메뉴 건너뛰기




Volumn 31, Issue 2, 2009, Pages 694-720

Combinatorial preconditioners for scalar elliptic finite-element problems

Author keywords

Combinatorial preconditioners; Finite elements; Preconditioning; Support preconditioners

Indexed keywords

ALGEBRAIC MULTIGRIDS; APPROXIMATION METHODS; COMBINATORIAL GRAPHS; DIAGONALLY DOMINANT; DIAGONALLY DOMINANT MATRIX; DIRECT SOLVERS; DISCRETIZATIONS; ELEMENT MATRIX; ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; FINITE ELEMENT; ILL-CONDITIONED; PRECONDITIONERS; PRECONDITIONING;

EID: 72449196960     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/060675940     Document Type: Article
Times cited : (5)

References (49)
  • 2
    • 72449146603 scopus 로고    scopus 로고
    • Using perturbed QR factorizations to solve linear least-squares problems
    • submitted.
    • H. Avron, E. Ng, and S. Toledo, Using perturbed QR factorizations to solve linear least-squares problems, SIAM J. Sci. Comput., submitted.
    • SIAM J. Sci. Comput.
    • Avron, H.1    Ng, E.2    Toledo, S.3
  • 3
    • 72449159475 scopus 로고    scopus 로고
    • Technical report, Tel-Aviv University, Israel, Apr. 2008
    • H. Avron, G. Shklarski, and S. Toledo, On element SDD approximability, Technical report, Tel-Aviv University, Israel, Apr. 2008, http://www.cs.tau.ac. il/-haima/element-approximability.pdf.
    • On Element SDD Approximability
    • Avron, H.1    Shklarski, G.2    Toledo, S.3
  • 4
    • 0002122998 scopus 로고
    • Optimally scaled matrices
    • F. L. Bauer, Optimally scaled matrices, Numer. Math., 5 (1963), pp. 73-87.
    • (1963) Numer. Math. , vol.5 , pp. 73-87
    • Bauer, F.L.1
  • 10
    • 55349087945 scopus 로고    scopus 로고
    • Solving elliptic finite element systems in near-linear time with support preconditioners
    • E. G. Boman, B. Hendrickson, and S. A. Vavasis, Solving elliptic finite element systems in near-linear time with support preconditioners, SIAM J. Numer. Anal., 46 (2008), pp. 3264-3284.
    • (2008) SIAM J. Numer. Anal. , vol.46 , pp. 3264-3284
    • Boman, E.G.1    Hendrickson, B.2    Vavasis, S.A.3
  • 12
    • 0028545288 scopus 로고
    • Minimizing the euclidean condition number
    • R. D. Braatz and M. Morari, Minimizing the Euclidean condition number, SIAM J.Control Optim., 32 (1994), pp. 1763-1768.
    • (1994) SIAM J.Control Optim. , vol.32 , pp. 1763-1768
    • Braatz, R.D.1    Morari, M.2
  • 15
    • 3042663694 scopus 로고    scopus 로고
    • Vaidya's preconditioners: Implementation and experimental study
    • D. Chen and S. Toledo, Vaidya's preconditioners: Implementation and experimental study, Electron. Trans. Numer. Anal., 16 (2003), pp. 30-49.
    • (2003) Electron. Trans. Numer. Anal. , vol.16 , pp. 30-49
    • Chen, D.1    Toledo, S.2
  • 16
    • 5444231904 scopus 로고    scopus 로고
    • Combinatorial characterization of the null spaces of symmetric H-matrices
    • D. Chen and S. Toledo, Combinatorial characterization of the null spaces of symmetric H-matrices, Linear Algebra Appl., 392 (2004), pp. 71-90.
    • (2004) Linear Algebra Appl. , vol.392 , pp. 71-90
    • Chen, D.1    Toledo, S.2
  • 17
    • 0002540389 scopus 로고
    • A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations
    • J. R. Bunch and D. J. Rose, eds., Academic Press, New York
    • P. Co-NCUS, G. H. GOLUB, AND D. P. 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, New York, 1976, pp. 309-332.
    • (1976) Sparse Matrix Computations , pp. 309-332
    • Co-Ncus, P.1    Golub, G.H.2    O'Leary, D.P.3
  • 19
    • 4043052305 scopus 로고    scopus 로고
    • New preconditioners for KKT systems of network flow problems
    • A. Frangioni AND C. GENTILE, New preconditioners for KKT systems of network flow problems, SIAM J. Optim., 14 (2004), pp. 894-913.
    • (2004) SIAM J. Optim. , vol.14 , pp. 894-913
    • Frangioni, A.1    Gentile, C.2
  • 20
    • 0002140542 scopus 로고    scopus 로고
    • Combinatorial preconditioners for sparse, symmetric, diagonally dominant linear systems
    • Ph.D. thesis, School of Computer Science, Carnegie Mellon University, Oct
    • K. D. Gremban, Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems, Ph.D. thesis, School of Computer Science, Carnegie Mellon University, Oct. 1996. Available as Technical Report CMU-CS-96 -123.
    • (1996) Available As Technical Report CMU-CS-96 -123
    • Gremban, K.D.1
  • 21
    • 0029218693 scopus 로고
    • Performance evaluation of a new parallel preconditioner, in Proceedings of the 9th International Parallel Processing Symposium
    • A longer ver sion is available as Technical Report CMU-CS-94-205, Carnegie-Mellon University
    • K. D. GREMBAN, G. L. MILLER, AND M. Zagha, Performance evaluation of a new parallel preconditioner, in Proceedings of the 9th International Parallel Processing Symposium, IEEE Computer Society, 1995, pp. 65-69. A longer ver sion is available as Technical Report CMU-CS-94-205, Carnegie-Mellon University.
    • (1995) IEEE Computer Society , pp. 65-69
    • Gremban, K.D.1    Miller, G.L.2    Zagha, M.3
  • 22
    • 0001748946 scopus 로고    scopus 로고
    • Algebraic multigrid methods based on element preconditioning
    • G. Haase, U. Langer, S. Reitzinger, AND J. Schicho, Algebraic multigrid methods based on element preconditioning, Int. J. Comput. Math., 78 (2001), pp. 575-598.
    • (2001) Int. J. Comput. Math. , vol.78 , pp. 575-598
    • Haase, G.1    Langer, U.2    Reitzinger, S.3    Schicho, J.4
  • 23
    • 0036532956 scopus 로고    scopus 로고
    • BoomerAMG: A parallel algebraic multigrid solver and pre-conditioner
    • V. E. Henson AND U. M. Yang, BoomerAMG: A parallel algebraic multigrid solver and pre-conditioner, Appl. Numer. Math., 41 (2002), pp. 155- 177.
    • (2002) Appl. Numer. Math. , vol.41 , pp. 155-177
    • Henson, V.E.1    Yang, U.M.2
  • 24
    • 0000135303 scopus 로고
    • Methods of conjugate gradients for solving linear systems
    • M. HESTENES AND E. STIEFEL, Methods of conjugate gradients for solving linear systems,J. Res. Natl. Bureau Standards, 49 (1952), pp. 409-436.
    • (1952) J. Res. Natl. Bureau Standards , vol.49 , pp. 409-436
    • Hestenes, M.1    Stiefel, E.2
  • 26
    • 0032131147 scopus 로고    scopus 로고
    • A fast and high quality multilevel scheme for partitioning irregular graphs
    • PII S1064827595287997
    • G. KARYPIS AND V. KUMAR, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359-392. (Pubitemid 128689516)
    • (1998) SIAM Journal of Scientific Computing , vol.20 , Issue.1 , pp. 359-392
    • Karypis, G.1    Kumar, V.2
  • 27
    • 79959646222 scopus 로고    scopus 로고
    • 1/6) time, parallel algorithm for solving planar Laplacians
    • SODA 2007, New Orleans, LA, January 7-9
    • 1/6) time, parallel algorithm for solving planar Laplacians, in Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, LA, January 7-9, 2007, N. Bansal, K. Pruhs, and C. Stein, eds., SIAM, Philadelphia, 2007, pp. 1002-10 11.
    • (2007) Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms , pp. 1002-1011
    • Koutis, I.1    Miller, G.L.2
  • 30
    • 84966241510 scopus 로고
    • An incomplete factorization technique for positive definite linear systems
    • T. MANTEUFFEL, An incomplete factorization technique for positive definite linear systems, Math. Comput., 34 (1980), pp. 473-497.
    • (1980) Math. Comput. , vol.34 , pp. 473-497
    • Manteuffel, T.1
  • 31
    • 72449125012 scopus 로고    scopus 로고
    • The MathWorks, Matlab version 7.2. software package, Jan. 2006
    • The MathWorks, Matlab version 7.2. software package, Jan. 2006.
  • 32
    • 84966225158 scopus 로고
    • An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix
    • J. A. MEIJERINK AND H. A. VAN DER Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31 (1977), pp. 148-162.
    • (1977) Math. Comput. , vol.31 , pp. 148-162
    • Meijerink, J.A.1    Vorst Der Van, H.A.2
  • 33
    • 0016555955 scopus 로고
    • Solution of sparse indefinite systems of linear equations
    • C. C. PAIGE AND M. A. SAUNDERS, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal., 12 (1975), pp. 617-629.
    • (1975) SIAM J. Numer. Anal. , vol.12 , pp. 617-629
    • Paige, C.C.1    Saunders, M.A.2
  • 34
    • 3142698698 scopus 로고    scopus 로고
    • A simple mesh generator in MATLAB
    • P.-O. Persson AND G. STRANG, A simple mesh generator in MATLAB, SIAM Rev., 46 (2004), pp. 329-345.
    • (2004) SIAM Rev. , vol.46 , pp. 329-345
    • Persson, P.-O.1    Strang, G.2
  • 35
    • 0000995669 scopus 로고    scopus 로고
    • An investigation of interior-point algorithms for the linear transportation problem
    • L. Portugal, F. Bastos, J. Júdice, J. Paixao, and T. Terlaky, An investigation of interior-point algorithms for the linear transportation problem, SIAM J. Sci. Comput., 17 (1996), pp. 1202-1223.
    • (1996) SIAM J. Sci. Comput. , vol.17 , pp. 1202-1223
    • Portugal, L.1    Bastos, F.2    Júdice, J.3    Paixao, J.4    Terlaky, T.5
  • 36
    • 0034147823 scopus 로고    scopus 로고
    • A truncatedprimal-infeasible dual-feasible interior point network flow method
    • L. F. Portugal, M. G. C. Resende, G. Veiga, and J. J. J udice, A truncatedprimal-infeasible dual-feasible interior point network flow method, Networks, 35 (2000), pp. 91-108.
    • (2000) Networks , vol.35 , pp. 91-108
    • Portugal, L.F.1    Resende, M.G.C.2    Veiga, G.3    Udice, J.J.J.4
  • 37
    • 0005722513 scopus 로고    scopus 로고
    • Algebraic multigrid and element preconditioning I
    • J. Kepler Universität (JKU) Linz, Dec.
    • S. REITZINGER, Algebraic multigrid and element preconditioning I, Technical report, J. Kepler Universität (JKU) Linz, Dec. 1998.
    • (1998) Technical Report
    • Reitzinger, S.1
  • 38
    • 0001024483 scopus 로고
    • An efficient implementation of the network interior-point method, in network flows and matching: The first DIMACS implementation challenge
    • D. Johnson and C. McGeoch, eds., AMS, New York
    • M. RESENSE AND G. VEIGA, An efficient implementation of the network interior-point method, in Network Flows and Matching: The First DIMACS Implementation Challenge, D. Johnson and C. McGeoch, eds., DIMACS Series in Discrete Mathematics and Computer Science 12, AMS, New York, 1993.
    • (1993) DIMACS Series in Discrete Mathematics and Computer Science , vol.12
    • Resense, M.1    Veiga, G.2
  • 39
    • 0012189842 scopus 로고
    • Regular incomplete factorizations of real positive definite matrices
    • Y. Robert, Regular incomplete factorizations of real positive definite matrices, Linear Algebra Appl., 48 (1982), pp. 105-117.
    • (1982) Linear Algebra Appl. , vol.48 , pp. 105-117
    • Robert, Y.1
  • 40
    • 0012065028 scopus 로고
    • Upper bounds for nearly optimal diagonal scaling of matrices
    • A. Shapiro, Upper bounds for nearly optimal diagonal scaling of matrices, Linear and Multilinear Algebra, 29 (1991), pp. 145-147.
    • (1991) Linear and Multilinear Algebra , vol.29 , pp. 145-147
    • Shapiro, A.1
  • 41
    • 61849092590 scopus 로고    scopus 로고
    • Rigidity in finite-element matrices: Sufficient conditions for the rigidity of structures and substructures
    • G. SHKLARSKI AND S. Toledo, Rigidity in finite-element matrices: Sufficient conditions for the rigidity of structures and substructures, SIAM J. Mat rix Anal. Appl., 30 (2008), pp. 7-40.
    • (2008) SIAM J. Mat Rix Anal. Appl. , vol.30 , pp. 7-40
    • Shklarski, G.1    Toledo, S.2
  • 46
    • 0038344596 scopus 로고
    • Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners, unpublished manuscript
    • A talk based on this manuscript was presented at the Minneapolis, MN, October
    • P. M. Vaidya, Solving linear equations with symmetric diagonally dominant matrices by constructing good preconditioners, unpublished manuscript. A talk based on this manuscript was presented at the IMA Workshop on Graph Theory and Sparse Matrix Computations, Minneapolis, MN, October 1991.
    • (1991) IMA Workshop on Graph Theory and Sparse Matrix Computations
    • Vaidya, P.M.1
  • 47
    • 0002090018 scopus 로고
    • Condition numbers and equilibration of matrices
    • A. van DER Sluis, Condition numbers and equilibration of matrices, Numer. Math., 14 (1969), pp. 14-23.
    • (1969) Numer. Math. , vol.14 , pp. 14-23
    • Van Der Sluis, A.1
  • 48
    • 0012126476 scopus 로고
    • Incomplete factorizations of matrices and connections with H-matrices
    • R. S. Varga, Saff E. B., and V. Mehrmann, Incomplete factorizations of matrices and connections with H-matrices, SIAM J. Numer. Anal., 17 (1980), pp. 787-793.
    • (1980) SIAM J. Numer. Anal. , vol.17 , pp. 787-793
    • Varga, R.S.1    Saff, E.B.2    Mehrmann, V.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.