Abstract
New circulant block-factorization preconditioners are introduced and studied. The general approach is first formulated for the case of block tridiagonal sparse matrices. Then estimates of the relative condition number for a model Dirichlet boundary value problem are derived. In the case ofy-periodic problems the circulant block-factorization preconditioner is shown to give an optimal convergence rate. Finally, using a proper imbedding of the original Dirichlet boundary value problem to ay-periodic one a preconditioner of optimal convergence rate for the general case is obtained. The total computational cost of the preconditioner isO (N logN) (based on FFT), whereN is the number of unknowns. That is, the algorithm is nearly optimal. Various numerical tests that demonstrate the features of the circulant block-factorization preconditioners are presented.
Zusammenfassung
Neue zyklische Matrixzerlegungen werden eingeführt und untersucht. Der allgemeine Ansatz wird für den Fall blocktridiagonaler schwachbesetzter Matrizen formuliert. Danach werden Abschätzungen der relativen Konditionszahl für ein Dirichlet-Modellproblem abgeleitet. Es wird gezeigt, daß die zyklische Matrixzerlegung im Falley-periodischer Aufgaben optimale Konvergenzraten liefert. Nach Einbettung des ursprünglichen Dirichlet-Problems in einey-periodische Aufgabe erhält man den allgemeinen Fall. Der Gesamtaufwand des Präkonditionierers beträgtO (N logN) gemäß des FFT-Aufwandes, wobeiN die Zahl der Unbekannten ist. Damit ist der Algorithmus fast optimal. Verschiedene numerische Tests zeigen die Eigenschaften der zyklischen Matrixzerlegung.
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References
Axelsson, O.: A survey of vectorizable preconditioning methods for large scale finite element matrices. In: Colloquium topics in applied numerical analysis, (Verwer, J. G., ed.), pp. 21–47. Syllabus 4, Center of Mathematics and Informatics (CMI), Amsterdam 1983.
Axelsson, O., Barker, V. A.: Finite element solution of boundary value problems: theory and computations. Orlando: Academic Press 1983.
Axelsson, O., Brinkkemper, S., Il'in, V. P.: On some versions of incomplete block-matrix factorization methods, Lin Alg. Appl.38, 3–15 (1984).
Axelsson, O., Polman, B.: On approximate factorization methods for block-matrices suitable for vector and parallel processors. Lin. Alg. Appl.77, 3–26 (1986).
Axelsson, O., Eijkhout, V. L.: Robust vectorizable preconditioners for three-dimensional elliptic difference equations with anisotropy. Algorithms and applications on vector and parallel computers, (te Riele, H. J. J., Dekker, Th. J., van der Vorst, H., eds.), pp. 279–306. Amsterdam: North-Holland 1987.
Bank, R. E.: Marching algorithms for elliptic boundary value problems. II: The variable coefficient case. SIAM J. Numer. Anal.14, 950–970 (1977).
Chan, R. H., Chan, T. F.: Circulant preconditioners for elliptic problems. J. Numerical Lin. Alg. Appl.1, 77–101 (1992).
Chan, T. F., Mathew, T. P.: The interface probing technique in domain decomposition. SIAM J. Matr. Anal. Appl.13, 212–238 (1992).
Chan, T. F., Vassilevski, P. S.: A framework for block-ILU factorizations using block-size reduction. Math. Comp. (in press).
Concus, P., Golub, G. H., Meurant, G.: Block preconditioning for the conjugate gradient method. SIAM J. Sci. Stat. Comput.6, 220–252 (1985).
D'yakonov, E. G.: On an iterative method for the solution of finite difference equations. Dokl. Acad. Nauk SSSR138, 522–525 (1961).
Davis, P. J.: Circulant matrices. New York: John Wiley 1979.
Golub, G. H., van Loan, C. F.: Matrix computations, 2nd edn. Baltimore: Johns Hopkins Univ. Press 1989.
Gunn, J. E.: The solution of difference equations by semi-explicit iterative techniques. SIAM J. Num. Anal.2, 24–45 (1965).
Gustafsson, I.: A class of first-order factorization methods. BIT18, 142–156 (1978).
Holmgren, S., Otto, K.: Iterative solution methods for block-tridiagonal systems of equations. SIAM J. Matr. Anal. Appl.13, 863–886 (1992).
Huckle, T.: Some aspects of circulant preconditioners. SIAM J. Sci. Comput.14, 531–541 (1993).
Huckle, T.: Circulant and skewcirculant matrices for solving Toeplitz matrix problems. SIAM J. Matr. Anal. Appl.13, 767–777 (1992).
Kettler, R.: Analysis and computations of relaxed schemes in robust multigrid and preconditioned conjugate gradient methods. In: Multigrid methods, Proceedings (Hackbusch, W., Trottenberg, U., eds.), pp. 502–534. Berlin Heidelberg New York Tokyo: Springer 1982 (Lecture Notes in Mathematics, Vol. 960).
van Loan, C.: Computational frameworks for the fast Fourier transform. Philadelphia: SIAM 1992.
Margenov, S. D., Lirkov, I. T.: Preconditioned conjugate gradient iterative algorithms for transputer based systems. In: Parallel and distributed processing (Boyanov, K., ed.), pp. 406–415. Sofia: BAS 1993.
Meurant, G.: The block-preconditioned conjugate gradient method on vector computers. BIT24, 623–633 (1984).
Meurant, G.: A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matr. Anal. Appl.13, 707–728 (1992).
Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math.74, 171–176 (1986).
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Lirkov, I.D., Margenov, S.D. & Vassilevski, P.S. Circulant block-factorization preconditioners for elliptic problems. Computing 53, 59–74 (1994). https://doi.org/10.1007/BF02262108
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DOI: https://doi.org/10.1007/BF02262108