The parameter choice rules for weighted Tikhonov regularization scheme

The well-known approach to solve the ill-posed problem is Tikhonov regularization scheme. But, the approximate solution of Tikhonov scheme may not contain all the details of the exact solution. To circumference this problem, weighted Tikhonov regularization has been introduced. In this article, we propose two a posteriori parameter choice rules to choose the regularization parameter for weighted Tikhonov regularization and establish the optimal rate of convergence $O\left({\delta }^{\frac{\alpha +1}{\alpha +2}}\right)$ for the scheme based on these proposed rules. The numerical results are documented to demonstrate the significance of the theoretical results.