%0 Journal Article
%T Using Frames in Steepest Descent-Based Iteration Method for Solving Operator Equations
%J Sahand Communications in Mathematical Analysis
%I University of Maragheh
%Z 2322-5807
%A Jamali, Hassan
%A Kolahdouz, Mohsen
%D 2021
%\ 05/01/2021
%V 18
%N 2
%P 97-109
%! Using Frames in Steepest Descent-Based Iteration Method for Solving Operator Equations
%K Hilbert space
%K Operator equation
%K Frame
%K Preconditioning
%K Steepest descent method
%K Convergence rate
%R 10.22130/scma.2020.123786.771
%X In 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.
%U https://scma.maragheh.ac.ir/article_244071_85781e3d1440a36d832856bfda468567.pdf