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The single-grid multilevel method and its applications
1. | Department of Mathematics, The Pennsylvania State Univeristy, University Park, PA 16802, United States |
References:
[1] |
D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H(div) and H(curl),, Numer. Math., 85 (2000), 197.
doi: 10.1007/PL00005386. |
[2] |
R. Blaheta, A multilevel method with correction by aggregation for solving discrete elliptic problems,, Apl. Mat., 31 (1986), 365.
|
[3] |
D. Braess, Towards algebraic multigrid for elliptic problems of second order,, Computing, 55 (1995), 379.
doi: 10.1007/BF02238488. |
[4] |
J. H. Bramble, "Multigrid Methods,", Pitman Research Notes in Mathematical Sciences, 294 (1993). Google Scholar |
[5] |
A. Brandt, Multi-level adaptive solutions to boundary-value problems,, Math. Comp., 31 (1977), 333.
doi: 10.1090/S0025-5718-1977-0431719-X. |
[6] |
A. Brandt, "Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics,", GMD-Studien [GMD Studies], 85 (1984). Google Scholar |
[7] |
A. Brandt, General highly accurate algebraic coarsening,, Elect. Trans. Numer. Anal., 10 (2000), 1.
|
[8] |
A. Brandt, J. Brannick, K. Kahl and I. Livshits, Bootstrap AMG,, SIAM J. Sci. Comput., 33 (2011), 612.
doi: 10.1137/090752973. |
[9] |
A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, In, (1985), 257.
|
[10] |
A. Brandt, S. F. McCormick and J. W. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, in, (1984).
|
[11] |
J. Brannick, Y. Chen, J. Krauss and L. Zikatanov, An algebraic multigrid method based on matching in graphs,, in, 91 (2013), 143.
doi: 10.1007/978-3-642-35275-1_15. |
[12] |
J. Brannick, Y. Chen and L. Zikatanov, An algebraic multilevel method for anisotropic elliptic equations based on graph partitioning,, Numer. Linear Algebra Appl., 19 (2012), 279.
doi: 10.1002/nla.1804. |
[13] |
J. Brannick and L. Zikatanov, Algebraic multigrid methods based on compatible relaxation and energy minimization,, in, 55 (2007), 15.
doi: 10.1007/978-3-540-34469-8_2. |
[14] |
M. Brezina, A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick and J. Ruge, Algebraic multigrid based on element interpolation (amge),, SIAM Journal on Scientific Computing, 22 (2001), 1570.
doi: 10.1137/S1064827598344303. |
[15] |
M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick and J. Ruge, Adaptive smoothed aggregation $(\alpha SA)$ multigrid,, SIAM Rev., 47 (2005), 317.
doi: 10.1137/050626272. |
[16] |
W. L. Briggs, V. E. Henson and S. F. McCormick, "A Multigrid Tutorial,", Second edition, (2000).
doi: 10.1137/1.9780898719505. |
[17] |
V. E. Bulgakov, Multi-level iterative technique and aggregation concept with semi-analytical preconditioning for solving boundary-value problems,, Communications in Numerical Methods in Engineering, 9 (1993), 649.
doi: 10.1002/cnm.1640090804. |
[18] |
T. Chartier, R. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, J. W. Ruge and P. S. Vassilevski, Spectral element agglomerate AMGe,, in, 55 (2007), 513.
doi: 10.1007/978-3-540-34469-8_64. |
[19] |
T. Chartier, R. D. Falgout, V. E. Henson, J. Jones, T. Manteuffel, S. McCormick, J. Ruge and P. S. Vassilevski, Spectral AMGe ($\rho$ AMGe),, SIAM J. Sci. Comput., 25 (2003), 1.
doi: 10.1137/S106482750139892X. |
[20] |
R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients,, The Computer Journal, 7 (1964), 149.
doi: 10.1093/comjnl/7.2.149. |
[21] |
M. Garland and N. Bell, CUSP: Generic parallel algorithms for sparse matrix and graph computations,, 2010., (). Google Scholar |
[22] |
L. Grasedyck, J. Xu and L. Wang, Algebraic multigrid methods based on auxiliary grids,, preprint, (2013). Google Scholar |
[23] |
W. Hackbusch, "Multigrid Methods and Applications,", Computational Mathematics, 4 (1985).
|
[24] |
V. E. Henson and P. S. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG,, Copper Mountain Conference (2000), 23 (2000), 629.
doi: 10.1137/S1064827500372997. |
[25] |
R. Hiptmair, Multigrid method for Maxwell's equations,, SIAM J. Numer. Anal., 36 (1999), 204.
doi: 10.1137/S0036142997326203. |
[26] |
X. Hu, P. S. Vasilevski and J. Xu, Comparative convergence analysis of nonlinear AMLI-cycle multigrid,, Submitted to SIAM Journal on Numerical Analysis., 51 (2013), 1349.
doi: 10.1137/110850049. |
[27] |
J. E. Jones and P. S. Vassilevski, AMGe based on element agglomeration,, SIAM J. Sci. Comput., 23 (2001), 109.
doi: 10.1137/S1064827599361047. |
[28] |
H. Kim, J. Xu and L. Zikatanov, A multigrid method based on graph matching for convection-diffusion equations,, Numer. Linear Algebra Appl., 10 (2003), 181.
doi: 10.1002/nla.317. |
[29] |
H. Kim, J. Xu and L. Zikatanov, Uniformly convergent multigrid methods for convection-diffusion problems without any constraint on coarse grids,, Adv. Comput. Math., 20 (2004), 385.
doi: 10.1023/A:1027378015262. |
[30] |
J. K. Kraus, An algebraic preconditioning method for $M$-matrices: Linear versus non-linear multilevel iteration,, Numer. Linear Algebra Appl., 9 (2002), 599.
doi: 10.1002/nla.281. |
[31] |
I. Lashuk and P. S. Vassilevski, On some versions of the element agglomeration AMGe method,, Numer. Linear Algebra Appl., 15 (2008), 595.
doi: 10.1002/nla.585. |
[32] |
O. Livne and A. Brandt, Lean algebraic multigrid (lamg): Fast graph laplacian linear solver,, SIAM J. Sci. Comput., 34 (2012), 499.
doi: 10.1137/110843563. |
[33] |
J. Mandel, M. Brezina and P. Vaněk, Energy optimization of algebraic multigrid bases,, Computing, 62 (1999), 205.
doi: 10.1007/s006070050022. |
[34] |
Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
|
[35] |
Y. Notay and A. Napov, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
|
[36] |
Y. Notay and P. S. Vassilevski, Recursive Krylov-based multigrid cycles,, Numer. Linear Algebra Appl., 15 (2008), 473.
doi: 10.1002/nla.542. |
[37] |
J. W. Ruge and K. Stüben, Algebraic multigrid,, in, 3 (1987), 73.
|
[38] |
Y. Saad, A flexible inner-outer preconditioned gmres algorithm,, SIAM Journal on Scientific Computing, 14 (1993), 461.
doi: 10.1137/0914028. |
[39] |
K. Stüben, An introduction to algebraic multigrid,, in, (2001), 413. Google Scholar |
[40] |
U. Trottenberg, C. Oosterlee and A. Schüller, "Multigrid,", Academic Press, (2001).
|
[41] |
P. Vaněk, J. Mandel and M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems,, International GAMM-Workshop on Multi-level Methods (Meisdorf, 56 (1996), 179.
doi: 10.1007/BF02238511. |
[42] |
P. S. Vassilevski, "Multilevel Block Factorization Preconditioners. Matrix-based Analysis and Algorithms for Solving Finite Element Equations,", Springer, (2008).
|
[43] |
W. L. Wan, T. F. Chan and B. Smith, An energy-minimizing interpolation for robust multigrid methods,, SIAM J. Sci. Comput., 21 (2000), 1632.
doi: 10.1137/S1064827598334277. |
[44] |
F. Wang and J. Xu, A crosswind block iterative method for convection-dominated problems,, SIAM J. Sci. Comput., 21 (1999), 620.
doi: 10.1137/S106482759631192X. |
[45] |
L. Wang, X. Hu, J. Cohen and J. Xu, A parallel auxiliary grid AMG method for GPU,, SIAM J. Sci. Comput., 35 (). Google Scholar |
[46] |
P. Wesseling, "An Introduction to Multigrid Methods,", Reprint of the 1992 edition, (1992). Google Scholar |
[47] |
J. Xu, Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems,, in, (1992), 106.
|
[48] |
J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids,, Computing, 56 (1996), 215.
doi: 10.1007/BF02238513. |
[49] |
J. Xu, Fast Poisson-based solvers for linear and nonlinear PDEs,, in, (2010), 2886.
|
[50] |
J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations,, Math. Comp., 68 (1999), 1429.
doi: 10.1090/S0025-5718-99-01148-5. |
[51] |
J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space,, J. Amer. Math. Soc., 15 (2002), 573.
doi: 10.1090/S0894-0347-02-00398-3. |
[52] |
J. Xu and L. Zikatanov, On an energy minimizing basis for algebraic multigrid methods,, Comput. Vis. Sci., 7 (2004), 121.
doi: 10.1007/s00791-004-0147-y. |
[53] |
L. Zikatanov, Two-sided bounds on the convergence rate of two-level methods,, Numer. Linear Alg. Appl., 15 (2008), 439.
doi: 10.1002/nla.556. |
show all references
References:
[1] |
D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H(div) and H(curl),, Numer. Math., 85 (2000), 197.
doi: 10.1007/PL00005386. |
[2] |
R. Blaheta, A multilevel method with correction by aggregation for solving discrete elliptic problems,, Apl. Mat., 31 (1986), 365.
|
[3] |
D. Braess, Towards algebraic multigrid for elliptic problems of second order,, Computing, 55 (1995), 379.
doi: 10.1007/BF02238488. |
[4] |
J. H. Bramble, "Multigrid Methods,", Pitman Research Notes in Mathematical Sciences, 294 (1993). Google Scholar |
[5] |
A. Brandt, Multi-level adaptive solutions to boundary-value problems,, Math. Comp., 31 (1977), 333.
doi: 10.1090/S0025-5718-1977-0431719-X. |
[6] |
A. Brandt, "Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics,", GMD-Studien [GMD Studies], 85 (1984). Google Scholar |
[7] |
A. Brandt, General highly accurate algebraic coarsening,, Elect. Trans. Numer. Anal., 10 (2000), 1.
|
[8] |
A. Brandt, J. Brannick, K. Kahl and I. Livshits, Bootstrap AMG,, SIAM J. Sci. Comput., 33 (2011), 612.
doi: 10.1137/090752973. |
[9] |
A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, In, (1985), 257.
|
[10] |
A. Brandt, S. F. McCormick and J. W. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, in, (1984).
|
[11] |
J. Brannick, Y. Chen, J. Krauss and L. Zikatanov, An algebraic multigrid method based on matching in graphs,, in, 91 (2013), 143.
doi: 10.1007/978-3-642-35275-1_15. |
[12] |
J. Brannick, Y. Chen and L. Zikatanov, An algebraic multilevel method for anisotropic elliptic equations based on graph partitioning,, Numer. Linear Algebra Appl., 19 (2012), 279.
doi: 10.1002/nla.1804. |
[13] |
J. Brannick and L. Zikatanov, Algebraic multigrid methods based on compatible relaxation and energy minimization,, in, 55 (2007), 15.
doi: 10.1007/978-3-540-34469-8_2. |
[14] |
M. Brezina, A. Cleary, R. Falgout, V. Henson, J. Jones, T. Manteuffel, S. McCormick and J. Ruge, Algebraic multigrid based on element interpolation (amge),, SIAM Journal on Scientific Computing, 22 (2001), 1570.
doi: 10.1137/S1064827598344303. |
[15] |
M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick and J. Ruge, Adaptive smoothed aggregation $(\alpha SA)$ multigrid,, SIAM Rev., 47 (2005), 317.
doi: 10.1137/050626272. |
[16] |
W. L. Briggs, V. E. Henson and S. F. McCormick, "A Multigrid Tutorial,", Second edition, (2000).
doi: 10.1137/1.9780898719505. |
[17] |
V. E. Bulgakov, Multi-level iterative technique and aggregation concept with semi-analytical preconditioning for solving boundary-value problems,, Communications in Numerical Methods in Engineering, 9 (1993), 649.
doi: 10.1002/cnm.1640090804. |
[18] |
T. Chartier, R. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, J. W. Ruge and P. S. Vassilevski, Spectral element agglomerate AMGe,, in, 55 (2007), 513.
doi: 10.1007/978-3-540-34469-8_64. |
[19] |
T. Chartier, R. D. Falgout, V. E. Henson, J. Jones, T. Manteuffel, S. McCormick, J. Ruge and P. S. Vassilevski, Spectral AMGe ($\rho$ AMGe),, SIAM J. Sci. Comput., 25 (2003), 1.
doi: 10.1137/S106482750139892X. |
[20] |
R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients,, The Computer Journal, 7 (1964), 149.
doi: 10.1093/comjnl/7.2.149. |
[21] |
M. Garland and N. Bell, CUSP: Generic parallel algorithms for sparse matrix and graph computations,, 2010., (). Google Scholar |
[22] |
L. Grasedyck, J. Xu and L. Wang, Algebraic multigrid methods based on auxiliary grids,, preprint, (2013). Google Scholar |
[23] |
W. Hackbusch, "Multigrid Methods and Applications,", Computational Mathematics, 4 (1985).
|
[24] |
V. E. Henson and P. S. Vassilevski, Element-free AMGe: General algorithms for computing interpolation weights in AMG,, Copper Mountain Conference (2000), 23 (2000), 629.
doi: 10.1137/S1064827500372997. |
[25] |
R. Hiptmair, Multigrid method for Maxwell's equations,, SIAM J. Numer. Anal., 36 (1999), 204.
doi: 10.1137/S0036142997326203. |
[26] |
X. Hu, P. S. Vasilevski and J. Xu, Comparative convergence analysis of nonlinear AMLI-cycle multigrid,, Submitted to SIAM Journal on Numerical Analysis., 51 (2013), 1349.
doi: 10.1137/110850049. |
[27] |
J. E. Jones and P. S. Vassilevski, AMGe based on element agglomeration,, SIAM J. Sci. Comput., 23 (2001), 109.
doi: 10.1137/S1064827599361047. |
[28] |
H. Kim, J. Xu and L. Zikatanov, A multigrid method based on graph matching for convection-diffusion equations,, Numer. Linear Algebra Appl., 10 (2003), 181.
doi: 10.1002/nla.317. |
[29] |
H. Kim, J. Xu and L. Zikatanov, Uniformly convergent multigrid methods for convection-diffusion problems without any constraint on coarse grids,, Adv. Comput. Math., 20 (2004), 385.
doi: 10.1023/A:1027378015262. |
[30] |
J. K. Kraus, An algebraic preconditioning method for $M$-matrices: Linear versus non-linear multilevel iteration,, Numer. Linear Algebra Appl., 9 (2002), 599.
doi: 10.1002/nla.281. |
[31] |
I. Lashuk and P. S. Vassilevski, On some versions of the element agglomeration AMGe method,, Numer. Linear Algebra Appl., 15 (2008), 595.
doi: 10.1002/nla.585. |
[32] |
O. Livne and A. Brandt, Lean algebraic multigrid (lamg): Fast graph laplacian linear solver,, SIAM J. Sci. Comput., 34 (2012), 499.
doi: 10.1137/110843563. |
[33] |
J. Mandel, M. Brezina and P. Vaněk, Energy optimization of algebraic multigrid bases,, Computing, 62 (1999), 205.
doi: 10.1007/s006070050022. |
[34] |
Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
|
[35] |
Y. Notay and A. Napov, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123.
|
[36] |
Y. Notay and P. S. Vassilevski, Recursive Krylov-based multigrid cycles,, Numer. Linear Algebra Appl., 15 (2008), 473.
doi: 10.1002/nla.542. |
[37] |
J. W. Ruge and K. Stüben, Algebraic multigrid,, in, 3 (1987), 73.
|
[38] |
Y. Saad, A flexible inner-outer preconditioned gmres algorithm,, SIAM Journal on Scientific Computing, 14 (1993), 461.
doi: 10.1137/0914028. |
[39] |
K. Stüben, An introduction to algebraic multigrid,, in, (2001), 413. Google Scholar |
[40] |
U. Trottenberg, C. Oosterlee and A. Schüller, "Multigrid,", Academic Press, (2001).
|
[41] |
P. Vaněk, J. Mandel and M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems,, International GAMM-Workshop on Multi-level Methods (Meisdorf, 56 (1996), 179.
doi: 10.1007/BF02238511. |
[42] |
P. S. Vassilevski, "Multilevel Block Factorization Preconditioners. Matrix-based Analysis and Algorithms for Solving Finite Element Equations,", Springer, (2008).
|
[43] |
W. L. Wan, T. F. Chan and B. Smith, An energy-minimizing interpolation for robust multigrid methods,, SIAM J. Sci. Comput., 21 (2000), 1632.
doi: 10.1137/S1064827598334277. |
[44] |
F. Wang and J. Xu, A crosswind block iterative method for convection-dominated problems,, SIAM J. Sci. Comput., 21 (1999), 620.
doi: 10.1137/S106482759631192X. |
[45] |
L. Wang, X. Hu, J. Cohen and J. Xu, A parallel auxiliary grid AMG method for GPU,, SIAM J. Sci. Comput., 35 (). Google Scholar |
[46] |
P. Wesseling, "An Introduction to Multigrid Methods,", Reprint of the 1992 edition, (1992). Google Scholar |
[47] |
J. Xu, Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems,, in, (1992), 106.
|
[48] |
J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids,, Computing, 56 (1996), 215.
doi: 10.1007/BF02238513. |
[49] |
J. Xu, Fast Poisson-based solvers for linear and nonlinear PDEs,, in, (2010), 2886.
|
[50] |
J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations,, Math. Comp., 68 (1999), 1429.
doi: 10.1090/S0025-5718-99-01148-5. |
[51] |
J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space,, J. Amer. Math. Soc., 15 (2002), 573.
doi: 10.1090/S0894-0347-02-00398-3. |
[52] |
J. Xu and L. Zikatanov, On an energy minimizing basis for algebraic multigrid methods,, Comput. Vis. Sci., 7 (2004), 121.
doi: 10.1007/s00791-004-0147-y. |
[53] |
L. Zikatanov, Two-sided bounds on the convergence rate of two-level methods,, Numer. Linear Alg. Appl., 15 (2008), 439.
doi: 10.1002/nla.556. |
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