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The convergence of Levenberg-Marquard method is discussed for the inverseproblem to reconstruct the storage modulus and loss modulus for the so calledscalar model by single interior measurement. The scalar model is the mostsimplest model for data analysis used as the modeling partial differentialequation in the diagnosing modality called the magnetic resonance elastographywhich is used to diagnose for instance lever cancer. The convergence of themethod is proved by showing that the measurement map which maps the aboveunknown moduli to the measured data satisfies the so called the tangential conecondition. The argument of the proof is quite general and in principle can beapplied to any similar inverse problem to reconstruct the unknown coefficientsof the model equation given as a partial differential equation of divergenceform by one single interior measurement. The performance of the method isnumerically tested for the two layered picewise homogneneous scalar model in arectangular domain.

Convergence of Lebenberg-Marquard method for the Inverse Problem with an Interior Measurement