A class of a posteriori parameter choice rules for filter-based regularization schemes

Regularization is a method for providing a stable approximate solution to ill-posed operator equations, and it involves the regularization parameter which plays an important role in the convergence of the method. In this article, we propose a class of a posteriori parameter choice rules for filter-based regularization methods and establish the optimal rate of convergence \(O(\delta ^{\frac{\nu }{\nu +1}})\) $O({\delta }^{\frac{\nu }{\nu +1}})$ from the proposed rules. We study these methods along with the proposed parameter choice rules in the context of pseudo-differential operator equations as well as the analytic continuation problem. The numerical implementation of the pseudo-differential operator equation and analytic continuation problem is discussed.