A numerical differentiation method and its application to reconstruction of discontinuity

In this paper, we discuss a classical ill-posed problem-numerical differentiation by the Tikhonov regularization. Based on the conditional stability estimate for this ill-posed problem, a new simple method for choosing regularization parameters is proposed. We show that it has an almost optimal convergence rate when the exact solution is in H-2. The advantages of our method are (1) we can get similar computational results with much less computation, in comparison with other methods, and (2) we can find the discontinuous points numerically.