Huygens' principle and iterative methods in inverse obstacle scattering

Advances in Computational Mathematics

Volumn 33, Issue 4, 2010, Pages 413-429

  Ivanyshyn, Olha ^a   Kress, Rainer ^a   Serranho, Pedro ^b,c  

^a UNIVERSITY OF GÖTTINGEN   (Germany)

^b INSTITUTO SUPERIOR TÉCNICO   (Portugal)

^c POLYTECHNIC INSTITUTE OF LEIRIA   (Portugal)

Author keywords

Inverse scattering; Nonlinear integral equations; Sound soft obstacle

Indexed keywords

EID: 77957716714     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-009-9135-6     Document Type: Article

References (40)

1. Cakoni, F., Colton, D.: Qualitative Methods in Inverse Scattering Theory: An Introduction. Springer, New York (2005).
2. Colton, D., Coyle, J., Monk, P.: Recent developments in inverse acoustic scattering theory. SIAM Rev. 42, 369-414 (2000).
3. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998).
4. Colton, D., Kress, R.: Using fundamental solutions in inverse scattering. Inverse Probl. 22, R49-R66 (2006).
5. Colton, D., Sleeman, B. D.: Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math. 31, 253-259 (1983).
6. 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).
7. Engl, H. W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996).
8. 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).
9. Ganesh, M., Graham, I.: A high-order algorithm for obstacle scattering in three dimensions. J. Comput. Phys. 198, 211-242 (2004).
10. Gintides, D.: Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality. Inverse Probl. 21, 1195-1205 (2005).
11. Harbrecht, H., Hohage, T.: Fast methods for three-dimensional inverse obstacle scattering problems. J. Integral Equ. Appl. 19, 237-260 (2007).
12. Hettlich, F.: Frechet derivatives in inverse obstacle scattering. Inverse Probl. 14, 209-210 (1998).
13. 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).
14. Hohage, T.: Iterative methods in inverse obstacle scattering: regularization theory of linear and nonlinear exponentially ill-posed problems. Dissertation, Linz (1999).
15. Ivanyshyn, O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Probl. Imaging 1, 609-622 (2007).
16. Ivanyshyn, O.: Nonlinear boundary integral equations in inverse scattering. Dissertation Göttingen (2007).
17. Ivanyshyn, O., Johansson, T.: Boundary integral equations for acoustical inverse sound-soft scattering. J. Inverse Ill-Posed Probl. 15, 1-14 (2007).
18. 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).
19. 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).
20. 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).
21. 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).
22. Ivanyshyn, O., Kress, R.: Inverse scattering for planar cracks via nonlinear integral equations. Math. Methods Appl. Sci. 31, 1221-1232 (2008).
23. 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).
24. 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).
25. 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).
26. Kress, R.: Acoustic scattering: scattering by obstacles. In: Pike, R., Sabatier, P. (eds.) Scattering, pp. 52-73. Academic, London (2001).
27. Kress, R.: Newton's method for inverse obstacle scattering meets the method of least squares. Inverse Probl. 19, 91-104 (2003).
28. Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21, 1207-1223 (2005).
29. Kress R., Serranho P.: A hybrid method for two-dimensional crack reconstruction. Inverse Probl. 21, 773-784 (2005).
30. Kress R., Serranho P.: A hybrid method for sound-hard obstacle reconstruction. J. Comput. Appl. Math. 204, 418-427 (2007).
31. Lee, Kuo-Ming: Inverse scattering via nonlinear integral equations for a Neumann crack Inverse Probl. 22, 1989-2000 (2006).
32. Potthast, R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Probl. 10, 431-447 (1994).
33. Potthast, R.: Point-Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall, London (2001).
34. Potthast, R.: On the convergence of a new Newton-type method in inverse scattering. Inverse Probl. 17, 1419-1434 (2001).
35. Potthast, R.: Sampling and probe methods-an algorithmical view. Computing 75, 215-235 (2005).
36. Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Probl. 22, R1-R47 (2006).
37. Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Probl. 22, 663-680 (2006).
38. Serranho, P.: A hybrid method for sound-soft obstacles in 3D. Inverse Probl. Imaging 1, 691-712 (2007).
39. Serranho, P.: A hybrid method for inverse obstacle scattering problems. Dissertation Göttingen (2007).
40. 3. Dissertation Göttingen (1990).