Numerical recovery of unknown source in a kinetic equation with absorption and scattering via an interior measurement

This study addresses an inverse source problem for a general stationary kinetic equation involving absorption and scattering terms. The primary objective is the simultaneous reconstruction of the phase space particle distribution and an unknown source in the model, utilizing boundary conditions defined by derivatives and additional interior measurements. In the theoretical part of the study, the uniqueness of the solution to the inverse problem is proved. For the numerical solution, a comprehensive computational framework is developed based on finite difference approximations for derivatives and the trapezoidal rule for the discretization of the integral terms. The reconstruction strategy employs bilinear interpolation techniques coupled with Tikhonov regularization. In addition, a Monte Carlo noise sensitivity analysis is carried out to assess the stability and robustness of the proposed method under perturbed data. Finally, the effectiveness of the method is illustrated through numerical examples.