:10.1007/BF01789259 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
