Abstract
With the analysis of the preceding chapters, we now are well prepared for studying inverse acoustic obstacle scattering problems. We recall that the direct scattering problem is, given information on the boundary of the scatterer and the nature of the boundary condition, to find the scattered wave and in particular its behavior at large distances from the scatterer, i.e., its far field. The inverse problem starts from this answer to the direct problem, i.e., a knowledge of the far field pattern, and asks for the nature of the scatterer. Of course, there is a large variety of possible inverse problems, for example, if the boundary condition is known, find the shape of the scatterer, or, if the shape is known, find the boundary condition, or, if the shape and the type of the boundary condition are known for a penetrable scatterer, find the space dependent coefficients in the transmission or resistive boundary condition, etc. Here, following the main guideline of our book, we will concentrate on one model problem for which we will develop ideas which in general can also be used to study a wider class of related problems. The inverse problem we consider is, given the far field pattern for one or several incident plane waves and knowing that the scatterer is sound-soft, to determine the shape of the scatterer. We want to discuss this inverse problem for frequencies in the resonance region, that is, for scatterers D and wave numbers k such that the wavelengths 2π∕k is less than or of a comparable size to the diameter of the scatterer. This inverse problem turns out to be nonlinear and improperly posed. Although both of these properties make the inverse problem hard to solve, it is the latter which presents the more challenging difficulties. The inverse obstacle problem is improperly posed since, as we already know, the determination of the scattered wave u s from a given far field pattern u ∞ is improperly posed. It is nonlinear since, given the incident wave u i and the scattered wave u s, the problem of finding the boundary of the scatterer as the location of the zeros of the total wave u i + u s is nonlinear.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York 1975.
Akduman, I., and Kress, R.: Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape. Radio Science 38, 1055–1064 (2003).
Alessandrini, G., and Rondi, L.: Determining a sound–soft polyhedral scatterer by a single far–field measurement. Proc. Amer. Math. Soc. 133, 1685–1691 (2005).
Altundag, A., and Kress, R.: On a two-dimensional inverse scattering problem for a dielectric. Applicable Analysis 91, 757–771 (2012).
Altundag, A., and Kress, R.: An iterative method for a two-dimensional inverse scattering problem for a dielectric. Jour. on Inverse and Ill-Posed Problem 20, 575–590 (2012).
Alves, C.J.S., and Ha-Duong, T.: On inverse scattering by screens. Inverse Problems 13, 1161–1176 (1997).
Angell, T.S., Colton, D., and Kirsch, A.: The three dimensional inverse scattering problem for acoustic waves. J. Diff. Equations 46, 46–58 (1982).
Angell, T.S., Kleinman, R.E., and Roach, G.F.: An inverse transmission problem for the Helmholtz equation. Inverse Problems 3, 149–180 (1987).
Aramini, R., Caviglia, G., Masa, A., and Piana, M.: The linear sampling method and energy conservation. Inverse Problems 26, 05504 (2010).
Arens, T.: Why linear sampling works. Inverse Problems 20, 163–173 (2004).
Arens, T., and Lechleiter, A.: The linear sampling method revisited. Jour. Integral Equations and Applications 21, 179–202 (2009).
Audibert, L., and Haddar, H.: A generalized formulation of the linear sampling method with exact characterization of targets in terms of far field measurements. Inverse Problems 30, 035011 (2015).
Bellis, C., Bonnet, M., and Cakoni, F.: Acoustic inverse scattering using topological derivative of far-field measurements-based L 2 cost functionals. Inverse Problems 29, 075012 (2013).
Bleistein, N.: Mathematical Methods for Wave Phenomena. Academic Press, Orlando 1984.
Blöhbaum, J.: Optimisation methods for an inverse problem with time-harmonic electromagnetic waves: an inverse problem in electromagnetic scattering. Inverse Problems 5, 463–482 (1989).
Bojarski, N.N.: Three dimensional electromagnetic short pulse inverse scattering. Spec. Proj. Lab. Rep. Syracuse Univ. Res. Corp., Syracuse 1967.
Bojarski, N.N.: A survey of the physical optics inverse scattering identity. IEEE Trans. Ant. Prop. AP-20, 980–989 (1982).
Bourgeois, L., Chaulet, N., and Haddar, H.: Stable reconstruction of generalized impedance boundary conditions. Inverse Problems 27, 095002 (2011).
Bourgeois, L., Chaulet, N., and Haddar, H.: On simultaneous identification of a scatterer and its generalized impedance boundary condition. SIAM J. Sci. Comput. 34, A1824–A1848 (2012).
Bourgeois, L., and Haddar, H.: Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Problems and Imaging 4, 19–38, (2010).
Burger, M.: A level set method for inverse problems. Inverse Problems 17, 1327–1355 (2001).
Cakoni, F., and 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., and Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin 2006.
Cakoni, F., Colton, D., and Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues. SIAM, Philadelphia 2016.
Cakoni, F., Hu, Y., and Kress, R.: Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging. Inverse Problems 30, 105009 (2014).
Cakoni, F., and Kress, R.: Integral equation methods for the inverse obstacle problem with generalized impedance boundary condition. Inverse Problems 29, 015005 (2013).
Catapano, I., Crocco, L., and Isernia, T.: On simple methods for shape reconstruction of unknown scatterers. IEEE Trans. Antennas Prop. 55, 1431–1436 (2007).
Cheng, J., and Yamamoto, M.: Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves. Inverse Problems 19, 1361–1384 (2003).
Colton, D., and Kirsch, A.: Karp’s theorem in acoustic scattering theory. Proc. Amer. Math. Soc. 103, 783–788 (1988).
Colton, D., and Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12, 383–393 (1996).
Colton, D., and Kress, R.: On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces. Math. Methods Applied Science 24, 1289–1303 (2001).
Colton, D., and Kress, R.: Using fundamental solutions in inverse scattering. Inverse Problems 22, R49–R66 (2006).
Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.
Colton, D., and Monk, P.: A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region. SIAM J. Appl. Math. 45, 1039–1053 (1985).
Colton, D., and 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. 46, 506–523 (1986).
Colton, D., and Monk, P.: The numerical solution of the three dimensional inverse scattering problem for time-harmonic acoustic waves. SIAM J. Sci. Stat. Comp. 8, 278–291 (1987).
Colton, D., and Sleeman, B.D.: Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math. 31, 253–259 (1983).
Colton, D., and Sleeman, B.D.: An approximation property of importance in inverse scattering theory. Proc. Edinburgh Math. Soc. 44, 449–454 (2001).
Devaney, A.J.: Mathematical Foundations of Imaging, Tomography and Wavefield Inversion. Cambridge University Press, Cambridge 2012.
Dorn, O., and Lesselier, D.: Level set methods for inverse scattering - some recent developments. Inverse Problems 25, 125001 (2009).
Farhat, C., Tezaur, R., and Djellouli, R.: On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method. Inverse Problems 18, 1229–1246 (2002).
Feijoo, G.R.: A new method in inverse scattering based on the topological derivative. Inverse Problems 20, 1819–1840 (2004).
Gerlach, T. and Kress, R.: Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Problems 12, 619–625 (1996).
Gilbarg, D., and Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin 1977.
Gintides, D.: Local uniqueness for the inverse scattering problem in acoustics via the Faber–Krahn inequality. Inverse Problems 21, 1195–1205 (2005).
Haas, M., and Lehner, G.: Inverse 2D obstacle scattering by adaptive iteration. IEEE Transactions on Magnetics 33, 1958–1961 (1997)
Hanke, M.: Why linear sampling really seems to work. Inverse Problems and Imaging 2, 373–395 (2008).
Hanke, M., Hettlich, F., and Scherzer, O.: The Landweber iteration for an inverse scattering problem. In: Proceedings of the 1995 Design Engineering Technical Conferences, Vol. 3, Part C (Wang et al, eds).
Harbrecht, H., and Hohage. T.: Fast methods for three-dimensional inverse obstacle scattering problems. Jour. Integral Equations and Appl. 19, 237–260 (2007).
Hettlich, F.: On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation. Inverse Problems 10, 129–144 (1994).
Hettlich, F.: Fréchet derivatives in inverse obstacle scattering. Inverse Problems 11, 371–382 (1995).
Hettlich, F.: An iterative method for the inverse scattering problem from sound-hard obstacles. In: Proceedings of the ICIAM 95, Vol. II, Applied Analysis (Mahrenholz and Mennicken, eds). Akademie Verlag, Berlin (1996).
Hettlich, F., and Rundell, W.: A second degree method for nonlinear inverse problem. SIAM J. Numer. Anal. 37, 587–620 (2000).
Hohage, T.: Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem. Inverse Problems 13, 1279–1299 (1997).
Hohage, T.: Iterative Methods in Inverse Obstacle Scattering: Regularization Theory of Linear and Nonlinear Exponentially Ill-Posed Problems. Dissertation, Linz 1999.
Ikehata, M.: Reconstruction of the shape of an obstacle from the scattering amplitude at a fixed frequency. Inverse Problems 14, 949–954 (1998).
Ikehata, M.: Reconstruction of obstacle from boundary measurements. Wave Motion 30, 205–223 (1999).
Imbriale, W.A., and Mittra, R.: The two-dimensional inverse scattering problem. IEEE Trans. Ant. Prop. AP-18, 633–642 (1970).
Isakov, V.: On uniqueness in the inverse transmission scattering problem. Comm. Part. Diff. Equa. 15, 1565–1587 (1990).
Isakov, V.: Inverse Problems for Partial Differential Equations. 2nd ed, Springer, Berlin 2006.
Ivanyshyn, O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems and Imaging 1, 609–622 (2007).
Ivanyshyn, O.: Nonlinear Boundary Integral Equations in Inverse Scattering. Dissertation, Göttingen, 2007.
Ivanyshyn Yaman, O.: Reconstruction of generalized impedance functions for 3D acoustic scattering. J. Comput. Phys., to appear.
Ivanyshyn, O., and Johansson, T.: Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle. J. Integral Equations Appl. 19, 289–308 (2007).
Ivanyshyn, O., and Johansson, T.: A coupled boundary integral equation method for inverse sound-soft scattering. In: Proceedings of waves 2007. The 8th international conference on mathematical and numerical aspects of waves, University of Reading, 153–155 (2007).
Ivanyshyn, O., and Kress, R.: Nonlinear integral equations in inverse obstacle scattering. In: Mathematical Methods in Scattering Theory and Biomedical Engineering, (Fotiatis and Massalas, eds). World Scientific, Singapore, 39–50 (2006).
Ivanyshyn, O., and Kress, R.: Inverse scattering for planar cracks via nonlinear integral equations. Math. Meth. Appl. Sciences 31, 1221–1232 (2007).
Ivanyshyn, O., and Kress, R.: Identification of sound-soft 3D obstacles from phaseless data. Inverse Problems and Imaging 4, 131–149 (2010).
Ivanyshyn, O., and Kress, R.: Inverse scattering for surface impedance from phase-less far field data. J. Comput. Phys. 230, 3443–3452 (2011).
Ivanyshyn, O., Kress, R., and Serranho, P.: Huygens’ principle and iterative methods in inverse obstacle scattering. Adv. Comput. Math. 33, 413–429 (2010).
Johansson, T., and 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).
Karp, S.N.: Far field amplitudes and inverse diffraction theory. In: Electromagnetic Waves (Langer, ed). Univ. of Wisconsin Press, Madison, 291–300 (1962).
Kirsch, A.: The domain derivative and two applications in inverse scattering. Inverse Problems 9, 81–96 (1993).
Kirsch, A.: Characterization of the shape of the scattering obstacle by the spectral data of the far field operator. Inverse Problems 14, 1489–1512 (1998).
Kirsch, A., and Grinberg, N.: The Factorization Method for Inverse Problems. Oxford University Press, Oxford, 2008.
Kirsch, A., and Kress, R.: On an integral equation of the first kind in inverse acoustic scattering. In: Inverse Problems (Cannon and Hornung, eds). ISNM 77, 93–102 (1986).
Kirsch, A., and Kress, R.: An optimization method in inverse acoustic scattering. In: Boundary elements IX, Vol 3. Fluid Flow and Potential Applications (Brebbia, Wendland and Kuhn, eds). Springer, Berlin, 3–18 (1987).
Kirsch, A., and Kress, R.: Uniqueness in inverse obstacle scattering. Inverse Problems 9, 285–299 (1993).
Kirsch, A., Kress, R., Monk, P., and Zinn, A.: Two methods for solving the inverse acoustic scattering problem. Inverse Problems 4, 749–770 (1988).
Kress, R.: Inverse scattering from an open arc. Math. Meth. in the Appl. Sci. 18, 267–293 (1995).
Kress, R.: Integral equation methods in inverse acoustic and electromagnetic scattering. In: Boundary Integral Formulations for Inverse Analysis (Ingham and Wrobel, eds). Computational Mechanics Publications, Southampton, 67–92 (1997).
Kress, R.: Newton’s Method for inverse obstacle scattering meets the method of least squares. Inverse Problems 19, 91–104 (2003).
Kress, R.: Linear Integral Equations. 3rd ed, Springer, Berlin 2014.
Kress, R.: Integral equation methods in inverse obstacle scattering with a generalized impedance boundary condition. In: Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan (Dick, Kuo and Wozniakowski, eds). Springer, New York, 721–740 (2018).
Kress, R.: Nonlocal impedance conditions in direct and inverse obstacle scattering. Inverse Problems 35, 024002 (2019).
Kress, R., and Päivärinta, L.: On the far field in obstacle scattering. SIAM J. Appl. Math. 59, 1413–1426 (1999).
Kress, R., and Rundell, W.: A quasi-Newton method in inverse obstacle scattering. Inverse Problems 10, 1145–1157 (1994).
Kress, R., and Rundell, W.: Inverse obstacle scattering with modulus of the far field pattern as data. In: Inverse Problems in Medical Imaging and Nondestructive Testing (Engl, Louis and Rundell , eds). Springer, Wien, 75–92 (1997).
Kress, R., and Rundell, W.: Inverse obstacle scattering using reduced data. SIAM J. Appl. Math. 59, 442–454 (1999).
Kress, R., and Rundell, W.: Inverse scattering for shape and impedance. Inverse Problems 17, 1075–1085 (2001).
Kress, R., and Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Problems 21, 1207–1223 (2005).
Kress, R., and Rundell, W.: Inverse scattering for shape and impedance revisited. Jour. Integral Equations and Appl. 30, 293–311 (2018).
Kress, R., and Serranho, P.: A hybrid method for two-dimensional crack reconstruction. Inverse Problems 21, 773–784 (2005)
Kress, R., and Serranho, P.: A hybrid method for sound-hard obstacle reconstruction. J. Comput. Appl. Math. 24, 418–427 (2007).
Kress, R., Tezel, N., and Yaman, F.: A second order Newton method for sound soft inverse obstacle scattering. Jour. Inverse and Ill-Posed Problems 17, 173–185 (2009).
Kress, R., and Zinn, A.: On the numerical solution of the three dimensional inverse obstacle scattering problem. J. Comp. Appl. Math. 42, 49–61 (1992).
Langenberg, K.J.: Applied inverse problems for acoustic, electromagnetic and elastic wave scattering. In: Basic Methods of Tomography and Inverse Problems (Sabatier, ed). Adam Hilger, Bristol and Philadelphia, 127–467 (1987).
Lax, P.D., and Phillips, R.S.: Scattering Theory. Academic Press, New York 1967.
Lee, K.M.: Inverse scattering via nonlinear integral equations for a Neumann crack. Inverse Problems 22, 1989–2000 ( 2006).
Leis, R.: Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York 1986.
Le Louër, F., and Rapún, M.-L.: Topological sensitivity for solving inverse multiple scattering problems in 3D electromagnetism. Part I : One step method. SIAM J. Imaging Sci. 10, 1291–1321 (2017).
Le Louër, F., and Rapún, M.-L.: Topological sensitivity for solving inverse multiple scattering problems in 3D electromagnetism. Part II : Iterative method. SIAM J. Imaging Sci. 11, 734–769 (2018).
Liu, C.: Inverse obstacle problem: local uniqueness for rougher obstacles and the identification of a ball. Inverse Problems 13, 1063–1069 (1997).
Liu, H., and Zou, J.: Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers. Inverse Problems 22, 515–524 (2006).
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge 2000.
Mönch, L.: A Newton method for solving the inverse scattering problem for a sound-hard obstacle. Inverse Problems 12, 309–323 (1996).
Mönch, L.: On the inverse acoustic scattering problem from an open arc: the sound-hard case. Inverse Problems 13, 1379–1392 (1997).
Moré, J.J.: The Levenberg–Marquardt algorithm, implementation and theory. In: Numerical analysis (Watson, ed). Springer Lecture Notes in Mathematics 630, Berlin, 105–116 (1977).
Natterer, F.: The Mathematics of Computerized Tomography. Teubner, Stuttgart and Wiley, New York 1986.
Osher, S., and Sethian, J. A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988).
Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin 1984.
Potthast, R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Problems 10, 431–447 (1994).
Potthast, R.: Fréchet Differenzierbarkeit von Randintegraloperatoren und Randwertproblemen zur Helmholtzgleichung und den zeitharmonischen Maxwellgleichungen. Dissertation, Göttingen 1994.
Potthast, R.: Domain derivatives in electromagnetic scattering. Math. Meth. in the Appl. Sci. 19, 1157–1175 (1996).
Potthast, R.: A fast new method to solve inverse scattering problems. Inverse Problems 12, 731–742 (1996).
Potthast, R.: A point-source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA J. Appl. Math 61, 119–140 (1998).
Potthast, R.: Stability estimates and reconstructions in inverse acoustic scattering using singular sources. J. Comp. Appl. Math. 114, 247–274 (2000).
Potthast, R.: On the convergence of a new Newton-type method in inverse scattering. Inverse Problems 17, 1419–1434 (2001).
Potthast, R.: Point-Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall, London 2001.
Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Problems 22, R1–R47 (2006).
Potthast, R. and Schulz, J.: A multiwave range test for obstacle reconstructions with unknown physical properties. J. Comput. Appl. Math. 205, 53–71 (2007).
Ramm, A.G.: Scattering by Obstacles. D. Reidel Publishing Company, Dordrecht 1986.
Roger, A.: Newton Kantorovich algorithm applied to an electromagnetic inverse problem. IEEE Trans. Ant. Prop. AP-29, 232–238 (1981).
Santosa, F.: A level set approach for inverse problems involving obstacles. ESAIM Control Optim. Calculus Variations 1, 17–33 (1996).
Schormann, C.: Analytische und numerische Untersuchungen bei inversen Transmissionsproblemen zur zeitharmonischen Wellengleichung. Dissertation, Göttingen 2000.
Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Problems 22, 663–680 (2006).
Serranho, P.: A hybrid method for sound-soft obstacles in 3D. Inverse Problems and Imaging 1, 691–712 (2007).
Sleeman, B. D.: The inverse problem of acoustic scattering. IMA. J. Appl. Math 29, 113–142 (1982).
Sokolowski, J., and Zochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim., 37, 1251–1272 (1999).
Stefanov, P., and Uhlmann, G.: Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering. Proc. Amer. Math. Soc. 132, 1351–1354 (2003).
Treves, F.: Basic Linear Partial Differential Equations. Academic Press, New York 1975.
Zinn, A.: On an optimisation method for the full- and limited-aperture problem in inverse acoustic scattering for a sound-soft obstacle. Inverse Problems 5, 239–253 (1989).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Colton, D., Kress, R. (2019). Inverse Acoustic Obstacle Scattering. In: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-030-30351-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-30351-8_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-30350-1
Online ISBN: 978-3-030-30351-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)