Stable numerical differentiation for the second order derivatives

Consider a numerical differential problem, which aims to compute the second order derivative of a function stably from its given noisy data. For this ill-posed problem, we introduce the Lavrent′ev regularization scheme by reformulating this differentiation problem as an integral equation of the first kind. The advantage of this proposed scheme is that we can give the regularizing solution by an explicit integral expression, therefore it is easy to be implemented. The a-priori and a-posterior choice strategies for the regularization parameter are considered, with convergence analysis and error estimate of the regularizing solution for noisy data based on the integral operator decomposition. The validity of the proposed scheme is shown by several numerical examples.