Abstract
The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.
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References
Cakoni, F., Colton, D.: Qualitative Methods in Inverse Scattering Theory: An Introduction. Springer, New York (2005)
Colton, D., Coyle, J., Monk, P.: Recent developments in inverse acoustic scattering theory. SIAM Rev. 42, 369–414 (2000)
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)
Colton, D., Kress, R.: Using fundamental solutions in inverse scattering. Inverse Probl. 22, R49–R66 (2006)
Colton, D., Sleeman, B.D.: Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math. 31, 253–259 (1983)
Deuflhard, P., Engl, H.W., Scherzer, O.: A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Probl. 14, 1081–1106 (1998)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996)
Farhat, C., Tezaur, R., Djellouli, R.: On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method. Inverse Probl. 18, 1229–1246 (2002)
Ganesh, M., Graham, I.: A high-order algorithm for obstacle scattering in three dimensions. J. Comput. Phys. 198, 211–242 (2004)
Gintides, D.: Local uniqueness for the inverse scattering problem in acoustics via the Faber–Krahn inequality. Inverse Probl. 21, 1195–1205 (2005)
Harbrecht, H., Hohage, T.: Fast methods for three-dimensional inverse obstacle scattering problems. J. Integral Equ. Appl. 19, 237–260 (2007)
Hettlich, F.: Frechet derivatives in inverse obstacle scattering. Inverse Probl. 14, 209–210 (1998)
Hohage, T.: Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem. Inverse Probl. 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)
Ivanyshyn, O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Probl. Imaging 1, 609–622 (2007)
Ivanyshyn, O.: Nonlinear boundary integral equations in inverse scattering. Dissertation Göttingen (2007)
Ivanyshyn, O., Johansson, T.: Boundary integral equations for acoustical inverse sound-soft scattering. J. Inverse Ill-Posed Probl. 15, 1–14 (2007)
Ivanyshyn, O., Johansson, T.: Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle. J. Integral Equ. Appl. 19, 289–308 (2007)
Ivanyshyn, O., 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, pp. 153–155. University of Reading (2007)
Ivanyshyn, O., Johansson, B.T.: Reconstruction of acoustically sound-hard obstacles from the far field using a boundary integral equation method. In: Charalambopoulos, A., Fotiadis, D.I., Polyzos, D. (eds.) Advanced Topics in Scattering and Biomedical Engineering, pp. 47–56 (2008)
Ivanyshyn, O., Kress, R.: Nonlinear integral equations in inverse obstacle scattering. In: Fotiatis, D.I., Massalas, C.V. (eds.) Mathematical Methods in Scattering Theory and Biomedical Engineering, pp. 39–50. World Scientific, Singapore (2006)
Ivanyshyn, O., Kress, R.: Inverse scattering for planar cracks via nonlinear integral equations. Math. Methods Appl. Sci. 31, 1221–1232 (2008)
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)
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, pp. 3–18. Springer, Berlin (1987)
Kress, R.: Integral equation methods in inverse acoustic and electromagnetic scattering. In: Ingham, D.B., Wrobel, L.C. (eds.) Boundary Integral Formulations for Inverse Analysis, pp. 67–92. Computational Mechanics Publications, Southampton (1997)
Kress, R.: Acoustic scattering: scattering by obstacles. In: Pike, R., Sabatier, P. (eds.) Scattering, pp. 52–73. Academic, London (2001)
Kress, R.: Newton’s method for inverse obstacle scattering meets the method of least squares. Inverse Probl. 19, 91–104 (2003)
Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21, 1207–1223 (2005)
Kress R., Serranho P.: A hybrid method for two-dimensional crack reconstruction. Inverse Probl. 21, 773–784 (2005)
Kress R., Serranho P.: A hybrid method for sound-hard obstacle reconstruction. J. Comput. Appl. Math. 204, 418–427 (2007)
Lee, Kuo-Ming: Inverse scattering via nonlinear integral equations for a Neumann crack Inverse Probl. 22, 1989–2000 (2006)
Potthast, R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Probl. 10, 431–447 (1994)
Potthast, R.: Point-Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall, London (2001)
Potthast, R.: On the convergence of a new Newton-type method in inverse scattering. Inverse Probl. 17, 1419–1434 (2001)
Potthast, R.: Sampling and probe methods—an algorithmical view. Computing 75, 215–235 (2005)
Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Probl. 22, R1–R47 (2006)
Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Probl. 22, 663–680 (2006)
Serranho, P.: A hybrid method for sound-soft obstacles in 3D. Inverse Probl. Imaging 1, 691–712 (2007)
Serranho, P.: A hybrid method for inverse obstacle scattering problems. Dissertation Göttingen (2007)
Wienert, L.: Die numerische Approximation von Randintegraloperatoren für die Helmholtzgleichung im IR3. Dissertation Göttingen (1990)
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Communicated by the guest editors Benny Hon, Jin Cheng and Masahiro Yamamoto.
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Ivanyshyn, O., Kress, R. & Serranho, P. Huygens’ principle and iterative methods in inverse obstacle scattering. Adv Comput Math 33, 413–429 (2010). https://doi.org/10.1007/s10444-009-9135-6
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DOI: https://doi.org/10.1007/s10444-009-9135-6