Numerical Method For Solving Inverse Source Problem For Poisson Equation

We consider an inverse problem for the Poisson equation -Delta u = f in the square Omega = (0, 1) x (0, 1) which consists of determining the source f from boundary measurements. Such problem is ill-posed. We restrict ourselves to a class of functions f(x(1), x(2)) = phi(1)(x(2))g(1)(x(1))+ phi(2)(x(2))g(2)(x(1)). To illustrate our method, we first assume that g(1) and g(2) are known functions with partial data at the boundary. For the reconstruction, we consider approximations by the Fourier series, therefore we obtain an ill-posed linear system which requires a regularization strategy. In the general case, we propose an iterative algorithm based on the full data at the boundary. Finally, some numerical results are presented to show the effectiveness of the proposed reconstruction algorithms.