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<abstract><p>The present study focuses on reconstructing the Young's modulus for the elasticity imaging inverse problem. It is a very interesting and challenging problem encountered in tumor detection where the variation of the elastic properties of soft tissues allows to distinguish between normal and diseased tissues. The Levenberg-Marquardt method is used to treat this ill-posed inverse problem and the non-convex minimization is changed into a convex one. We get an explicit expression for computing the descent direction. The proposed technique with a constant and space dependant coefficients and for various real materials is examined. The obtained results of the 2D and 3D view for the reconstructed Young's modulus are agree with those of the exact coefficients. The proposed algorithm is implemented for different levels of noise in the data.</p></abstract>