# American Institute of Mathematical Sciences

August  2013, 7(3): 987-1005. doi: 10.3934/ipi.2013.7.987

## The single-grid multilevel method and its applications

 1 Department of Mathematics, The Pennsylvania State Univeristy, University Park, PA 16802, United States

Received  October 2012 Revised  February 2013 Published  September 2013

In this paper, we propose the single-grid multilevel (SGML) method for large-scale linear systems discretized from partial differential equations. The SGML method combines the methodologies of both the geometric and the algebraic multigrid methods. It uses the underlying geometric information from the finest grid. A simple and isotropic coarsening strategy is applied to explicitly control the complexity of the hierarchical structure, and smoothers are chosen based on the property of the model problem and the underlying grid information to complement the coarsening and maintain overall efficiency. Additionally, the underlying grid is used to design an efficient parallel algorithm in order to parallelize the SGML method. We apply the SGML method on the Poisson problem and the convection diffusion problem as examples, and we present the numerical results to demonstrate the performance of the SGML method.
Citation: Jinchao Xu. The single-grid multilevel method and its applications. Inverse Problems & Imaging, 2013, 7 (3) : 987-1005. doi: 10.3934/ipi.2013.7.987
##### 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. Google Scholar [2] R. Blaheta, A multilevel method with correction by aggregation for solving discrete elliptic problems,, Apl. Mat., 31 (1986), 365. Google Scholar [3] D. Braess, Towards algebraic multigrid for elliptic problems of second order,, Computing, 55 (1995), 379. doi: 10.1007/BF02238488. Google Scholar [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. Google Scholar [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. Google Scholar [8] A. Brandt, J. Brannick, K. Kahl and I. Livshits, Bootstrap AMG,, SIAM J. Sci. Comput., 33 (2011), 612. doi: 10.1137/090752973. Google Scholar [9] A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, In, (1985), 257. Google Scholar [10] A. Brandt, S. F. McCormick and J. W. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, in, (1984). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] W. L. Briggs, V. E. Henson and S. F. McCormick, "A Multigrid Tutorial,", Second edition, (2000). doi: 10.1137/1.9780898719505. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [25] R. Hiptmair, Multigrid method for Maxwell's equations,, SIAM J. Numer. Anal., 36 (1999), 204. doi: 10.1137/S0036142997326203. Google Scholar [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. Google Scholar [27] J. E. Jones and P. S. Vassilevski, AMGe based on element agglomeration,, SIAM J. Sci. Comput., 23 (2001), 109. doi: 10.1137/S1064827599361047. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] J. Mandel, M. Brezina and P. Vaněk, Energy optimization of algebraic multigrid bases,, Computing, 62 (1999), 205. doi: 10.1007/s006070050022. Google Scholar [34] Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123. Google Scholar [35] Y. Notay and A. Napov, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123. Google Scholar [36] Y. Notay and P. S. Vassilevski, Recursive Krylov-based multigrid cycles,, Numer. Linear Algebra Appl., 15 (2008), 473. doi: 10.1002/nla.542. Google Scholar [37] J. W. Ruge and K. Stüben, Algebraic multigrid,, in, 3 (1987), 73. Google Scholar [38] Y. Saad, A flexible inner-outer preconditioned gmres algorithm,, SIAM Journal on Scientific Computing, 14 (1993), 461. doi: 10.1137/0914028. Google Scholar [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). Google Scholar [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. Google Scholar [42] P. S. Vassilevski, "Multilevel Block Factorization Preconditioners. Matrix-based Analysis and Algorithms for Solving Finite Element Equations,", Springer, (2008). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [48] J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids,, Computing, 56 (1996), 215. doi: 10.1007/BF02238513. Google Scholar [49] J. Xu, Fast Poisson-based solvers for linear and nonlinear PDEs,, in, (2010), 2886. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [2] R. Blaheta, A multilevel method with correction by aggregation for solving discrete elliptic problems,, Apl. Mat., 31 (1986), 365. Google Scholar [3] D. Braess, Towards algebraic multigrid for elliptic problems of second order,, Computing, 55 (1995), 379. doi: 10.1007/BF02238488. Google Scholar [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. Google Scholar [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. Google Scholar [8] A. Brandt, J. Brannick, K. Kahl and I. Livshits, Bootstrap AMG,, SIAM J. Sci. Comput., 33 (2011), 612. doi: 10.1137/090752973. Google Scholar [9] A. Brandt, S. McCormick and J. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, In, (1985), 257. Google Scholar [10] A. Brandt, S. F. McCormick and J. W. Ruge, Algebraic multigrid (AMG) for sparse matrix equations,, in, (1984). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] W. L. Briggs, V. E. Henson and S. F. McCormick, "A Multigrid Tutorial,", Second edition, (2000). doi: 10.1137/1.9780898719505. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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). Google Scholar [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. Google Scholar [25] R. Hiptmair, Multigrid method for Maxwell's equations,, SIAM J. Numer. Anal., 36 (1999), 204. doi: 10.1137/S0036142997326203. Google Scholar [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. Google Scholar [27] J. E. Jones and P. S. Vassilevski, AMGe based on element agglomeration,, SIAM J. Sci. Comput., 23 (2001), 109. doi: 10.1137/S1064827599361047. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [33] J. Mandel, M. Brezina and P. Vaněk, Energy optimization of algebraic multigrid bases,, Computing, 62 (1999), 205. doi: 10.1007/s006070050022. Google Scholar [34] Y. Notay, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123. Google Scholar [35] Y. Notay and A. Napov, An aggregation-based algebraic multigrid method,, Electronic Transactions on Numerical Analysis, 37 (2010), 123. Google Scholar [36] Y. Notay and P. S. Vassilevski, Recursive Krylov-based multigrid cycles,, Numer. Linear Algebra Appl., 15 (2008), 473. doi: 10.1002/nla.542. Google Scholar [37] J. W. Ruge and K. Stüben, Algebraic multigrid,, in, 3 (1987), 73. Google Scholar [38] Y. Saad, A flexible inner-outer preconditioned gmres algorithm,, SIAM Journal on Scientific Computing, 14 (1993), 461. doi: 10.1137/0914028. Google Scholar [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). Google Scholar [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. Google Scholar [42] P. S. Vassilevski, "Multilevel Block Factorization Preconditioners. Matrix-based Analysis and Algorithms for Solving Finite Element Equations,", Springer, (2008). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [48] J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids,, Computing, 56 (1996), 215. doi: 10.1007/BF02238513. Google Scholar [49] J. Xu, Fast Poisson-based solvers for linear and nonlinear PDEs,, in, (2010), 2886. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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