Kim, K.;Leem, K.H.;Pelekanos, G.

  • Received : 2016.01.18
  • Accepted : 2016.01.30
  • Published : 2016.01.30


In this paper, we propose an efficient regularization technique (The Method of Regularized Ratios) for the reconstruction of the shape of a rigid elastic scatterer from far field measurements. The approach used is based on the factorization method and creates via Picard's condition ratios, baptized Regularized Ratios, that serve to effectively remove unwanted singular values that may lead to poor reconstructions. This is achieved through the use of a sophisticated algorithm that progressively adjusts an initially set moderate tolerance. 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.




  1. C.J.S Alves and R. Kress, 2002 On the far field operator in elastic obstacle scattering IMA J. Appl. Math. 67 1-21
  2. K. Anagnostopoulos and A. Charalambopoulos. The linear sampling method for the transmission problem in 2D anisotropic elasticity Inverse Problems (2006) 22, 553-577.
  3. F. Cakoni and D. Colton Qualitative Methods in Inverse Scattering Theory: An Introduction.(Springer-Verlag), 1988.
  4. D. Colton Partial Differential Equations: An Introduction. (New York:Random House, Inc.), 1988.
  5. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. (New York:Springer-Verlag), 1992.
  6. D. Colton and A. Kirsch, A simple method for solving the inverse scattering problems in the resonance region. Inverse Problems (1996) 12, 383-393.
  7. D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems (1999) 13, 1477-93
  8. D. Natrosvilli, Z. Tediashvili, Mixed type direct and inverse scattering problems, in Problems and Methods in Mathematical Physics (eds. Elschner J, Gohberg I and Silbermann B), Operator Theory: Advances and Applications(2001) 121 (Birkhauser, Basel) 366-389
  9. P. C. Hansen Rank-Deficient and Descrete Ill-Posed Problems. SIAM, Philadelphia, 1998.
  10. P. C. Hansen The L-Curve and its use in the numerical treatment of inverse problems. Computational Inverse Problems in Electrocardiology. WIT Press 119-142 (2001)
  11. G. Hu, A. Kirsch and M. Sini, Some Inverse Problem arising from elastic scattering by rigid obstacles, Inverse Problems (2013) 29, 1-21
  12. A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems (1998) 14, 1489-1512.
  13. K.Kim, K.H. Leem and G. Pelekanos An Alternative to Tikhonov Regularization for Linear Sampling Methods, Acta Applicandae Mathematicae (2010) 112, 171-180.
  14. V. D. Kupradze, Dynamical Problems in Elasticity, in "Progress in Solid Mechanics". North-Holland, Amsterdam (1963).
  15. G. Pelekanos and V. Sevroglou, Shape reconstrution of a 2D-elastic penetrable object via the L-curve method. J. Inv. Ill-Posed problems (2006) 14, No.4, 1-16.
  16. Pelekanos G and Sevroglou V 2003 Inverse scattering by penetrable objects in twodimensional elastodynamics J. Comp. Appl. Math. 151 129-140
  17. Sevroglou V and Pelekanos G 2001 An inversion algorithm in two-dimensional elasticity J. Math. Anal. Appl. 263 277-293
  18. Sevroglou V and Pelekanos G 2002 Two dimensional elastic Herglotz functions and their applications in inverse scattering Journal of Elasticity 68 123-144
  19. Sevroglou V 2005 The far-field operator for penetrable and absorbing obstacles in 2D inverse elastic scattering Inverse Problems 17 717-738