ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Linear Sampling Reconstructions Using a Singular Perturbation Technique

Journal of Applied Nonlinear Dynamics 1(3) (2012) 227--237 | DOI:10.5890/JAND.2012.07.001

Keehwan Kim $^{1}$ , Koung Hee Leem $^{2}$, George Pelekanos $^{2}$

$^{1}$ Department of Mathematics, Yeungnam University, 719-749, Gyeongsangbuk-do, South Korea

$^{2}$ Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, IL 62026, USA

Abstract

In this paper, we propose a new and efficient regularization technique (Singular Perturbation Technique-SPT) for the reconstruction of the shape of a scatterer from far field measurements. The approach used is based on the linear sampling method and uses singular perturbations of the identity along with Neumann series approximations to recover the shape of the unknown object. In comparison with the well established Tikhonov-Morozov regularization techniques our new algorithm appears to be more computationally efficient as it doesn’t require computation of the regularization parameter for each point in the grid. Moreover reconstructions obtained using (SPT) compare well with those obtained using other existing regularization methods.

References

1.  [1] Colton, D. and Kirsch, A. (1996), A simple method for solving the inverse scattering problems in the resonance region, Inverse Problems, 12, 383-393.
2.  [2] Colton, D., Piana, M. and Potthast, R. (1999), A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13, 1477-1493.
3.  [3] Wahba, G. (1990), Spline Models for Observational Data, SIAM, Philadelphia.
4.  [4] Viloche Bazán, F.S. (2008), Fixed-point iterations in determining the Tikhonov regularization parameter, Inverse Problems, 24, 1-15.
5.  [5] Viloche Bazán, F.S. and Francisco, J.B. (2009), An improved Fixed-point algorithm for determining a Tikhonov regularization parameter, Inverse Problems, 25.
6.  [6] Viloche Bazán, F.S., Francisco, J.B., Leem, K.H. and Pelekanos, G. (2012),Maximum product criterion as Tikhonov parameter choice-rule for Kirsch's factorization method, Journal of Computational and Applied Mathematics. Doi:10.1016/j.cam.2012.05.008
7.  [7] Hansen,P.C. (1998). Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia.
8.  [8] Pelekanos, G. and Sevroglou, V. (2006), Shape reconstruction of a 2D-elastic penetrable object via the L-curve method, J. Inv. Ill-Posed Problems, 14, 1-16.
9.  [9] Colton, D. and Kress, R. (1992), Inverse Acoustic and Electromagnetic Scattering Theory, Springer- Verlag, New York.
10.  [10] McGahan, R. and Kleinman, R. (1999), Third annual special session on image reconstruction using real data, part 1. IEEE Antennas Propag. Mag., 41, 34-36.
11.  [11] Caconi, F. and Colton, D. (2003), On the mathematical basis of the linear sampling method, Georg. Math. J., 10, 911-925.
12.  [12] Sergeev, V.O. (1971), Regularization of Voltera equations of the first kind, Dokl. Akad. Nauk SSSR, 197, 531-534.
13.  [13] Denisov, A.M. (1975) , The approximate solution of a Voltera equation of the first kind,USSR Comput. Math. Math. Phys., 15, 237-239.
14.  [14] Lavrent'ev, M.M. (1962),O nekotorykh nekorrektnykh zadachakh matematicheskoui fiziki , Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk. English transl. by Robert J. Sacker: Some improperly posed problems of mathematical physics, Springer-Verlag, Berlin, 1967.
15.  [15] Hörmander, L. (1994), The Analysis of Linear Partial Differential Operators III: Pseudo-differential operators, Springer-Verlag, Berlin.
16.  [16] Pöschel, J. and Trubowitz, E. (1986), Inverse Spectral Theory, Pure and Applied Mathematics, 130, Academic Press, New York.
17.  [17] Leem, K.H., Pelekanos, G. and Viloche Bazán, F.S. (2010), Fixed-point iterations in determining a Tikhonov regularization parameter in Kirsch's factorization method, Applied Math. Comput., 216, 3747- 3753.
18.  [18] Hansen, P.C. (1994), Regularization Tools: A Matlab Package for Analysis and Solution of Discreet Ill-Posed Problems, Numer. Algo., 6, 1-35.