Study on Robustness of Incomplete Cholesky Factorization using Preconditioning for Conjugate Gradient Method

불완전분해법을 전처리로 하는 공액구배법의 안정화에 대한 연구

  • 고진환 (한국과학기술원 기계공학과) ;
  • 이병채 (한국과학기술원 기계공학과)
  • Published : 2003.02.01


The preconditioned conjugate gradient method is an efficient iterative solution scheme for large size finite element problems. As preconditioning method, we choose an incomplete Cholesky factorization which has efficiency and easiness in implementation in this paper. The incomplete Cholesky factorization mettled sometimes leads to breakdown of the computational procedure that means pivots in the matrix become minus during factorization. So, it is inevitable that a reduction process fur stabilizing and this process will guarantee robustness of the algorithm at the cost of a little computation. Recently incomplete factorization that enhances robustness through increasing diagonal dominancy instead of reduction process has been developed. This method has better efficiency for the problem that has rotational degree of freedom but is sensitive to parameters and the breakdown can be occurred occasionally. Therefore, this paper presents new method that guarantees robustness for this method. Numerical experiment shows that the present method guarantees robustness without further efficiency loss.


Conjugate Gradient Method;Incomplete Cholesky Factorization;Preconditioning Method


  1. Saint-Georges, P., Warzee, G.., Beauwnes, R., and Notay, Y., 1999, 'Problem-Dependant Preconditioners for Iterative Solvers in FE Elastostatics,' Computers and Structures, Vol. 73, pp.33-43
  2. Hladic, I., Reed, M. B., Swoboda, G., 1997, 'Robust Preconditioners for Linear Elasticity FEM Analyses,' International Journal for Numerical Methods in Engineering, Vol. 40, pp. 2109-2127<2109::AID-NME163>3.0.CO;2-1
  3. Chow, E. and Saad, Y., 1997, 'Experimental Study of ILU Preconditioners for Indefinite Matrices,' Journal of Computational and Applied Mathematics, Vol. 86, pp. 387-414
  4. Buleev, N. I., 1960, 'A Numerical Method for the Solution of Two-Dimensional and Three-Dimensional Equation of Diffusion,' Math. Sb., Vol. 51, pp. 227-238
  5. Axelsson, O., 1972, 'A Generalized SSOR Method,' BIT, Vol. 13, pp. 443-467
  6. Gustafsson, I., 1978, 'A Class of First Order Factorization Methods,' BIT, Vol. 18, pp. 142-156
  7. Saint-Georges, P., Warzee, G.., Beauwnes, R., and Notay, Y., 1996, 'High-Performance PCG Solvers for FEM Structural Analysis,' International Journal for Numerical Methods in Engineering, Vol. 39, pp. 1313-1340<1313::AID-NME906>3.0.CO;2-J
  8. Beauwens, R. and Wilmet, R., 1989, 'Conditioning Analysis of Positive Definite Matrices by Approximate Factorizations,' Journal of Computational and Applied Mathematics, Vol. 26, pp. 257-269
  9. Axelsson, O. and Kolotilina, I., 1994, ',Diagonal Compensated Reduction and Related Preconditioning Methods,' Numerical Linear Algebra Application, Vol. 1, pp. 155-177
  10. Meijerink, J. A. and Van der vorst, H. A., 1977, 'An Iterative Solution Method for Linear System of Which the Coefficient Matrix is a Symmetric M-matrix,' Mathematical Computation, Vol. 31, pp. 148-162
  11. Kershaw, D. S., 1978, 'The Incomplete Cholesky-Conjugate Gradient Method for the Iterative Solution of Systems of Linear Equation,' Journal of Computational Physics, Vol. 26, pp. 43-65
  12. George, A., 1981, Computer Solution of Large Sparse Positive Definite System, Prentice-Hall