Optimization Method for Estimating the Inverse Source Term in Elliptic Equation

In this work, we investigate an inverse source problem for determining the unknown source term in one and two dimensional space of a linear elliptic equation. First, the inverse problem is formulated into an optimization problem with the Tikhonov regularization method. Then, the existence and uniqueness of the solution for the direct problem are proved. Second, the existence of the optimal solution is proved. Then, the convexity of the optimization problem is shown in order to ensure the uniqueness of the optimal solution. Moreover, the conjugate gradient method is applied to reconstruct the source term. Finally, to show the efficiency of the suggested approach, we give some numerical results in one and two dimensional space.