Geodesic
On the two-level preconditioning in least squares method
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2014), pp. 7-15.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper an approach to construct algebraic two-level preconditioners for the matrices of normal systems arising in data fitting by least squares method with piecewise linear basis functions is proposed. The approach is based on using hierarchical grids with their subdivision into substructures and corresponding partition of the matrices. Estimates for condition numbers of preconditioned matrices are obtained.
Keywords: least squares method, normal system, condition number. 
@article{UZERU_2014_1_a1,
     author = {Yu. R. Akopian and R. Z. Hovhannisyan},
     title = {On the two-level preconditioning in least squares method},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {7--15},
     publisher = {mathdoc},
     number = {1},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2014_1_a1/}
}
TY  - JOUR
AU  - Yu. R. Akopian
AU  - R. Z. Hovhannisyan
TI  - On the two-level preconditioning in least squares method
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 2014
SP  - 7
EP  - 15
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UZERU_2014_1_a1/
LA  - en
ID  - UZERU_2014_1_a1
ER  - 
%0 Journal Article
%A Yu. R. Akopian
%A R. Z. Hovhannisyan
%T On the two-level preconditioning in least squares method
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 2014
%P 7-15
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UZERU_2014_1_a1/
%G en
%F UZERU_2014_1_a1
Yu. R. Akopian; R. Z. Hovhannisyan. On the two-level preconditioning in least squares method. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2014), pp. 7-15. http://geodesic.mathdoc.fr/item/UZERU_2014_1_a1/

[1] A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996 | MR | Zbl

[2] J.M. Ortega, W.G. Poole Jr., An Introduction to Numerical Methods for Differential Equations, Pitman Publishing, Marshfield, MA, 1981 | MR | Zbl

[3] D.S. Watkins, Fundamentals of Matrix Computations, John Wiley $\$ Sons, Inc, NY, 2002 | MR

[4] R.Z. Hovhannisyan, “On the Behaviour of the Condition Number for Matrices of Normal Systems on the Sequences of Grids”, Mathematics in Higher School, 8:2 (2012), 41–51 (in Russian)

[5] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer, 2007 | MR | Zbl

[6] O. Axelsson, P. Vassilevski, “Algebraic Multilevel Preconditioning Methods. Part I”, Numerische Mathematik, 56 (1989), 157–177 | DOI | MR | Zbl

[7] Yu.R. Hakopian, Yu. A. Kuznetsov, “Algebraic Multigrid/Substructuring Preconditioners on Triangular Grids”, Soviet Journal of Numerical Analysis and Mathematical Modelling, 6:6 (1991), 453–483 | DOI | MR | Zbl

[8] O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[9] O. Axelsson, I. Gustafsson, “Preconditioning and Two-Level Multigrid Methods of Arbitrary Degree of Approximation”, Mathematics of Computation, 40:161 (1983), 219–242 | DOI | MR | Zbl

[10] O. Axelsson, Yu. Hakopian, “Algebraic Multilevel Preconditioners for Perturbed Finite Element Matrices”, East-West Journal of Numerical Mathematics, 5:1 (1997), 1–22 | MR | Zbl

[11] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1986