A numerical study of optimized sparse preconditioners
 A. M. Bruaset,
 A. Tveito
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Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.
Our conclusion is that the wellknown Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete twodimensional Poisson equation withn unknowns,O(n ^{1/4}) andO(n ^{1/2}) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate.
By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.
 E. Arge, M. Dæhlen, and A. Tveito,Box spline interpolation; a computational study J. Comput. Appl. Math., 44 (1992), pp. 303–329.
 S. F. Ashby,Polynomial preconditioning for conjugate gradient methods, Department of Computer Science, University of Illinois at UrbanaChampaign, Illinois, Ph.D. thesis, 1987. (Report No. UIUCDCSR871355.)
 S. F. Ashby,Minimax polynomial preconditioning for Hermitian linear systems SIAM J. Matrix Anal., 12 (1991), pp. 766–789.
 S. F. Ashby, M. J. Holst, T. A. Manteuffel, and P. E. Saylor,The role of the inner product in stopping criteria for conjugate gradient iterations, Report UCRLJC112586, Comp. & Math. Research Division, Lawrence Livermore National Lab., 1992.
 S. F. Ashby, T. A. Manteuffel, and J. S. Otto,A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems SIAM J. Sci. Stat. Comput., 13 (1992), pp. 1–29.
 S. F. Ashby, T. A. Manteuffel, and P. E. Saylor,Adaptive polynomial preconditioning for Hermitian linear systems BIT, 29 (1989), pp. 583–609.
 S. F. Ashby, T. A. Manteuffel, and P. E. Saylor,A taxonomy for conjugate gradient methods SIAM J. Numer. Anal., 27 (1990), pp. 1542–1568.
 O. Axelsson and G. Lindskog,On the eigenvalue distribution of a class of preconditioning methods Numer. Math., 48 (1986), pp. 479–498.
 O. Axelsson and G. Lindskog,On the rate of convergence of the preconditioned conjugate gradient method Numer. Math., 48 (1986), pp. 499–523.
 P. N. Brown and A. C. Hindmarsh,Matrixfree methods for stiff systems of ODE's SIAM J. Numer. Anal., 23 (1986), pp. 610–638.
 T. F. Chan,Fourier analysis of relaxed incomplete factorization preconditioners SIAM J. Sci. Stat. Comput., 12 (1991), pp. 668–680.
 T. F. Chan and H. C. Elman,Fourier analysis of iterative methods for elliptic problems SIAM Review, 31 (1989), pp. 20–49.
 P. Concus, G. H. Golub, and D. O'Leary,A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, 1976, pp. 309–332.
 S. D. Conte and C. de Boor,Elementary Numerical Analysis, McGrawHill, 1981.
 J. E. Dennis Jr. and H. Wolkowicz,Sizing and leastchange secant methods SIAM J. Numer. Anal., 30 (1993), pp. 1291–1314.
 J. M. Donato and T. C. Chan,Fourier analysis of incomplete factorization preconditioners for threedimensional anisotropic problems SIAM J. Sci. Stat. Comput., 13 (1992), pp. 319–338.
 P. F. Dubois, A. Greenbaum, and G. H. Rodrigue,Approximating the inverse of a matrix for use in iterative algorithms on vector processors Computing, 22 (1979), pp. 257–268.
 A. Greenbaum,Comparison of splittings used with the conjugate gradient algorithm Numer. Math., 33 (1979), pp. 181–194.
 A. Greenbaum and G. H. Rodrigue,Optimal preconditioners of a given sparsity pattern BIT, 29 (1989), pp. 610–634.
 I. Gustafsson,A class of first order factorization methods BIT, 18 (1978), pp. 142–156.
 A. Jennings,Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method J. Inst. Maths. Applics. 20 (1977), pp. 61–72.
 O. G. Johnson, C. A. Micchelli, and G. Paul,Polynomial preconditioners for conjugate gradient calculations SIAM J. Numer. Anal. 20 (1983), pp. 362–376.
 I. E. Kaporin,New convergence results and preconditioning strategies for the conjugate gradient method, Preprint, Dept. of Comp. Math. and Cyb., Moscow State University, 1992.
 The Mathworks,ProMatlab User's Guide, The Mathworks, 1990.
 J. A. Meijerink and H. A. van der Vorst,An iterative solution method for linear systems of which the coefficient matrix is a symmetric Mmatrix Math. Comp., 31 (1977), pp. 148–162.
 D. P. O'Leary,Yet another polynomial preconditioner for the conjugate gradient algorithm Linear Algebra Appl., 154/56 (1991), pp. 377–388.
 G. Pini and G. Gambolati,Is a simple diagonal scaling the best preconditioner for conjugate gradients on supercomputers? Adv. Water Resources, 13 (1990), pp. 147–153.
 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C. The Art of Scientific Computing, Cambridge University Press, 1988.
 Z. Strakoš,On the real convergence rate of the conjugate gradient method Linear Algebra Appl., 154/56 (1991), pp. 535–549.
 A. van der Sluis and H. A. van der Vorst,The rate of convergence of conjugate gradients Numer. Math., 48 (1986), pp. 543–560.
 R. Winther,Some superlinear convergence results for the conjugate gradient method SIAM J. Numer. Anal., 17 (1980), pp. 14–17.
 Title
 A numerical study of optimized sparse preconditioners
 Journal

BIT Numerical Mathematics
Volume 34, Issue 2 , pp 177204
 Cover Date
 19940601
 DOI
 10.1007/BF01955867
 Print ISSN
 00063835
 Online ISSN
 15729125
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 65F10
 15A06
 65F90
 65K10
 Conjugate gradient method
 preconditioning
 incomplete factorization
 polynomial preconditioner
 matrixfree method
 Fourier analysis
 Industry Sectors
 Authors

 A. M. Bruaset ^{(1)}
 A. Tveito ^{(1)}
 Author Affiliations

 1. SINTEF, P.O. Box 124, Blindern, N0314, Oslo, Norway