  A new numerical method to solve an inverse source problem for the radiative
transfer equation involving the absorption and scattering terms, with
incomplete data, is proposed. No restrictive assumption on those absorption and
scattering coefficients is imposed. The original inverse source problem is
reduced to boundary value problem for a system of coupled partial differential
equations of the first order. The unknown source function is not a part of this
system. Next, we write this system in the fully discrete form of finite
differences. That discrete problem is solved via the quasi-reversibility
method. We prove the existence and uniqueness of the regularized solution.
Especially, we prove the convergence of regularized solutions to the exact one
as the noise level in the data tends to zero via a new discrete Carleman
estimate. Numerical simulations demonstrate good performance of this method
even when the data is highly noisy.
