中文摘要：

考虑一个由函数的测量数据求解其二阶导数的数值微分问题。这是一个经典的不适定问题，测量数据的微小扰动将引起其导数的急剧变化。将该问题表示为第一类的积分方程，并引入Lavrentiev正则化方法对其进行求解，获得了二阶数值微分的稳定化算法。另外，基于积分方程算子的性质，进一步给出了正则化解的收敛性以及正则化参数的选取策略。

英文摘要：

Consider a numerical differential problem, which aims to compute the second order derivative of a function from its measured data in this paper. This is a classical ill - posed problem, which means that a small perturbation in the measured data will cause a huge error in its derivative. By expressing this problem as an integral equation of the first kind and solving it by the Lavrentiev regularization method, we get a stable algorithm of the second order numerical derivatives. Moreover, the convergence of the regularizing solution and the choosing strategy of the regularization parameter are presented based on the properties of the integral equation operator.