Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematics and Computer Sciences, Vali-e-Asr University of Rasanjan, Rafsanjan, Iran.

Abstract

In this paper, by using the concept of frames, two iterative methods are constructed to solve the operator equation $ Lu=f $ where $ L:H\rightarrow 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.

Keywords

[1] G. Beylkin, R.R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical    algorithms I, Comm. Pure and Appl. Math., 44 (1991), pp. 141-183.

[2] C. Brezinski, Projection Methods for System of Equations, Elsevier, Amsterdam, (1997).

[3] P. G. Casazza, The art of frame theory, Taiwanese J. of Math., 4 (2000), pp. 129-201.

[4] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, (2003).

[5] A. Cohen, Numerical Analysis of Wavelet Methods, Elsevier, (2003).

[6] A. Cohen and W. DeVore, Adaptive wavelet methods for elliptic operator equations:    convergence rates, Math. of comp., 233 (2001), pp. 27-75.

[7] S. Dahlke, M. Foransier and T. Raasch, Adaptive frame methods for operator equations, Advances in comp. Math., 1 (2007), pp. 27-63.

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

[9] H. Jamali and A.A. Hemmat, Approximated solutions to operator equations based on the frame bounds, Comm. Math. App., 3 (2012), pp. 253-259.

[10] H. Jamali and A.M. Kolahdouz, Modified frame-based Richardson iterative method and its convergence acceleration by Chebyshev polynomials, U.P.B. Sci. Bull., Series A., 80 (2018), pp. 83-92.

[11] T.J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell Publishing Company, (1969).

[12] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS press, New York, (2000).