Application of frames in Chebyshev and conjugate gradient methods

Jamali, H.; Afroomand, E.

Application of frames in Chebyshev and conjugate gradient methods

Document Type : Research Paper

Authors

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

Abstract

‎Given a frame of a separable Hilbert space $H$‎, ‎we present some‎ ‎iterative methods for solving an operator equation $Lu=f$‎, ‎where $L$ is a bounded‎, ‎invertible and symmetric‎ ‎operator on $H$‎. ‎We present some algorithms‎ ‎based on the knowledge of frame bounds‎, ‎Chebyshev method and conjugate gradient method‎, ‎in order to give some‎ ‎approximated solutions to the problem‎. ‎Then we investigate the‎ ‎convergence and optimality of them.

Keywords

Main Subjects

65-XX Numerical Analysis

References

G. Beylkin, R.R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math. 44 (1991), no. 1, 141--183.

C. Brezinski, Projection Methods for System of Equations, Elsevier, Amsterdam, 1997.

P.G. Casazza, The art of frame theory, Taiwanese J. math. 4 (2000), no. 2, 129--201.

C.C. Cheny, Introduction to Approximation Theory, McGraw Hill, New York, 1996.

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, 2003.

A. Cohen, Numerical Analysis of Wavelet Methods, Elsevier, 2003.

A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelets methods II-beyond the elliptic case, Found. Comput. Math. 2 (2002), no. 1, 203--245.

A. Cohen and W. DeVore, Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp. 70 (2001), no. 233, 27--75.

S. Dahlke, M. Foransier and T. Raasch, Adaptive frame methods for operator equations, Adv. Comput. Math. 27 (2007), no. 1, 27--63.

W. Dahmen and R. Schneider, A composite wavelet bases for operator equations, Math. Comp. 68 (1999), no. 1, 1533--1567.

W. Dahmen and R. Schneider, Wavelets on manifolds I. Construction and domain decomposition, SIAM J. Math. Anal. 31 (1999), no. 1, 184--230.

S.P. Frankle, Convergence rates of iterative treatments of partial differential equations, Math. Tables Aids. Comput. 4 (1950), no. 1, 65--75.

G.H. Golub and R.S. Varga, Chebyshev semi-iteration and second-order Richardson iterative methods, Numer. Math. 3 (1961), no. 1, 147--168.

A.A. Hemmat and H. Jamali, Adaptive Galerkin frame methods for solving operator equation, U. P. B. Sci. Bull. Series A. 73 (2011), no. 1, 129--138.

M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, Sect. B 49 (1952), no. 1, 409--436.

L.F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Proc. Roy. Soc. London Ser. A 210 (1910), no. 1, 307--357.

R.V. Sauthwell, Relaxation Methods in Theoretical Physics, Oxford Univ. Press, Oxford, 1946.

H.D. Simon, The Lanczos algorithm with partial reorthogonalization, Math. Comp. 42 (1983), no. 1, 115--142.

R. Stevenson, Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal. 41 (2003), no. 1, 1047--1100.