Numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform

Abstract

In this article, we discuss a classical ill-posed problem on numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform (DCT). After implementing an eigenvalue decomposition of the second-order difference matrix in the penalty we obtain an explicit minimizer of the Tikhonov functional in a linear combination of DCT-2 components. The choice of the regularization parameter thus can be properly chosen after calling a single variable bounded nonlinear function minimization. Moreover, we propose two approaches to weaken the Gibbs phenomena appearing near the boundary of the reconstructed derivatives. Several numerical examples are provided to show the computational efficiency of our proposed algorithm. Finally, the extension to the multidimensional numerical differentiation leads the approach to a wider scope.

Keywords:

• (multidimensional) numerical differentiation
• Tikhonov regularization
• parameter choice rules
• Gibbs phenomena

AMS Subject Classifications::

• 65D25
• 65J20

Acknowledgements

We are grateful to anonymous referees for careful reading and helpful comments, as well as pointing out that ref. 16,19 that have improved the presentation of the manuscript essentially. This work is supported by the Shanghai Science and Technology Commission Grant: 11ZR1402800. The second author is a research fellow of the Alexander von Humboldt Foundation. He thanks Nan Chen (Fudan University) and Sergei V Pereverzev (RICAM) for careful reading of the manuscript and for many useful comments which improved the presentation.