ALGEBRAIC MULTIGRID SMOOTHING PROPERTY OF KACZMARZ'S RELAXATION FOR GENERAL RECTANGULAR LINEAR SYSTEMS

In this paper we analyze the smoothing property from classical Algebraic Multigrid theory, for general rectangular systems of linear equations. We prove it for Kaczmarz's projection algorithm in the consistent case and obtain in this way a generalization of the classical well-known result by A. Brandt. We then extend this result for the Kaczmarz Extended algorithm in the inconsistent case.

[1]  A. Brandt.  Algebraic multigrid theory: The symmetric case , 1986 .

[2]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[3]  C. Popa.  Characterization of the solutions set of inconsistent least-squares problems by an extended Kaczmarz algorithm , 1999 .

[4]  K. Stüben.  Algebraic multigrid (AMG): experiences and comparisons , 1983 .

[5]  K. Tanabe.  Projection method for solving a singular system of linear equations and its applications , 1971 .

[6]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[7]  Marco Donatelli,et al.  A V-cycle Multigrid for multilevel matrix algebras: proof of optimality , 2007, Numerische Mathematik.

[8]  Andrzej Stachurski,et al.  Parallel Optimization: Theory, Algorithms and Applications , 2000, Parallel Distributed Comput. Pract..

[9]  Constantin Popa,et al.  Least-squares solution of overdetermined inconsistent linear systems using kaczmarz's relaxation , 1995, Int. J. Comput. Math..

[10]  Ulrich Rüde,et al.  Towards an algebraic multigrid method for tomographic image reconstruction -- improving convergence of ART , 2006 .

[11]  Stefano Serra Capizzano,et al.  V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems , 2004, SIAM J. Matrix Anal. Appl..

[12]  Stefano Serra Capizzano,et al.  Numerische Mathematik Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences , 2002 .

[13]  StübenKlaus.  Algebraic multigrid (AMG) , 1983 .

[14]  Y. Censor,et al.  Parallel Optimization:theory , 1997 .

[15]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[16]  C. Popa.  Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deficient least-squares problems , 1998 .