In this dissertation, we solve two inverse problems. The first one is the inverse source problem for the Helmholtz equation that governs the wave propagating in anisotropic media. The second one is to recover the initial condition for parabolic equations from the lateral Cauchy data.Regarding to the first problem, we propose a numerical method to compute a source function from the external measurement of the wave field generated by that source. We derive an equation which is independent of the unknown source. However, this equation is not a standard partial differential equation. A method to solve it is not yet available. By truncating the Fourier series of the wavefield with respect to a special basis, we can approximate that equation by a system of elliptic partial differential equations. The solution to this "approximate'' system directly yields the desired source function. We solve that system of elliptic equations by the quasi-reversibility method. The convergence of this method is proved in this dissertation via a new Carleman estimate. Regarding to the second problem, we find the initial condition for parabolic equations from the Cauchy lateral data of their solutions. We employ a technique similar to the one mentioned in the previous paragraph. We split our method into two stages. In the first stage, we establish an additional equation for the solution to the parabolic equation. Solving this equation is challenging. The theory to solve it is not yet available. Hence, in the second stage, we approximate this equation by an elliptic system. This system is solved by the quasi-reversibility method. The convergence of the quasi-reversibility method as the measurement noise goes to zero is proved. We present the implementation of our algorithm in details and verify our method by showing some numerical examples. The convergence of the quasi-reversibility method in both problems are proved using Carlerman estimates. These estimates are discussed in this dissertation.