메뉴 건너뛰기




Volumn 21, Issue 2, 2014, Pages 194-220

Theoretical bounds for algebraic multigrid performance: Review and analysis

Author keywords

Algebraic multigrid; Convergence bounds; Predictive theory

Indexed keywords


EID: 84893733926     PISSN: 10705325     EISSN: 10991506     Source Type: Journal    
DOI: 10.1002/nla.1930     Document Type: Article
Times cited : (37)

References (54)
  • 1
    • 84966233568 scopus 로고
    • Multi-level adaptive solutions to boundary-value problems
    • Brandt A. Multi-level adaptive solutions to boundary-value problems. Mathematics of Computation 1977 31(138):333-390.
    • (1977) Mathematics of Computation , vol.31 , Issue.138 , pp. 333-390
    • Brandt, A.1
  • 2
    • 84893764974 scopus 로고
    • Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical Report, Institute for Computational Studies, Colorado State University
    • Brandt A, McCormick SF, Ruge JW. Algebraic multigrid (AMG) for automatic multigrid solution with application to geodetic computations. Technical Report, Institute for Computational Studies, Colorado State University, 1982.
    • (1982)
    • Brandt, A.1    McCormick, S.F.2    Ruge, J.W.3
  • 3
    • 0002438068 scopus 로고
    • Algebraic multigrid (AMG) for sparse matrix equations
    • Evans DJ (ed.). Cambridge University Press: Cambridge
    • Brandt A, McCormick SF, Ruge JW. Algebraic multigrid (AMG) for sparse matrix equations. In Sparsity and Its Applications, Evans DJ (ed.). Cambridge University Press: Cambridge, 1984; 257-284.
    • (1984) Sparsity and Its Applications , pp. 257-284
    • Brandt, A.1    McCormick, S.F.2    Ruge, J.W.3
  • 4
    • 0001885411 scopus 로고
    • Algebraic multigrid
    • Frontiers in Applied Mathematics. SIAM: Philadelphia, PA
    • Ruge JW, Stüben K. Algebraic multigrid. In Multigrid Methods, Vol.3, Frontiers in Applied Mathematics. SIAM: Philadelphia, PA, 1987; 73-130.
    • (1987) Multigrid Methods , vol.3 , pp. 73-130
    • Ruge, J.W.1    Stüben, K.2
  • 6
    • 77949646375 scopus 로고    scopus 로고
    • When does two-grid optimality carry over to the V-cycle?
    • Napov A, Notay Y. When does two-grid optimality carry over to the V-cycle? Numerical Linear Algebra with Applications 2010 17(2-3):273-290.
    • (2010) Numerical Linear Algebra with Applications , vol.17 , Issue.2-3 , pp. 273-290
    • Napov, A.1    Notay, Y.2
  • 13
    • 0001822668 scopus 로고
    • Multigrid methods: fundamental algorithms, model problem analysis and applications
    • Hackbusch W, Trottenberg U (eds), Lecture Notes in Mathematics. Springer-Verlag: Berlin
    • Stüben K, Trottenberg U. Multigrid methods: fundamental algorithms, model problem analysis and applications. In Multigrid Methods, Vol.960, Hackbusch W, Trottenberg U (eds), Lecture Notes in Mathematics. Springer-Verlag: Berlin, 1982; 1-176.
    • (1982) Multigrid Methods , vol.960 , pp. 1-176
    • Stüben, K.1    Trottenberg, U.2
  • 14
    • 0343959849 scopus 로고
    • Algebraic formalisation of the multigrid method in the symmetric and positive definite case-a convergence estimation for the V-cycle
    • Institute of Mathematics and Its Applications Conference Series New Series. Oxford University Press: New York
    • Maitre JF, Musy F. Algebraic formalisation of the multigrid method in the symmetric and positive definite case-a convergence estimation for the V-cycle. In Multigrid Methods for Integral and Differential Equations (Bristol, 1983), Vol.3, Institute of Mathematics and Its Applications Conference Series New Series. Oxford University Press: New York, 1985; 213-223.
    • (1985) Multigrid Methods for Integral and Differential Equations (Bristol, 1983) , vol.3 , pp. 213-223
    • Maitre, J.F.1    Musy, F.2
  • 15
    • 0022107998 scopus 로고
    • Sharp estimates for multigrid rates of convergence with general smoothing and acceleration
    • Bank RE, Douglas CC. Sharp estimates for multigrid rates of convergence with general smoothing and acceleration. SIAM Journal on Numerical Analysis 1985 22(4):617-633.
    • (1985) SIAM Journal on Numerical Analysis , vol.22 , Issue.4 , pp. 617-633
    • Bank, R.E.1    Douglas, C.C.2
  • 16
    • 0001685657 scopus 로고
    • An algebraic interpretation of multigrid methods
    • McCormick SF. An algebraic interpretation of multigrid methods. SIAM Journal on Numerical Analysis 1982 19(3):548-560.
    • (1982) SIAM Journal on Numerical Analysis , vol.19 , Issue.3 , pp. 548-560
    • McCormick, S.F.1
  • 17
    • 0021417959 scopus 로고
    • Multigrid methods for variational problems: further results
    • McCormick SF. Multigrid methods for variational problems: further results. SIAM Journal on Numerical Analysis 1984 21(2):255-263.
    • (1984) SIAM Journal on Numerical Analysis , vol.21 , Issue.2 , pp. 255-263
    • McCormick, S.F.1
  • 18
    • 0022107991 scopus 로고
    • Multigrid methods for variational problems: general theory for the V-cycle
    • McCormick SF. Multigrid methods for variational problems: general theory for the V-cycle. SIAM Journal on Numerical Analysis 1985 22(4):634-643.
    • (1985) SIAM Journal on Numerical Analysis , vol.22 , Issue.4 , pp. 634-643
    • McCormick, S.F.1
  • 19
    • 0001927250 scopus 로고
    • Algebraic multigrid theory: the symmetric case
    • July
    • Brandt A. Algebraic multigrid theory: the symmetric case. Applied Mathematics and Computation July1986 19: 23-56.
    • (1986) Applied Mathematics and Computation , vol.19 , pp. 23-56
    • Brandt, A.1
  • 20
    • 84966225192 scopus 로고
    • Convergence estimates for product iterative methods with applications to domain decomposition
    • Bramble JH, Pasciak JE, Wang J, Xu J. Convergence estimates for product iterative methods with applications to domain decomposition. Mathematics of Computation 1991 57: 1-21.
    • (1991) Mathematics of Computation , vol.57 , pp. 1-21
    • Bramble, J.H.1    Pasciak, J.E.2    Wang, J.3    Xu, J.4
  • 21
    • 0036016141 scopus 로고    scopus 로고
    • The method of alternating projections and the method of subspace corrections in Hilbert space
    • Xu J, Zikatanov L. The method of alternating projections and the method of subspace corrections in Hilbert space. Journal of the American Mathematical Society 2002 15(3):573-597.
    • (2002) Journal of the American Mathematical Society , vol.15 , Issue.3 , pp. 573-597
    • Xu, J.1    Zikatanov, L.2
  • 22
    • 45549111807 scopus 로고
    • Algebraic study of multigrid methods for symmetric, definite problems
    • Mandel J. Algebraic study of multigrid methods for symmetric, definite problems. Applied Mathematics and Computation 1988 25(1, part I):39-56.
    • (1988) Applied Mathematics and Computation , vol.25 , Issue.1 PART I , pp. 39-56
    • Mandel, J.1
  • 23
    • 0008313962 scopus 로고    scopus 로고
    • Parallel multigrid methods
    • ICASE LaRC Interdisciplinary Series in Science and Engineering. Kluwer Acad. Publ.: Dordrecht
    • Jones JE, McCormick SF. Parallel multigrid methods. In Parallel Numerical Algorithms (Hampton, VA, 1994), Vol.4, ICASE LaRC Interdisciplinary Series in Science and Engineering. Kluwer Acad. Publ.: Dordrecht, 1997; 203-224.
    • (1997) Parallel Numerical Algorithms (Hampton, VA, 1994) , vol.4 , pp. 203-224
    • Jones, J.E.1    McCormick, S.F.2
  • 24
    • 0036532956 scopus 로고    scopus 로고
    • BoomerAMG: a parallel algebraic multigrid solver and preconditioner
    • Developments and trends in iterative methods for large systems of equations-in memoriam Rüdiger Weiss (Lausanne, 2000).
    • Henson VE, Yang UM. BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Applied Numerical Mathematics. An IMACS Journal 2002 41(1):155-177. Developments and trends in iterative methods for large systems of equations-in memoriam Rüdiger Weiss (Lausanne, 2000).
    • (2002) Applied Numerical Mathematics. An IMACS Journal , vol.41 , Issue.1 , pp. 155-177
    • Henson, V.E.1    Yang, U.M.2
  • 25
    • 70249096845 scopus 로고    scopus 로고
    • Auxiliary space AMG for H(curl) problems
    • Langer U, Discacciati M, Keyes DE, Widlund OB, Zulehner W, Barth TJ, Griebel M, Keyes DE, Nieminen RM, Roose D, Schlick T (eds), Lecture Notes in Computational Science and Engineering. Springer Berlin: Heidelberg
    • Kolev TV, Vassilevski PS. Auxiliary space AMG for H(curl) problems. In Domain Decomposition Methods in Science and Engineering XVII, Vol.60, Langer U, Discacciati M, Keyes DE, Widlund OB, Zulehner W, Barth TJ, Griebel M, Keyes DE, Nieminen RM, Roose D, Schlick T (eds), Lecture Notes in Computational Science and Engineering. Springer Berlin: Heidelberg, 2008; 147-154.
    • (2008) Domain Decomposition Methods in Science and Engineering XVII , vol.60 , pp. 147-154
    • Kolev, T.V.1    Vassilevski, P.S.2
  • 27
    • 0142055288 scopus 로고    scopus 로고
    • An algebraic multigrid method for finite element discretizations with edge elements
    • Reitzinger S, Schöberl J. An algebraic multigrid method for finite element discretizations with edge elements. Numerical Linear Algebra with Applications 2002 9: 223-238.
    • (2002) Numerical Linear Algebra with Applications , vol.9 , pp. 223-238
    • Reitzinger, S.1    Schöberl, J.2
  • 32
    • 0030388574 scopus 로고    scopus 로고
    • Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems
    • Vaněk P, Mandel J, Brezina M. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing. Archives for Scientific Computing 1996 56(3):179-196.
    • (1996) Computing. Archives for Scientific Computing , vol.56 , Issue.3 , pp. 179-196
    • Vaněk, P.1    Mandel, J.2    Brezina, M.3
  • 34
    • 77953814525 scopus 로고    scopus 로고
    • Compatible relaxation and coarsening in algebraic multigrid
    • Brannick JJ, Falgout RD. Compatible relaxation and coarsening in algebraic multigrid. SIAM Journal on Scientific Computing 2010 32(3):1393-1416.
    • (2010) SIAM Journal on Scientific Computing , vol.32 , Issue.3 , pp. 1393-1416
    • Brannick, J.J.1    Falgout, R.D.2
  • 39
    • 0003810147 scopus 로고
    • Mathematics and its Applications, Kluwer Academic Publishers Group: Dordrecht, Translated from the 1989 Russian original by N. B. Urusova and revised by the author.
    • Shaǐdurov VV. Multigrid Methods for Finite Elements, Mathematics and its Applications, Vol.318. Kluwer Academic Publishers Group: Dordrecht, 1995. Translated from the 1989 Russian original by N. B. Urusova and revised by the author.
    • (1995) Multigrid Methods for Finite Elements , vol.318
    • Shaǐdurov, V.V.1
  • 40
    • 84966216042 scopus 로고
    • Convergence of multi-grid iterations applied to difference equations
    • April
    • Hackbusch W. Convergence of multi-grid iterations applied to difference equations. Mathematics of Computation April1980 34(150):425-440.
    • (1980) Mathematics of Computation , vol.34 , Issue.150 , pp. 425-440
    • Hackbusch, W.1
  • 41
    • 33846798182 scopus 로고    scopus 로고
    • A V-cycle multigrid for multilevel matrix algebras: proof of optimality
    • Aricò A, Donatelli M. A V-cycle multigrid for multilevel matrix algebras: proof of optimality. Numerische Mathematik 2007 105(4):511-547.
    • (2007) Numerische Mathematik , vol.105 , Issue.4 , pp. 511-547
    • Aricò, A.1    Donatelli, M.2
  • 42
    • 84893795196 scopus 로고
    • On the convergence of multi-grid iteration. Technical Report 79-4, Mathematisches Institut, Universität zu Köln
    • Hackbusch W. On the convergence of multi-grid iteration. Technical Report 79-4, Mathematisches Institut, Universität zu Köln, 1979.
    • (1979)
    • Hackbusch, W.1
  • 43
    • 0005589275 scopus 로고
    • Survey of convergence proofs for multigrid iteration
    • Frehse J, Pallasche D, Trottenberg U (eds). North Holland Publishing Company: Amsterdam, New York
    • Hackbusch W. Survey of convergence proofs for multigrid iteration. In Special Topics in Applied Mathematics, Frehse J, Pallasche D, Trottenberg U (eds). North Holland Publishing Company: Amsterdam, New York, 1980; 151-164.
    • (1980) Special Topics in Applied Mathematics , pp. 151-164
    • Hackbusch, W.1
  • 45
    • 22144438010 scopus 로고    scopus 로고
    • Algebraic multigrid and algebraic multilevel methods: a theoretical comparison
    • Notay Y. Algebraic multigrid and algebraic multilevel methods: a theoretical comparison. Numerical Linear Algebra with Applications 2005 12(5-6):419-451.
    • (2005) Numerical Linear Algebra with Applications , vol.12 , Issue.5-6 , pp. 419-451
    • Notay, Y.1
  • 48
    • 4043106376 scopus 로고    scopus 로고
    • Multigrid solvers and multilevel optimization strategies
    • Cong J, Shinnerl JR (eds), Combinatorial Optimization. Kluwer Academic Publishers: Dordrecht, The Netherlands
    • Brandt A, Ron D. Multigrid solvers and multilevel optimization strategies. In Multilevel Optimization in VLSICAD, Vol.14, Cong J, Shinnerl JR (eds), Combinatorial Optimization. Kluwer Academic Publishers: Dordrecht, The Netherlands, 2003; 1-69.
    • (2003) Multilevel Optimization in VLSICAD , vol.14 , pp. 1-69
    • Brandt, A.1    Ron, D.2
  • 49
    • 84861387731 scopus 로고    scopus 로고
    • An algebraic multigrid method with guaranteed convergence rate
    • Napov A, Notay Y. An algebraic multigrid method with guaranteed convergence rate. SIAM Journal on Scientific Computing 2012 34(2):A1079-A1109.
    • (2012) SIAM Journal on Scientific Computing , vol.34 , Issue.2
    • Napov, A.1    Notay, Y.2
  • 50
    • 84885013804 scopus 로고    scopus 로고
    • Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs
    • Brannick J, Chen Y, Kraus J, Zikatanov L. Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs. SIAM Journal on Numerical Analysis 2013 51(3):1805-1827.
    • (2013) SIAM Journal on Numerical Analysis , vol.51 , Issue.3 , pp. 1805-1827
    • Brannick, J.1    Chen, Y.2    Kraus, J.3    Zikatanov, L.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.