Convergence of Levenberg–Marquardt method for the inverse problem with an interior measurement

The convergence of Levenberg–Marquardt method is discussed for the inverse problem to reconstruct the storage modulus and loss modulus for the so-called scalar model by a single interior measurement. The scalar model is the most simplest model for data analysis used as the modeling partial differential equation in the diagnosing modality called the magnetic resonance elastography which is used to diagnose for instance lever cancer. The convergence of the method is proved by showing that the measurement map which maps the above unknown moduli to the measured data satisfies the so-called the tangential cone condition. The argument of the proof is quite general and in principle can be applied to any similar inverse problem to reconstruct the unknown coefficients of the model equation given as a partial differential equation of divergence form by one single interior measurement. The performance of the method is numerically tested for the two-layered piecewise homogeneous scalar models in a rectangular domain and a circular domain.