A numerical differentiation method based on legendre expansion with super order Tikhonov regularization

The aim of this paper is to develop a method based on Legendre expansion to compute numerical derivatives of a function from its perturbed data. The Tikhonov regularization combined with a new penalty term is used to deal with the ill posed-ness of the problem. It has been shown that the solution process is uniform for various smoothness of functions. Moreover, the convergence rates can be obtained self-adaptively when we choose the regularization parameter by a discrepancy principle. Numerical tests show that the method gives good results. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

The paper introduces a method based on Legendre expansion to calculate numerical derivatives of a function from perturbed data, utilizing Tikhonov regularization and a new penalty term to address the ill-posedness of the problem. It has been demonstrated that the solution process is consistent across various levels of function smoothness. Additionally, self-adaptive convergence rates can be achieved by selecting the regularization parameter through a discrepancy principle. Numerical tests indicate that the method produces satisfactory results.