Abstract
We give a survey of the mathematical basis of inverse scattering theory, concentrating on the case of time-harmonic acoustic waves. After an introduction and historical remarks, we give an outline of the direct scattering problem. This is then followed by sections on uniqueness results in inverse scattering theory and iterative and decomposition methods to reconstruct the shape and material properties of the scattering object. We conclude by discussing qualitative methods in inverse scattering theory, in particular the linear sampling method and its use in obtaining lower bounds on the constitutive parameters of the scattering object.
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References
Alessandrini, G., Rondi, L.: Determining a sound–soft polyhedral scatterer by a single far–field measurement. Proc. Am. Math. Soc. 133, 1685–1691 (2005)
Arens, T.: Why linear sampling works. Inverse Prob. 20, 163–173 (2004)
Arens, T., Lechleiter, A.: The linear sampling method revisited. J. Integral Eqn. Appl. 21, 179–202 (2009)
Bukhgeim, A.: Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill–Posed Prob. 16, 19–33 (2008)
Cakoni, F., Colton, D.: A uniqueness theorem for an inverse electromagnetic scattering problem in inhomogeneous anisotropic media. Proc. Edinb. Math. Soc. 46, 293–314 (2003)
Cakoni, F., Colton, D.: The determination of the surface impedance of a partially coated obstacle from far field data. SIAM J. Appl. Math. 64, 709–723 (2004)
Cakoni, F., Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin (2006)
Cakoni, F., Colton, D., Haddar, H.: The computation of lower bounds for the norm of the index of refraction in anisotropic media from far field data. J. Integral Eqn. Appl. 21, 203–227 (2009)
Cakoni, F., Colton, D., Haddar, H.: The interior transmission problem for regions with cavities. SIAM J. Math. Anal. 42, 145–162 (2010)
Cakoni, F., Colton, D., Haddar, H.: On the determination of Dirichlet and transmission eigenvalues from far field data. Comput. Rend. Math. 348, 379–383 (2010)
Cakoni, F., Colton, D., Monk, P.: The electromagnetic inverse scattering problem for partly coated Lipschitz domains. Proc. R. Soc. Edinb. 134A, 661–682 (2004)
Cakoni, F., Colton, D., Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM.
Cakoni, F., Fares, M., Haddar, H.: Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects. Inverse Prob. 22, 845–867 (2006)
Cakoni, F., Gintides, D., Haddar, H.: The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42, 237–255 (2010)
Cakoni, F., Haddar, H.: A variational approach for the solution of the electro-magnetic interior transmission problem for anisotropic media. Inverse Prob. Imaging 1, 443–456 (2007)
Cakoni, F., Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Appl. Anal. 89, 29–47 (2010)
Colton, D., Haddar, H.: An application of the reciprocity gap functional to inverse scattering theory. Inverse Prob. 21, 383–398 (2005)
Colton, D., Haddar, H., Monk, P.: The linear sampling method for solving the electromagnetic inverse scattering problem. SIAM J. Sci. Comput. 24, 719–731 (2002)
Colton, D., Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Prob. 12, 383–393 (1996)
Colton, D., Kirsch, A., Päivärinta, L.: Far field patterns for acoustic waves in an inhomogeneous medium. SIAM J. Math. Anal. 20, 1472–1483 (1989)
Colton, D., Kress, R.: Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium. SIAM J. Appl. Math. 55, 1724–1735 (1995)
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)
Colton, D., Kress, R.: On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces. Math. Methods Appl. Sci. 24, 1289–1303 (2001)
Colton, D., Monk, P.: A novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region II. SIAM J. Appl. Math. 26, 506–523 (1986)
Colton, D., Monk, P.: The inverse scattering problem for acoustic waves in an inhomogeneous medium. Quart. J. Mech. Appl. Math. 41, 97–125 (1988)
Colton, D., Monk, P.: Target identification of coated objects. IEEE Trans. Antennas Prop. 54, 1232–1242 (2006)
Colton, D., Päivärinta, L.: The uniqueness of a solution to an inverse scattering problem for electromagnetic waves. Arch. Ration. Mech. Anal. 119, 59–70 (1992)
Colton, D., Päivärinta, L., Sylvester, J.: The interior transmission problem. Inverse Probl. Imaging 1, 13–28 (2007)
Colton, D., Piana, M., Potthast, R.: A simple method using Mozorov’s discrepancy principle for solving inverse scattering problems. Inverse Prob. 13, 1477–1493 (1997)
Colton, D., Sleeman, B.: An approximation property of importance in inverse scattering theory. Proc. Edinburgh. Math. Soc. 44, 449–454 (2001)
Farhat, C., Tezaur, R., Djellouli, R.: On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method. Inverse Prob. 18, 1229–1246 (2002)
Gintides, D.: Local uniqueness for the inverse scattering problem in acoustics via the Faber–Krahn inequality. Inverse Prob. 21, 1195–1205 (2005)
Gylys–Colwell, F.: An inverse problem for the Helmholtz equation. Inverse Prob. 12, 139–156 (1996)
Haddar, H., Monk, P.: The linear sampling method for solving the electromagnetic inverse medium problem. Inverse Prob. 18, 891–906 (2002)
Hähner, P.: A periodic Faddeev–type solution operator. J. Diff. Eqn. 128, 300–308 (1996)
Hähner, P.: On the uniqueness of the shape of a penetrable anisotropic obstacle. J. Comput. Appl. Math. 116, 167–180 (2000)
Hähner, P.: Electromagnetic wave scattering. In: Pike, R., Sabatier, P. (eds.) Scattering. Academic, New York (2002)
Harbrecht, H., Hohage, T.: Fast methods for three-dimensional inverse obstacle scattering problems. J. Integral Eqn. Appl. 19, 237–260 (2007)
Hohage, T.: Iterative methods in inverse obstacle scattering: regularization theory of linear and nonlinear exponentially ill-posed problems. Dissertation, Linz (1999)
Isakov, V.: On the uniqueness in the inverse transmission scattering problem. Commun. Partial Diff. Eqns. 15, 1565–1587 (1988)
Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, Berlin (1996)
Ivanyshyn, O.: Nonlinear boundary integral equations in inverse scattering. Dissertation, Gäottingen (2007)
Ivanyshyn, O., Kress, R.: Nonlinear integral equations in inverse obstacle scattering. In: Fotiatis M (ed) Mathematical Methods in Scattering Theory and Biomedical Engineering. World Scientific, Singapore, pp. 39–50 (2006)
Ivanyshyn, O., Kress, R.: Identification of sound-soft 3D obstacles from phaseless data. Inverse Prob. Imaging 4, 131–149 (2010)
Johansson, T., Sleeman, B.: Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern. IMA J. Appl. Math. 72, 96–112 (2007)
Jones, D.S.: Acoustic and Electromagnetic Waves. Clarendon, Oxford (1986)
Kirsch, A.: The domain derivative and two applications in inverse scattering. Inverse Prob. 9, 81–86 (1993)
Kirsch, A.: Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Prob. 14, 1489–1512 (1998)
Kirsch, A.: Factorization of the far field operator for the inhomogeneous medium case and an application in inverse scattering theory. Inverse Prob. 15, 413–429 (1999)
Kirsch, A.: An integral equation approach and the interior transmission problem for Maxwell’s equations. Inverse Prob. Imaging 1, 159–179 (2007)
Kirsch, A.: On the existence of transmission eigenvalues. Inverse Prob. Imaging 3, 155–172 (2009)
Kirsch, A., Grinberg, N.: The Factorization Method for Inverse Problems. Oxford University Press, Oxford (2008)
Kirsch, A., Kress, R.: An optimization method in inverse acoustic scattering. In: Brebbia, C.A. et al. (eds.) Boundary Elements IX. Fluid Flow and Potential Applications, vol. 3. Springer, Berlin (1987)
Kirsch, A., Kress, R.: Uniqueness in inverse obstacle scattering. Inverse Prob. 9, 285–299 (1993)
Kleinman, R., van den Berg, P.: A modified gradient method for two dimensional problems in tomography. J. Comput. Appl. Math. 42, 17–35 (1992)
Kress, R.: Electromagnetic waves scattering. In: Pike, R., Sabatier, P. (eds.) Scattering. Academic, New York (2002)
Kress, R.: Newton’s Method for inverse obstacle scattering meets the method of least squares. Inverse Prob. 19, 91–104 (2003)
Kress, R., Rundell, W.: Inverse scattering for shape and impedance. Inverse Prob. 17, 1075–1085 (2001)
Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Prob. 21, 1207–1223 (2005)
Langenberg, K.: Applied inverse problems for acoustic, electromagnetic and elastic wave scattering. In: Sabatier, P. (ed.) Basic Methods of Tomography and Inverse Problems. Adam Hilger, Bristol (1987)
Lax, P.D., Phillips, R.S.: Scattering Theory. Academic, New York (1967)
Liu, H.: A global uniqueness for formally determined electromagnetic obstacle scattering. Inverse Prob. 24, 035018 (2008)
Liu, H., Zou, J.: Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers. Inverse Prob. 22, 515–524 (2006)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)
Morse, P.M., Ingard, K.U.: Linear acoustic theory. In: Faugge, S. (ed.) Encyclopedia of Physics. Springer, Berlin (1961)
Mäuller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin (1969)
Nachman, A.: Reconstructions from boundary measurements. Ann. Math. 128, 531–576 (1988)
Nédélec, J.C.: Acoustic and Electromagnetic Equations. Springer, Berlin (2001)
Novikov, R.: Multidimensional inverse spectral problems for the equation \(-\Delta \psi +\) \(\left (v(x)-\right.\) \(\left.Eu(x)\right )\psi = 0\). Trans. Funct. Anal. Appl. 22, 263–272 (1988)
Ola, P., Päivärinta, L., Somersalo, E.: An inverse boundary value problem in electrodynamics. Duke Math. J. 70, 617–653 (1993)
Ola, P., Somersalo, E.: Electromagnetic inverse problems and generalized Sommerfeld potentials. SIAM J. Appl. Math. 56, 1129–1145 (1996)
Päivärinta, L., Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal. 40, 738–753 (2008)
Piana, M.: On uniqueness for anisotropic inhomogeneous inverse scattering problems. Inverse Prob. 14, 1565–1579 (1998)
Potthast, R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Prob. 10, 431–447 (1994)
Potthast, R.: Point-Sources and Multipoles in Inverse Scattering Theory. Chapman and Hall, London (2001)
Potthast, R.: On the convergence of a new Newton-type method in inverse scattering. Inverse Prob. 17, 1419–1434 (2001)
Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Prob. 22, R1–R47 (2006)
Ramm, A.: Recovery of the potential from fixed energy scattering data. Inverse Prob. 4, 877–886 (1988)
Rjasanow, S., Steinbach, O.: The Fast Solution of Boundary Integral Equations. Springer, Berlin (2007)
Roger, R.: Newton Kantorovich algorithm applied to an electromagnetic inverse problem. IEEE Trans. Antennas Prop. 29, 232–238 (1981)
Rondi, L.: Unique determination of non-smooth sound-soft scatterers by finitely many far-field measurements. Indiana Univ. Math. J. 52, 1631–1662 (2003)
Rynne, B.P., Sleeman, B.D.: The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J. Math. Anal. 22, 1755–1762 (1991)
Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Prob. 22, 663–680 (2006)
Serranho, P.: A hybrid method for inverse obstacle scattering problems. Dissertation, Gäottingen (2007)
Serranho, P.: A hybrid method for sound-soft obstacles in 3D. Inverse Prob. Imaging 1, 691–712 (2007)
Stefanov, P., Uhlmann, G.: Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering. Proc. Am. Math. Soc. 132, 1351–1354 (2003)
Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987)
van den Berg, R., Kleinman, R.: A contrast source inversion method. Inverse Prob. 13, 1607–1620 (1997)
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Colton, D., Kress, R. (2015). Inverse Scattering. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0790-8_48
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DOI: https://doi.org/10.1007/978-1-4939-0790-8_48
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