Reconstruction of source function in a parabolic equation using partial boundary measurements

In this paper, we present the analytical and numerical study of the optimization approach for determining the space-dependent source function in the parabolic inverse source problem using partial boundary measurements. The Lagrangian approach for the solution of the optimization problem is presented, and optimality conditions are derived. The proof of the Fr\'echet differentiability of the regularized Tikhonov functional and the existence result for the solution of the inverse source problem are established. A local stability estimate for the unknown source term is also presented. The numerical examples justify the theoretical investigations using the conjugate gradient method (CGM) in 2D and 3D tests with noisy data.