A Lie-Group Adaptive Method for Imaging a Space-Dependent Rigidity Coefficient in an Inverse Scattering Problem of Wave Propagation

We are concerned with the reconstruction of an unknown spacedependent rigidity coefficient in a wave equation. This problem is known as one of the inverse scattering problems. Based on a two-point Lie-group equation we develop a Lie-group adaptive method (LGAM) to solve this inverse scattering problem through iterations, which possesses a special character that by using only two boundary conditions and two initial conditions, as those used in the direct problem, we can effectively reconstruct the unknown rigidity function by a self-adaption between the local in time differential governing equation and the global in time algebraic Lie-group equation. The accuracy and efficiency of the present LGAM are assessed by comparing the imaged results with some postulated exact solutions. By means of LGAM, it is quite versatile to handle the wave inverse scattering problem for the image of the rigidity coefficient without needing any extra information from the wave motion.

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