University of MaraghehSahand Communications in Mathematical Analysis2322-580718220210501Using Frames in Steepest Descent-Based Iteration Method for Solving Operator Equations9710924407110.22130/scma.2020.123786.771ENHassanJamaliDepartment of Mathematics, Faculty of Mathematics and Computer Sciences, Vali-e-Asr University of Rasanjan, Rafsanjan, Iran.MohsenKolahdouzDepartment of Mathematics, Faculty of Mathematics and Computer Sciences, Vali-e-Asr University of Rasanjan, Rafsanjan, Iran.Journal Article20200330In this paper, by using the concept of frames, two iterative methods are constructed to solve the operator equation $ Lu=f $ where $ L:Hrightarrow H $ is a bounded, invertible and self-adjoint linear operator on a separable Hilbert space $ H $. These schemes are analogous with steepest descent method which is applied on a preconditioned equation obtained by frames instead. We then investigate their convergence via corresponding convergence rates, which are formed by the frame bounds. We also investigate the optimal case, which leads to the exact solution of the equation. The first scheme refers to the case where $H$ is a real separable Hilbert space, but in the second scheme, we drop this assumption.https://scma.maragheh.ac.ir/article_244071_85781e3d1440a36d832856bfda468567.pdf