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In this paper a nonlinear inversion method is presented for determining the mass density of an elastic inclusion from the knowledge of how the inclusion scatters known incident elastic waves. The algorithm employed is an extension of the multiplicative regularized contrast source inversion method (MR-CSI) to elasticity. This method involves alternate determination of the mass density contrast and the contrast sources (the product of the contrast and the fields) in each iterative step. The simple updating schemes of the method allow the introduction of an extra regularization term to the cost functional as a multiplicative constraint. This so-called MR-CSI method (MR-CSI) has been proven to be very effective for the acoustic and electromagnetic inverse scattering problems. Numerical examples demonstrate that the MR-CSI method shows excellent edge preserving properties by robustly handling noisy data very well, even for more complicated elastodynamic problems.

In this paper scattering problems for the rigid body and the cavity in twodimensional linear elasticity are considered. In each case the corresponding far-field scattering amplitudes are presented and the Herglotz condition and Herglotz wavefunctions are introduced. A pair of integral equations are constructed in the far-field region. The properties of the Herglotz functions are used to derive solvability conditions and to built approximate far-field equations. A method for solving inverse scattering problems is proposed, and the support of the scattering obstacle is found by noting the unboundedness of the L 2 -norm of the Herglotz densities as an interior point approaches the boundary of the scattering object from inside the scatterer. Illustration of the unboundedness property on the boundary is carried out for rigid circular cylinders and cavities. Numerical results for rigid bodies are also given, showing the applicability of this method.  2001 Academic Press

In this paper, we propose and analyze two new direct factorization methods for solving inverse scattering problems. Both direct factorization methods are built upon the mathematically justified factorization method developed by Kirsch. The first one is naturally derived from a recent direct sampling method by replacing the corresponding far-field operator F in the indicator function by the factorized far-field operator (F * F) 1/4 . The second one is based on a truncated Neumann series approximation of the inverse of an appropriately scaled factorized far-field operator. Both direct factorization methods are shown to be stable with respect to noise and mathematically equivalent. Numerical results with both synthetic and real experimental data are presented to illustrate the promising accuracy of our proposed direct factorization methods in comparison to Kirsch's factorization method and the direct sampling method.

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In this paper we discuss an inversion algorithm which combined with the L-curve criterion for the selection of the regularization parameter, effectively yields shape reconstructions of penetrable obstacles. The required scattered elastic field is generated by either a P or S-incident wave. In particular the improved variant of the linear sampling method (LSM), the so called (F * F ) 1/4 -method is studied for the two dimensional elastic transmission case. For our reconstructions we assume that the far field data are noisy and we employ the L-curve for the selection of the regularization parameter. The location of the vertex of the L-curve yields an appropriate value of the regularization parameter. Furthermore, the L-curve approach does not require a priori knowledge of the noise level, and hence combined with the LSM can be used for real world reconstructions, in which noise in the data is unknown.

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