[petsc-users] Convergence of AMG
Smith, Barry F.
bsmith at mcs.anl.gov
Sun Oct 28 15:55:27 CDT 2018
> On Oct 28, 2018, at 12:16 PM, Manav Bhatia <bhatiamanav at gmail.com> wrote:
>
> Hi,
>
> I am attempting to solve a Mindlin plate bending problem with AMG solver in petsc. This test case is with a mesh of 300x300 elements and 543,606 dofs.
>
> The discretization includes 6 variables (u, v, w, tx, ty, tz), but only three are relevant for plate bending (w, tx, ty).
>
> I am calling the solver with the following options:
>
> -pc_type gamg -pc_gamg_threshold 0. --node-major-dofs -mat_block_size 6 -ksp_rtol 1.e-8 -ksp_monitor -ksp_converged_reason -ksp_view
>
> And the convergence behavior is shown below, along with the ksp_view information. Based on notes in the manual, this seems to be subpar convergence rate. At the end of the solution the norm of each variable is :
>
> var: 0: u : norm: 5.505909e-18
> var: 1: v : norm: 7.639640e-18
> var: 2: w : norm: 3.901464e-03
> var: 3: tx : norm: 4.403576e-02
> var: 4: ty : norm: 4.403576e-02
> var: 5: tz : norm: 1.148409e-16
What do you mean by var: 2: w : norm etc? Is this the norm of the error for that variable, the norm of the residual, something else? How exactly are you calculating it?
Thanks
Barry
>
> I tried different values of -ksp_rtol from 1e-1 to 1e-8 and this does not make a lot of difference in the norms of (w, tx, ty).
>
> I do provide the solver with 6 rigid-body vectors to approximate the null-space of the problem. Without these the solver shows very poor convergence.
>
> I would appreciate advice on possible strategies to improve this behavior.
>
> Thanks,
> Manav
>
> 0 KSP Residual norm 1.696304497261e+00
> 1 KSP Residual norm 1.120485505777e+00
> 2 KSP Residual norm 8.324222302402e-01
> 3 KSP Residual norm 6.477349534115e-01
> 4 KSP Residual norm 5.080936471292e-01
> 5 KSP Residual norm 4.051099646638e-01
> 6 KSP Residual norm 3.260432664653e-01
> 7 KSP Residual norm 2.560483838143e-01
> 8 KSP Residual norm 2.029943986124e-01
> 9 KSP Residual norm 1.560985741610e-01
> 10 KSP Residual norm 1.163720702140e-01
> 11 KSP Residual norm 8.488411085459e-02
> 12 KSP Residual norm 5.888041729034e-02
> 13 KSP Residual norm 4.027792209980e-02
> 14 KSP Residual norm 2.819048087304e-02
> 15 KSP Residual norm 1.904674196962e-02
> 16 KSP Residual norm 1.289302447822e-02
> 17 KSP Residual norm 9.162203296376e-03
> 18 KSP Residual norm 7.016781679507e-03
> 19 KSP Residual norm 5.399170865328e-03
> 20 KSP Residual norm 4.254385887482e-03
> 21 KSP Residual norm 3.530831740621e-03
> 22 KSP Residual norm 2.946780747923e-03
> 23 KSP Residual norm 2.339361361128e-03
> 24 KSP Residual norm 1.815072489282e-03
> 25 KSP Residual norm 1.408814185342e-03
> 26 KSP Residual norm 1.063795714320e-03
> 27 KSP Residual norm 7.828540233117e-04
> 28 KSP Residual norm 5.683910750067e-04
> 29 KSP Residual norm 4.131151010250e-04
> 30 KSP Residual norm 3.065608221019e-04
> 31 KSP Residual norm 2.634114273459e-04
> 32 KSP Residual norm 2.198180137626e-04
> 33 KSP Residual norm 1.748956510799e-04
> 34 KSP Residual norm 1.317539710010e-04
> 35 KSP Residual norm 9.790121566055e-05
> 36 KSP Residual norm 7.465935386094e-05
> 37 KSP Residual norm 5.689506626052e-05
> 38 KSP Residual norm 4.413136619126e-05
> 39 KSP Residual norm 3.512194236402e-05
> 40 KSP Residual norm 2.877755408287e-05
> 41 KSP Residual norm 2.340080556431e-05
> 42 KSP Residual norm 1.904544450345e-05
> 43 KSP Residual norm 1.504723478235e-05
> 44 KSP Residual norm 1.141381950576e-05
> 45 KSP Residual norm 8.206151384599e-06
> 46 KSP Residual norm 5.911426091276e-06
> 47 KSP Residual norm 4.233669089283e-06
> 48 KSP Residual norm 2.898052944223e-06
> 49 KSP Residual norm 2.023556779973e-06
> 50 KSP Residual norm 1.459108043935e-06
> 51 KSP Residual norm 1.097335545865e-06
> 52 KSP Residual norm 8.440457332262e-07
> 53 KSP Residual norm 6.705616854004e-07
> 54 KSP Residual norm 5.404888680234e-07
> 55 KSP Residual norm 4.391368084979e-07
> 56 KSP Residual norm 3.697063014621e-07
> 57 KSP Residual norm 3.021772094146e-07
> 58 KSP Residual norm 2.479354520792e-07
> 59 KSP Residual norm 2.013077841968e-07
> 60 KSP Residual norm 1.553159612793e-07
> 61 KSP Residual norm 1.400784224898e-07
> 62 KSP Residual norm 9.707453662195e-08
> 63 KSP Residual norm 7.263173080146e-08
> 64 KSP Residual norm 5.593723572132e-08
> 65 KSP Residual norm 4.448788809586e-08
> 66 KSP Residual norm 3.613992590778e-08
> 67 KSP Residual norm 2.946099051876e-08
> 68 KSP Residual norm 2.408053564170e-08
> 69 KSP Residual norm 1.945257374856e-08
> 70 KSP Residual norm 1.572494535110e-08
>
>
> KSP Object: 4 MPI processes
> type: gmres
> restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
> happy breakdown tolerance 1e-30
> maximum iterations=10000, initial guess is zero
> tolerances: relative=1e-08, absolute=1e-50, divergence=10000.
> left preconditioning
> using PRECONDITIONED norm type for convergence test
> PC Object: 4 MPI processes
> type: gamg
> type is MULTIPLICATIVE, levels=6 cycles=v
> Cycles per PCApply=1
> Using externally compute Galerkin coarse grid matrices
> GAMG specific options
> Threshold for dropping small values in graph on each level = 0. 0. 0. 0.
> Threshold scaling factor for each level not specified = 1.
> AGG specific options
> Symmetric graph false
> Number of levels to square graph 1
> Number smoothing steps 1
> Coarse grid solver -- level -------------------------------
> KSP Object: (mg_coarse_) 4 MPI processes
> type: preonly
> maximum iterations=10000, initial guess is zero
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_coarse_) 4 MPI processes
> type: bjacobi
> number of blocks = 4
> Local solve is same for all blocks, in the following KSP and PC objects:
> KSP Object: (mg_coarse_sub_) 1 MPI processes
> type: preonly
> maximum iterations=1, initial guess is zero
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_coarse_sub_) 1 MPI processes
> type: lu
> out-of-place factorization
> tolerance for zero pivot 2.22045e-14
> using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
> matrix ordering: nd
> factor fill ratio given 5., needed 1.
> Factored matrix follows:
> Mat Object: 1 MPI processes
> type: seqaij
> rows=6, cols=6, bs=6
> package used to perform factorization: petsc
> total: nonzeros=36, allocated nonzeros=36
> total number of mallocs used during MatSetValues calls =0
> using I-node routines: found 2 nodes, limit used is 5
> linear system matrix = precond matrix:
> Mat Object: 1 MPI processes
> type: seqaij
> rows=6, cols=6, bs=6
> total: nonzeros=36, allocated nonzeros=36
> total number of mallocs used during MatSetValues calls =0
> using I-node routines: found 2 nodes, limit used is 5
> linear system matrix = precond matrix:
> Mat Object: 4 MPI processes
> type: mpiaij
> rows=6, cols=6, bs=6
> total: nonzeros=36, allocated nonzeros=36
> total number of mallocs used during MatSetValues calls =0
> using nonscalable MatPtAP() implementation
> using I-node (on process 0) routines: found 2 nodes, limit used is 5
> Down solver (pre-smoother) on level 1 -------------------------------
> KSP Object: (mg_levels_1_) 4 MPI processes
> type: chebyshev
> eigenvalue estimates used: min = 0.099971, max = 1.09968
> eigenvalues estimate via gmres min 0.154032, max 0.99971
> eigenvalues estimated using gmres with translations [0. 0.1; 0. 1.1]
> KSP Object: (mg_levels_1_esteig_) 4 MPI processes
> type: gmres
> restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
> happy breakdown tolerance 1e-30
> maximum iterations=10, initial guess is zero
> tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
> left preconditioning
> using PRECONDITIONED norm type for convergence test
> estimating eigenvalues using noisy right hand side
> maximum iterations=2, nonzero initial guess
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_levels_1_) 4 MPI processes
> type: sor
> type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
> linear system matrix = precond matrix:
> Mat Object: 4 MPI processes
> type: mpiaij
> rows=54, cols=54, bs=6
> total: nonzeros=2916, allocated nonzeros=2916
> total number of mallocs used during MatSetValues calls =0
> using I-node (on process 0) routines: found 11 nodes, limit used is 5
> Up solver (post-smoother) same as down solver (pre-smoother)
> Down solver (pre-smoother) on level 2 -------------------------------
> KSP Object: (mg_levels_2_) 4 MPI processes
> type: chebyshev
> eigenvalue estimates used: min = 0.171388, max = 1.88526
> eigenvalues estimate via gmres min 0.0717873, max 1.71388
> eigenvalues estimated using gmres with translations [0. 0.1; 0. 1.1]
> KSP Object: (mg_levels_2_esteig_) 4 MPI processes
> type: gmres
> restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
> happy breakdown tolerance 1e-30
> maximum iterations=10, initial guess is zero
> tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
> left preconditioning
> using PRECONDITIONED norm type for convergence test
> estimating eigenvalues using noisy right hand side
> maximum iterations=2, nonzero initial guess
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_levels_2_) 4 MPI processes
> type: sor
> type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
> linear system matrix = precond matrix:
> Mat Object: 4 MPI processes
> type: mpiaij
> rows=642, cols=642, bs=6
> total: nonzeros=99468, allocated nonzeros=99468
> total number of mallocs used during MatSetValues calls =0
> using nonscalable MatPtAP() implementation
> using I-node (on process 0) routines: found 47 nodes, limit used is 5
> Up solver (post-smoother) same as down solver (pre-smoother)
> Down solver (pre-smoother) on level 3 -------------------------------
> KSP Object: (mg_levels_3_) 4 MPI processes
> type: chebyshev
> eigenvalue estimates used: min = 0.164216, max = 1.80637
> eigenvalues estimate via gmres min 0.0376323, max 1.64216
> eigenvalues estimated using gmres with translations [0. 0.1; 0. 1.1]
> KSP Object: (mg_levels_3_esteig_) 4 MPI processes
> type: gmres
> restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
> happy breakdown tolerance 1e-30
> maximum iterations=10, initial guess is zero
> tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
> left preconditioning
> using PRECONDITIONED norm type for convergence test
> estimating eigenvalues using noisy right hand side
> maximum iterations=2, nonzero initial guess
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_levels_3_) 4 MPI processes
> type: sor
> type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
> linear system matrix = precond matrix:
> Mat Object: 4 MPI processes
> type: mpiaij
> rows=6726, cols=6726, bs=6
> total: nonzeros=941796, allocated nonzeros=941796
> total number of mallocs used during MatSetValues calls =0
> using nonscalable MatPtAP() implementation
> using I-node (on process 0) routines: found 552 nodes, limit used is 5
> Up solver (post-smoother) same as down solver (pre-smoother)
> Down solver (pre-smoother) on level 4 -------------------------------
> KSP Object: (mg_levels_4_) 4 MPI processes
> type: chebyshev
> eigenvalue estimates used: min = 0.163283, max = 1.79611
> eigenvalues estimate via gmres min 0.0350306, max 1.63283
> eigenvalues estimated using gmres with translations [0. 0.1; 0. 1.1]
> KSP Object: (mg_levels_4_esteig_) 4 MPI processes
> type: gmres
> restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
> happy breakdown tolerance 1e-30
> maximum iterations=10, initial guess is zero
> tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
> left preconditioning
> using PRECONDITIONED norm type for convergence test
> estimating eigenvalues using noisy right hand side
> maximum iterations=2, nonzero initial guess
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_levels_4_) 4 MPI processes
> type: sor
> type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
> linear system matrix = precond matrix:
> Mat Object: 4 MPI processes
> type: mpiaij
> rows=41022, cols=41022, bs=6
> total: nonzeros=2852316, allocated nonzeros=2852316
> total number of mallocs used during MatSetValues calls =0
> using nonscalable MatPtAP() implementation
> using I-node (on process 0) routines: found 3432 nodes, limit used is 5
> Up solver (post-smoother) same as down solver (pre-smoother)
> Down solver (pre-smoother) on level 5 -------------------------------
> KSP Object: (mg_levels_5_) 4 MPI processes
> type: chebyshev
> eigenvalue estimates used: min = 0.157236, max = 1.7296
> eigenvalues estimate via gmres min 0.0317897, max 1.57236
> eigenvalues estimated using gmres with translations [0. 0.1; 0. 1.1]
> KSP Object: (mg_levels_5_esteig_) 4 MPI processes
> type: gmres
> restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
> happy breakdown tolerance 1e-30
> maximum iterations=10, initial guess is zero
> tolerances: relative=1e-12, absolute=1e-50, divergence=10000.
> left preconditioning
> using PRECONDITIONED norm type for convergence test
> estimating eigenvalues using noisy right hand side
> maximum iterations=2, nonzero initial guess
> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
> left preconditioning
> using NONE norm type for convergence test
> PC Object: (mg_levels_5_) 4 MPI processes
> type: sor
> type = local_symmetric, iterations = 1, local iterations = 1, omega = 1.
> linear system matrix = precond matrix:
> Mat Object: () 4 MPI processes
> type: mpiaij
> rows=543606, cols=543606, bs=6
> total: nonzeros=29224836, allocated nonzeros=29302596
> total number of mallocs used during MatSetValues calls =0
> has attached near null space
> using I-node (on process 0) routines: found 45644 nodes, limit used is 5
> Up solver (post-smoother) same as down solver (pre-smoother)
> linear system matrix = precond matrix:
> Mat Object: () 4 MPI processes
> type: mpiaij
> rows=543606, cols=543606, bs=6
> total: nonzeros=29224836, allocated nonzeros=29302596
> total number of mallocs used during MatSetValues calls =0
> has attached near null space
> using I-node (on process 0) routines: found 45644 nodes, limit used is 5
>
>
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