Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

A MULTILEVEL BLOCK INCOMPLETE CHOLESKY PRECONDITIONER FOR SOLVING NORMAL EQUATIONS IN LINEAR LEAST SQUARES PROBLEMS

  • Jun, Zhang (Department of Energy Office of Science, Japanese Research Organization for Information Science & Technology (RIST)) ;
  • Tong, Xiao (Laboratory for High Performance Scientific Computing and Computer Simulation, Department of Computer, University of Kentucky)
  • Published : 2003.01.01

⟨ Previous Next ⟩

Abstract

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

Keywords

References

  1. Int. J. Numer. Methods Engrg v.20 A robust incomplete chpleski-conhugate grasient alforithm M. A. Ahiz;A. Jennings
  2. SIAM J. Matriz Anal. Appl. v.12 Minimax polynomial preconditioning for Hermitian linear systms S, Ashby
  3. Comput. Methods App;. Mech. Engrg. v.64 Algorithms and experiments for structural mechanics on high-performance architectures M. W. Berry;R. J. Plemmons
  4. Numerical Methods for Least-Squares Problems A, Bjorck
  5. Elect. Trans. Numer. Anal. v.8 Preconditioners for least squares problems by LU factorization A. Bjorck;J. Y. Tuan
  6. ORSA J. Comput. v.2 Future development of a primal-dual interior point method I. Chio;C. L. Monna;D. Shanno
  7. Augmented Lagrange Methods: Applocations to the Numerical Solution of Boundary-Value Problems M. Fortin;R. Glowinski
  8. Linear Algrbra Appl. v.88;89 A note on two SOR methods for sparse least squares problems R. Freund
  9. Scientific Computing: An Introduction with Parallel Computing G. H. Golub;J. M. Ortega
  10. (3rd ed.) v.5 Matrix Computations G. H. Golub;C. F. van Loan
  11. SIAM J. Sci. Statist. Comput. v.5 Sparse orthogonal schemes for structural optimization using force method M. T. Heath;R. J. Plemmons;R. C. Ward
  12. J. Research Nat. Bur. Standards v.49 Methods of conjugate gradients for solving linear sustems M. R. Hestenes;E. Stiefel
  13. SIAM J. Sci. Statist. Comput. v.5 Incomplete methods for solving$A^T$ Ax = b A, Jenings;M. A. Ajiz
  14. ACM Trans. Math. Software v.21 An incomplete Cholesky factorization M. T. Jones;P. E. Plassmann
  15. Science v.200 Geodesy: dealing with an enormous computer task E. G. Kolata
  16. SIAM J. Sci. Comput. v.21 Incomplete Cholesky factorization with limited memory C. Lin;J. J. More
  17. Math. Comput. v.34 An incomplete factorization technique for positive definte linear systems T. A. Mantenffel
  18. Math. Comput. v.31 An iterative solution method for linear systems of which the coefficient matrix is a summetric M-matrix J. A. Meijerink;H. A. van der Vorst
  19. ACM Trans. Math. Software v.8 LSQR: an algorithm for sparse linear equations and sparse least squares C. C. Paige;M. A. Saunders
  20. Technical Report CSD-TR 464, Departmant of Computer Science PARVEC workshop on very large squares problems and supercomputers J. R. Rice
  21. J. Comput. Appl. Math. v.24 Preconditioning techniques for nonsymmetric and indefinite linear systems Y. Saad
  22. Numer. Linear Algebra Appl. v.1 ILUT: a dual threshold incomplete LU preconditioner Y. Saad
  23. Interative Methods for Sparse Lineat Systems Y. Saad
  24. SIAM J. Sci. Comput. v.20 BILUM: block versions of multielimination and multilevel ILU preconditiner for general sparse linear systems Y. Saad;J. Zhang
  25. SIAM J. Matrix Anal. Appl. v.21 BILUTM: a domain-based multilevel multilevel block ILUT preconditioner for general sparse metrices Y. Saad;J. Zhang
  26. Paral. Comput. v.28 Patallel two level block ILU proconditioning techniques for solving large sparse linear systems C. Shen;J. Zhang
  27. Basic Decompositions v.I Matrix Algorithms G. W. Stewart
  28. Ph. D. Thesis Incomplete Factorization Preconditioning for Linear Least Squares Problems X. Wang
  29. SIAM J. Sci. Comput. v.18 CIMGS: an incomplete orthogonal factorization preconditioner X, Wang;K. A. Gallivan;R. Bramley