Robust parameter-free algebraic multilevel preconditioning

Numerical Linear Algebra with Applications

Volumn 9, Issue 6-7, 2002, Pages 409-428

  Notay, Y ^a  

^a UNIVERSITÉ LIBRE DE BRUXELLES   (Belgium)

Author keywords

Convergence; Iterative methods; Preconditioning

Indexed keywords

EID: 0036391779     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.294     Document Type: Conference Paper

References (49)

1. Hackbusch W. Multi-grid Methods and Applications. Springer: Berlin, 1985.
2. Trottenberg U., Oosterlee CW., Schüller A. Multigrid. Academic Press: London, 2001.
3. Wesseling P. An Introduction to Multigrid Methods. Wiley: Chichester, 1992.
4. Bank RE. Smith RK. An algebraic multilevel multigraph algorithm. SIAM Journal on Scientific Computing 2002; 23:1572-1592.
5. Braess D. Towards algebraic multigrid for elliptic problems of second order. Computing 1995; 55:379-393.
6. Brezina M, Cleary AJ, Falgout RD, Henson VE, Jones JE. Manteuffel TA, McCormick SF, Ruge JW. Algebraic multigrid based on element interpolation (AMGe). SIAM Journal on Scientific Computing 2000; 22: 1570-1592.
7. Jones JE, Vassilevski PS. AMGe based on element agglomeration. SIAM Journal on Scientific Computing 2001; 23:109-133.
8. Ruge JW, Stüben K. Algebraic multigrid (AMG). In Multigrid Methods, McCormick SF (ed.), Frontiers in Applied Mathematics, vol. 3, SIAM: Philadelphia, 1987; 73-130.
9. Stüben K. An Introduction to Algebraic Multigrid. In Trottenberg V, Oosterlee CW. Schuller (eds). Academic Press: London, 2001; 413-532 (Appendix A).
10. Vaněk P, Mandel J, Brezina M. Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 1996; 56:179-196.
11. Axelsson P. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
12. Saad Y. Iterative Methods for Sparse Linear Systems. PWS Publishing: New York, 1996.
13. Bank RE, Wagner C. Multilevel ILU decomposition. Numerische Mathematik 1999; 82:543-576.
14. Botta EFF, Wubs FW. Matrix renumbering ILU: an effective algebraic multilevel ILU preconditioner for sparse matrices. SIAM Journal on Matrix Analysis and Applications 1999; 20:1007-1026.
15. Notay Y, Quld Amar Z. A nearly optimal preconditioning based on recursive red-black orderings. Numerical Linear Algebra with Applications 1997; 4:369-391.
16. Saad Y. ILUM: a parallel multi-elimination ILU preconditioner for general sparse matrices. SIAM Journal on Scientific Computing 1996; 17:830-847.
17. Saad Y, Suchomel B. ARMS: An algebraic recursive multilevel solver for general sparse linear systems. Technical Report umsi-99-107-REVIS, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, 2001.
18. Saad Y, Zhang J. BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. SIAM Journal on Scientific Computing 1999; 20:2103-2121.
19. van der Ploeg A, Botta EFF, Wubs FW. Nested grids ILU-decomposition (NGILU). Journal of Computational and Applied Mathematics 1996; 66:515-526.
20. Bank RE, Smith RK. The incomplete factorization multigraph algorithm. SIAM Journal on Scientific Computing 1999; 20:1349-1364.
21. Reusken A. On the approximate cyclic reduction preconditioner. SIAM Journal on Scientific Computing 1999; 21:565-590.
22. Axelsson O. The method of diagonal compensation of reduced matrix entries and multilevel iteration. Journal of Computational and Applied Mathematics 1991; 38:31-43.
23. Axelsson O, Eijkhout V. Analysis of recursive 5-point/9-point factorization method. In Preconditioned Conjugate Gradient Methods, Axelsson O, Kolotilina LYu (eds), Lectures Notes in Mathematics, vol. 1457. Springer: Berlin, Heidelberg, New York, 1990; 154-173.
24. Axelsson O, Eijkhout V. The nested recursive two level factorization for nine-point difference matrices. SIAM Journal on Scientific Computing 1991; 12:1373-1400.
25. Axelsson O, Neytcheva M. Algebraic multilevel iterations for Stieltjes matrices. Numerical Linear Algebra with Applications 1994; 1:213-236.
26. Axelsson O, Padiy A. On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems. SIAM Journal on Scientific Computing 1999; 20:1807-1830.
27. Axelsson O, Vassilevski PS. Algebraic multilevel preconditioning methods, I. Numerische Mathematik 1989; 56:157-177.
28. Axelsson O, Vassilevski PS. Algebraic multilevel preconditioning methods, II. SIAM Journal on Numerical Analysis 1990; 27:1569-1590.
29. Axelsson O, Vassilevski PS. Variable-step multilevel preconditioning methods. I. self adjoint and positive definite elliptic problems. Numerical Linear Algebra with Applications 1994; 1:75-101.
30. Margenov S. Semi-coarsening AMLI algorithms for elasticity problems. Numerical Linear Algebra with Applications 1999; 5:347-362.
31. Margenov S, Vassilevski P. Algebraic multilevel preconditioning of anisotropic elliptic problems. SIAM Journal on Scientific Computing 1994; 15:1026-1037.
32. Notay Y. Using approximate inverses in algebraic multilevel methods. Numerische Mathematik 1998; 80: 397-417.
33. Notay Y. Optimal V cycle algebraic multilevel preconditioning. Numerical Linear Algebra with Applications 1998; 5:441-459.
34. Notay Y. Optimal order preconditioning of finite difference matrices. SIAM Journal on Scientific Computing 2000; 21:1991-2007.
35. Notay Y. A robust algebraic multilevel preconditioner for non symmetric M-matrices. Numerical Linear Algebra with Applications 2000; 7:243-267.
36. Reusken A. A multigrid method based on incomplete Gaussian elimination. Numerical Linear Algebra with Applications 1996; 3:369-390.
37. Vassilevski P. Nearly optimal iterative methods for solving finite element elliptic equations based on the multilevel splitting of the matrix. Technical Report # 1989-09, Institute for Scientific Computation, University of Wyoming, Laramie, U.S.A., 1989.
38. Vassilevski P. Hybrid V-cycle algebraic multilevel preconditioned. Mathematics of Computation 1992; 58: 489-512.
39. Vassilevski P. On two ways of stabilizing the hierarchical basis multilevel methods. SIAM Review 1997; 39: 18-53.
40. Axelsson O. On multigrid methods of the two-level type. In Multigrid Methods, Hackbusch W, Trottenberg U (eds), Lectures Notes in Mathematics, vol. 960. Springer: Berlin, Heidelberg, New York. 1981; 352-367.
41. Axelsson O, Gustafsson I. Preconditioning and two-level multigrid methods of arbitrary degree of approximation. Mathematics of Computation 1983; 40:214-242.
42. Bank RE, Dupont TF. Analysis of a two-level scheme for solving finite element equations. Technical Report CNA-159, Center for Numerical Analysis, The University of Texas at Austin, TX, U.S.A., 1980.
43. Eijkhout V, Vassilevski P. The role of the strengthened c.b.s. inequality in multilevel methods. SIAM Review 1991; 33:405-419.
44. Axelsson O, Vassilevski PS. A black-box generalized conjugate gradient solver with inner iterations and variable-step preconditioning. SIAM Journal on Matrix Analysis and Applications 1991; 12:625-644.
45. Golub GH, Ye Q. Inexact preconditioned conjugate gradient method with inner-outer iterations. SIAM Journal on Scientific Computing 1999; 21:1305-1320.
46. Notay Y. Flexible conjugate gradients. SIAM Journal on Scientific Computing 2000; 22:1444-1460.
47. Saad Y. A flexible inner-outer preconditioned GMRES algorithm. SIAM Journal on Scientific Computing 1993; 14:461-469.
48. van der Vorst H, Vuik C. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications 1994; 1:369-386.
49. Elman H, Silvester D. Fast non-symmetric iterations and preconditioning for Navier-Stokes equations. SIAM Journal on Scientific Computing 1996; 17:33-46.