메뉴 건너뛰기




Volumn 24, Issue 6, 2003, Pages 2076-2092

On solving block-structured indefinite linear systems

Author keywords

Augmented lagrangian; Condition number; Indefinite linear systems; Schur complement; Singularity; Spectrum

Indexed keywords

INVERSE PROBLEMS; LAGRANGE MULTIPLIERS; MATRIX ALGEBRA; OPTIMIZATION; POLES AND ZEROS; PROBLEM SOLVING; QUADRATIC PROGRAMMING;

EID: 0347577906     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827500375096     Document Type: Article
Times cited : (149)

References (50)
  • 1
    • 0036057097 scopus 로고    scopus 로고
    • A backward error analysis of a null space algorithm in sparse quadratic programming
    • M. Arioli and L. Baldini, A backward error analysis of a null space algorithm in sparse quadratic programming, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 425-442.
    • (2001) SIAM J. Matrix Anal. Appl. , vol.23 , pp. 425-442
    • Arioli, M.1    Baldini, L.2
  • 3
    • 0009924116 scopus 로고    scopus 로고
    • Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices
    • Z. Bai and G. H. Golub, Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices, Ann. Numer. Math., 4 (1997), pp. 29-38.
    • (1997) Ann. Numer. Math. , vol.4 , pp. 29-38
    • Bai, Z.1    Golub, G.H.2
  • 4
    • 34249955392 scopus 로고
    • A class of iterative methods for solving saddle point problems
    • R. E. Bank, B. D. Welfert, and H. Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math., 55 (1990), pp. 645-666.
    • (1990) Numer. Math. , vol.55 , pp. 645-666
    • Bank, R.E.1    Welfert, B.D.2    Yserentant, H.3
  • 5
    • 0347942419 scopus 로고
    • Solution of equality-constrained quadratic programming problems by a projection iterative method
    • M. Benzi, Solution of equality-constrained quadratic programming problems by a projection iterative method, Rend. Mat. Appl. (7), 13 (1993), pp. 275-296.
    • (1993) Rend. Mat. Appl. , vol.13 , Issue.7 , pp. 275-296
    • Benzi, M.1
  • 6
    • 0043044851 scopus 로고    scopus 로고
    • Bounds for the entries of matrix functions with applications to preconditioning
    • M. Benzi and G. H. Golub, Bounds for the entries of matrix functions with applications to preconditioning, BIT, 39 (1999), pp. 417-438.
    • (1999) BIT , vol.39 , pp. 417-438
    • Benzi, M.1    Golub, G.H.2
  • 8
    • 0001341650 scopus 로고    scopus 로고
    • Analysis of the inexact Uzawa algorithm for saddle point problems
    • J. H. Bramble, J. E. Pasciak, and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072-1092.
    • (1997) SIAM J. Numer. Anal. , vol.34 , pp. 1072-1092
    • Bramble, J.H.1    Pasciak, J.E.2    Vassilev, A.T.3
  • 10
    • 0015200767 scopus 로고
    • Direct methods for solving symmetric indefinite systems of linear equations
    • J. R. Bunch and B. N. Parlett, Direct methods for solving symmetric indefinite systems of linear equations, SIAM J. Numer. Anal., 8 (1971), pp. 639-655.
    • (1971) SIAM J. Numer. Anal. , vol.8 , pp. 639-655
    • Bunch, J.R.1    Parlett, B.N.2
  • 11
    • 0021479428 scopus 로고
    • Deflated decomposition of solutions of nearly singular systems
    • T. F. Chan, Deflated decomposition of solutions of nearly singular systems, SIAM J. Numer. Anal., 21 (1984), pp. 738-754.
    • (1984) SIAM J. Numer. Anal. , vol.21 , pp. 738-754
    • Chan, T.F.1
  • 12
    • 84966216063 scopus 로고
    • The numerical solution of equality-constrainted quadratic programming problems
    • N. Dyn and W. E. Ferguson, The numerical solution of equality-constrainted quadratic programming problems, Math. Comp., 41 (1983), pp. 165-170.
    • (1983) Math. Comp. , vol.41 , pp. 165-170
    • Dyn, N.1    Ferguson, W.E.2
  • 13
    • 0033279287 scopus 로고    scopus 로고
    • Preconditioning for the steady-state Navier-Stokes equations with low viscosity
    • H. C. Elman, Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput., 20 (1999), pp. 1299-1316.
    • (1999) SIAM J. Sci. Comput. , vol.20 , pp. 1299-1316
    • Elman, H.C.1
  • 14
    • 0028741633 scopus 로고
    • Inexact and preconditioned Uzawa algorithms for saddle point problems
    • H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645-1661.
    • (1994) SIAM J. Numer. Anal. , vol.31 , pp. 1645-1661
    • Elman, H.C.1    Golub, G.H.2
  • 15
    • 0002160502 scopus 로고    scopus 로고
    • Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations
    • H. Elman and D. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput., 17 (1996), pp. 33-46.
    • (1996) SIAM J. Sci. Comput. , vol.17 , pp. 33-46
    • Elman, H.1    Silvester, D.2
  • 16
    • 0036003235 scopus 로고    scopus 로고
    • Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations
    • H. C. Elman, D. J. Silvester, and A. J. Wathen, Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90 (2002), pp. 641-664.
    • (2002) Numer. Math. , vol.90 , pp. 641-664
    • Elman, H.C.1    Silvester, D.J.2    Wathen, A.J.3
  • 17
    • 0000382130 scopus 로고    scopus 로고
    • Minimum residual methods for augmented systems
    • B. Fischer, A. Ramage, D. J. Silvester, and A. J. Wathen, Minimum residual methods for augmented systems, BIT, 38 (1998), pp. 527-543.
    • (1998) BIT , vol.38 , pp. 527-543
    • Fischer, B.1    Ramage, A.2    Silvester, D.J.3    Wathen, A.J.4
  • 18
    • 0347331935 scopus 로고
    • An ideal penalty function for constrained optimization
    • Academic Press, New York
    • R. Fletcher, An ideal penalty function for constrained optimization, in Nonlinear Programming, 2, Academic Press, New York, 1974, pp. 121-163.
    • (1974) Nonlinear Programming , vol.2 , pp. 121-163
    • Fletcher, R.1
  • 20
    • 0346681533 scopus 로고
    • Numerical Analysis report NA/167, University of Dundee, Department of Mathematics and Computer Science, Dundee, UK
    • R. Fletcher and T. Johnson, On the Stability of Null-Space Methods for KKT Systems, Numerical Analysis report NA/167, University of Dundee, Department of Mathematics and Computer Science, Dundee, UK, 1995.
    • (1995) On the Stability of Null-Space Methods for KKT Systems
    • Fletcher, R.1    Johnson, T.2
  • 21
    • 21144480332 scopus 로고
    • Newton methods for large-scale linear equality-constrained minimization
    • A. Forsgren and W. Murray, Newton methods for large-scale linear equality-constrained minimization, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 560-587.
    • (1993) SIAM J. Matrix Anal. Appl. , vol.14 , pp. 560-587
    • Forsgren, A.1    Murray, W.2
  • 23
    • 0026123179 scopus 로고
    • Inertia-controlling methods for general quadratic programming
    • P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, Inertia-controlling methods for general quadratic programming, SIAM Rev., 33 (1991), pp. 1-36.
    • (1991) SIAM Rev. , vol.33 , pp. 1-36
    • Gill, P.E.1    Murray, W.2    Saunders, M.A.3    Wright, M.H.4
  • 26
    • 0003093333 scopus 로고
    • Matrices, moments and quadrature
    • Pitman Res. Notes Math. Ser. 3, Longman Sci. Tech., Harlow, UK
    • G. H. Golub and G. Meurant, Matrices, moments and quadrature, in Numerical Analysis 1993, Pitman Res. Notes Math. Ser. 3, Longman Sci. Tech., Harlow, UK, 1993, pp. 105-156.
    • (1993) Numerical Analysis 1993 , pp. 105-156
    • Golub, G.H.1    Meurant, G.2
  • 27
    • 0002211239 scopus 로고    scopus 로고
    • An iteration for indefinite systems and its application to the Navier-Stokes equations
    • G. H. Golub and A. J. Wathen, An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19 (1998), pp. 530-539.
    • (1998) SIAM J. Sci. Comput. , vol.19 , pp. 530-539
    • Golub, G.H.1    Wathen, A.J.2
  • 28
    • 0022061431 scopus 로고
    • On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem
    • N. I. M. Gould, On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem, Math. Programming, 32 (1985), pp. 90-99.
    • (1985) Math. Programming , vol.32 , pp. 90-99
    • Gould, N.I.M.1
  • 29
    • 0003470761 scopus 로고    scopus 로고
    • Iterative methods for solving linear systems
    • SIAM, Philadelphia, 1997
    • A. Greenbaum, Iterative Methods for Solving Linear Systems, Frontiers Appl. Math., 17, SIAM, Philadelphia, 1997.
    • (1997) Frontiers Appl. Math. , vol.17
    • Greenbaum, A.1
  • 30
    • 0000183429 scopus 로고    scopus 로고
    • On optimization techniques for solving nonlinear inverse problems
    • E. Haber, U. M. Ascher, and D. Oldenburg, On optimization techniques for solving nonlinear inverse problems, Inverse Problems, 16 (2000), pp. 1263-1280.
    • (2000) Inverse Problems , vol.16 , pp. 1263-1280
    • Haber, E.1    Ascher, U.M.2    Oldenburg, D.3
  • 32
    • 0004151494 scopus 로고
    • Cambridge University Press, Cambridge, UK
    • R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
    • (1985) Matrix Analysis
    • Horn, R.A.1    Johnson, C.R.2
  • 34
    • 0034310998 scopus 로고    scopus 로고
    • A contextualized historical analysis of the Kuhn-Tucker theorem in nonlinear programming: The impact of World War II
    • T. H. Kjeldsen, A contextualized historical analysis of the Kuhn-Tucker theorem in nonlinear programming: The impact of World War II, Historia Math., 27 (2000), pp. 331-361.
    • (2000) Historia Math. , vol.27 , pp. 331-361
    • Kjeldsen, T.H.1
  • 35
    • 11744343493 scopus 로고    scopus 로고
    • An optimal preconditioner for a class of saddle point problems with a penalty term
    • A. Klawonn, An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), pp. 540-552.
    • (1998) SIAM J. Sci. Comput. , vol.19 , pp. 540-552
    • Klawonn, A.1
  • 36
    • 0034890938 scopus 로고    scopus 로고
    • A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity
    • A. Klawonn and O. B. Widlund, A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity, SIAM J. Sci. Comput., 22 (2000), pp. 1199-1219.
    • (2000) SIAM J. Sci. Comput. , vol.22 , pp. 1199-1219
    • Klawonn, A.1    Widlund, O.B.2
  • 37
    • 0036205721 scopus 로고    scopus 로고
    • On block preconditioners for nonsymmetric saddle point problems
    • P. Krzyzanowski, On block preconditioners for nonsymmetric saddle point problems, SIAM J. Sci. Comput., 23 (2001), pp. 157-169.
    • (2001) SIAM J. Sci. Comput. , vol.23 , pp. 157-169
    • Krzyzanowski, P.1
  • 39
    • 0034497397 scopus 로고    scopus 로고
    • A note on preconditioning for indefinite linear systems
    • M. F. Murphy, G. H. Golub, and A. J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21 (2000), pp. 1969-1972.
    • (2000) SIAM J. Sci. Comput. , vol.21 , pp. 1969-1972
    • Murphy, M.F.1    Golub, G.H.2    Wathen, A.J.3
  • 41
    • 0033296578 scopus 로고    scopus 로고
    • Linear algebra methods in a mixed approximation of magnetostatic problems
    • I. Perugia, V. Simoncini, and M. Arioli, Linear algebra methods in a mixed approximation of magnetostatic problems, SIAM J. Sci. Comput., 21 (1999), pp. 1085-1101.
    • (1999) SIAM J. Sci. Comput. , vol.21 , pp. 1085-1101
    • Perugia, I.1    Simoncini, V.2    Arioli, M.3
  • 43
    • 0001464667 scopus 로고
    • A preconditioned iterative method for saddlepoint problems
    • T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 887-904.
    • (1992) SIAM J. Matrix Anal. Appl. , vol.13 , pp. 887-904
    • Rusten, T.1    Winther, R.2
  • 44
  • 45
    • 0346681528 scopus 로고
    • On the implicit deflation of nearly singular systems of linear equations
    • G. W. Stewart, On the implicit deflation of nearly singular systems of linear equations, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 136-140.
    • (1981) SIAM J. Sci. Statist. Comput. , vol.2 , pp. 136-140
    • Stewart, G.W.1
  • 48
    • 0024018037 scopus 로고
    • A framework for equilibrium equations
    • G. Strang, A framework for equilibrium equations, SIAM Rev., 30 (1988), pp. 283-297.
    • (1988) SIAM Rev. , vol.30 , pp. 283-297
    • Strang, G.1
  • 50
    • 0001071831 scopus 로고    scopus 로고
    • The convergence of iterative solution methods for symmetric and indefinite linear systems
    • Pitman Res. Notes Math. Ser. 380, Addison Wesley Longman Sci. Tech., Harlow, UK
    • A. J. Wathen, B. Fischer, and D. Silvester, The convergence of iterative solution methods for symmetric and indefinite linear systems, in Numerical Analysis 1997, Pitman Res. Notes Math. Ser. 380, Addison Wesley Longman Sci. Tech., Harlow, UK, 1997, pp. 230-243.
    • (1997) Numerical Analysis 1997 , pp. 230-243
    • Wathen, A.J.1    Fischer, B.2    Silvester, D.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.