This work addresses inverse problems encountered in the mathematical modeling of thermal conduction, with a focus on the Cauchy problem, a classic example of an ill-posed inverse problem. Due to uncertainties in conditions and parameters, reconstructing inaccessible information from partial data is often unstable and computationally expensive. As a result, a regularization approach is essential. We propose a numerical method to stabilize the problem by reformulating it as an optimal control problem with regularization. To overcome the limitations of first-order methods and efficiently handle large-scale data, we introduce the Limited Memory BFGS optimization algorithm to ensure stability and convergence. Numerical experiments in two dimensions are performed using the finite element method to approximate the direct and adjoint problems at each iteration.