×

zbMATH — the first resource for mathematics

On imaging obstacles inside inhomogeneous media. (English) Zbl 1131.78004
Summary: We show that an obstacle inside a known inhomogeneous medium can be determined from measurements of the scattering amplitude at one frequency, without a priori knowledge of the boundary condition. We also show that an obstacle inside a known inhomogeneous anisotropic conducting medium can be determined from electrostatic current and voltage measurements on the boundary of a domain containing the obstacle. Moreover, two obstacles with boundary measurements which are merely comparable as operators must be identical. The first part of the paper gives an extension of the factorization method which may be of independent interest and also yields a new reconstruction procedure.

MSC:
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. sc. norm. super. Pisa cl. sci. (4), 2, 151-218, (1975) · Zbl 0315.47007
[2] Brühl, M., Explicit characterization of inclusions in electrical impedance tomography, SIAM J. math. anal., 32, 6, 1327-1341, (2001) · Zbl 0980.35170
[3] Colton, D.; Kress, R., Inverse acoustic and electromagnetic scattering theory, (1992), Springer-Verlag Berlin · Zbl 0760.35053
[4] Costabel, M., Boundary integral operators on Lipschitz domains: elementary results, SIAM J. math. anal., 19, 3, 613-626, (1988) · Zbl 0644.35037
[5] Gâtel, Y.; Yafaev, D., On solutions of the Schrödinger equation with radiation conditions at infinity: the long-range case, Ann. inst. Fourier (Grenoble), 49, 5, 1581-1602, (1999) · Zbl 0939.35050
[6] Gebauer, B., The factorization method for real elliptic problems, Z. anal. anwend., 25, 81-102, (2006) · Zbl 1091.35115
[7] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equation of second order, (1977), Springer-Verlag Berlin · Zbl 0691.35001
[8] Grinberg, N., The operator factorization method in inverse obstacle scattering, Integral equations operator theory, 54, 333-348, (2006) · Zbl 1091.78008
[9] Grinberg, N.; Kirsch, A., The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. inverse ill-posed probl., 10, 171-185, (2002) · Zbl 0999.35108
[10] Hörmander, L., The analysis of linear partial differential operators III, (1985), Springer-Verlag Berlin
[11] Hyvönen, N., Characterizing inclusions in optical tomography, Inverse probl., 20, 737-751, (2004) · Zbl 1065.65146
[12] Isakov, V.; Nachman, A., Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. amer. math. soc., 347, 9, 3375-3390, (1995) · Zbl 0849.35148
[13] Jerison, D.; Kenig, C., Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of math. (2), 121, 463-494, (1985) · Zbl 0593.35119
[14] Kaipio, J.; Somersalo, E., Statistical and computational inverse problems, Appl. math. sci., vol. 160, (2005), Springer-Verlag New York · Zbl 1068.65022
[15] Kato, T., Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. pure appl. math., 12, 403-425, (1959) · Zbl 0091.09502
[16] Kirsch, A., An introduction to the mathematical theory of inverse problems, (1996), Springer-Verlag Berlin · Zbl 0865.35004
[17] Kirsch, A., Characterization of the scattering obstacle by the spectral data of the far field operator, Inverse problems, 14, 1489-1512, (1998) · Zbl 0919.35147
[18] Kirsch, A., New characterizations of solutions in inverse scattering theory, Appl. anal., 76, 319-350, (2000) · Zbl 1028.35167
[19] Kirsch, A., The factorization method for a class of inverse elliptic problems, Math. nachr., 278, 3, 258-277, (2005) · Zbl 1067.35148
[20] Kirsch, A.; Päivärinta, L., On recovering obstacles inside inhomogeneities, Math. meth. appl. sci., 21, 619-651, (1998) · Zbl 0915.35116
[21] Lax, P.D.; Phillips, R.S., Scattering theory, (1967), Academic Press New York · Zbl 0214.12002
[22] Majda, A., A representation formula for the scattering operator and the inverse problem for arbitrary bodies, Comm. pure appl. math., 30, 165-194, (1977) · Zbl 0335.35076
[23] McLean, W., Strongly elliptic systems and boundary integral equations, (2000), Cambridge Univ. Press Cambridge · Zbl 0948.35001
[24] Nachman, A., Inverse scattering at fixed energy, () · Zbl 0947.81562
[25] Nachman, A., Global uniqueness for a two-dimensional boundary value problem, Ann. of math. (2), 143, 71-96, (1996) · Zbl 0857.35135
[26] Reed, M.; Simon, B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, (1975), Academic Press New York · Zbl 0308.47002
[27] Reed, M.; Simon, B., Methods of modern mathematical physics. IV. analysis of operators, (1978), Academic Press New York · Zbl 0401.47001
[28] Schechter, M., Spectra of partial differential operators, (1971), North-Holland Amsterdam · Zbl 0225.35001
[29] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta math., 3, 247-302, (1964) · Zbl 0128.09101
[30] Simon, B., Schrödinger semigroups, Bull. amer. math. soc. (N.S.), 7, 3, 447-526, (1982) · Zbl 0524.35002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.