Modeling, analysis, and numerical methods for shape identification of moving geometric inverse source and potential of parabolic problems

In this paper, geometric reconstructions of moving discontinuities of source and potential for parabolic problems are studied. We consider modeling, analysis, and numerical method for time-varying shape identification in the unsteady diffusion process. We detect the shape of a space-time tube for the source and potential from exterior boundary measurements. The existence of minimizers for models with perimeter regularization is analyzed. Moving shape sensitivity analysis is performed and a time-varying level set method is developed for numerical reconstruction. Numerical results in 3D and 4D including one time dimension are presented to demonstrate the effectiveness of the algorithm.