A fast multiscale Galerkin method for the first kind ill‐posed integral equations via Tikhonov regularization

Abstract

In this paper, we develop a fast multiscale Galerkin method to solve the Fredholm integral equations of the first kind via Tikhonov regularization. The method leads to fast solutions of discrete regularization methods. We obtain optimal convergence rates for approximate solutions with an a priori parameter choice and a kind of discrepancy principle. Finally, numerical experiments are given to illustrate the efficiency of the method.

Keywords:

• ill-posed integral equations
• discrete regularization methods
• multiscale Galerkin methods

2000 AMS Subject Classification :

• 65J10
• 65R20

Acknowledgements

Z. Chen and H. Yang was supported in part by the Natural Science Foundation of China under grants 10771224 and 10371137, the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008 and Guangdong Provincial Natural Science Foundation of China under grant 1011170. G. Nelakanti was supported in part by National Board of Higher Mathematics, India, and the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008.