Document Type: Research Paper

Authors

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

Abstract

In this paper, we design some iterative schemes for solving operator equation $ Lu=f $, where $ L:H\rightarrow H $ is a bounded, invertible and self-adjoint operator on a separable Hilbert space $ H $. In this concern,  Richardson and Chebyshev iterative methods are two outstanding as well as long-standing ones. They can be implemented in different ways via different concepts.
In this paper, these schemes exploit the almost recently developed notion of g-frames which result in modified convergence rates compared with early computed ones in corresponding classical formulations.
In fact, these convergence rates are formed by the lower and upper bounds of the given g-frame. Therefore, we can determine any convergence rate by considering an appropriate g-frame.

Keywords

Main Subjects

[1] A. Askari Hemmat and H. Jamali, Adaptive Galerkin frame methods for solving operator equation, U.P.B. Sci. Bull., Series A, 73 (2011), pp. 129-138.

[2] K. Atkinson, W. Han, Theoretical Numerical Analysis, Springer, Third edition, 2009.

[3] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory, Cambridge, Third edition, 2007.

[4] C.C. Cheny, Introduction to Approximation Theory, McGraw Hill, New York, 1966.

[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003.

[6] S. Dahlke, M. Fornasier, and T. Raasch, Adaptive frame methods for elliptic operator equations, Advances in comp. Math., 27 (2007), pp. 27-63.

[7] H. Jamali and S. Ghaedi, Applications of frames of subspaces in Richardson and Chebyshev methods for solving operator equations, Math. Commun., 22 (2017), pp. 13-23 .

[8] H. Jamali and N. Momeni, Application of g-frames in conjugate gradient, Adv. Pure Appl. Math., 7 (2016), pp. 205-212.

[9] A. Najati, M. H. Faroughi, and A. Rahimi, G-frames and stability of g-frames in Hilbert spaces, Methods Funct. Anal. Topology, 14 (2008), pp. 271-286.

[10] Y. Saad, Iterative methods for Sparse Linear Systems, PWS press, New York, 2000.

[11] R. Stevenson, Adaptive solution of operator equations using wawelet frames, SIAM J. Numer. Anal., 41 (2003), pp. 1074-1100.

[12] W. Sun, G-frames and G-Riesz Bases, J. Math. Anal. Appl., 322 (2006), pp. 437-452.

[13] W. Sun, Stability of g-frames, J. Math. Anal. Appl., 326 (2007), pp. 858-868.