pyamg.aggregation

Aggregation-based AMG.

pyamg.aggregation.adaptive_sa_solver(A, initial_candidates=None, symmetry='hermitian', pdef=True, num_candidates=1, candidate_iters=5, improvement_iters=0, epsilon=0.1, max_levels=10, max_coarse=10, aggregate='standard', prepostsmoother=('gauss_seidel', {'sweep': 'symmetric'}), smooth=('jacobi', {}), strength='symmetric', coarse_solver='pinv', eliminate_local=(False, {'thresh': 1.0}), keep=False, **kwargs)

Create a multilevel solver using Adaptive Smoothed Aggregation (aSA).

Parameters:

Acsr_array, bsr_array

Square matrix in CSR or BSR format.

initial_candidatesNone, array

If a matrix, then this forms the basis for the first m candidates. Also in this case, the initial setup stage is skipped, because this provides the first candidate(s). If None, then a random initial guess and relaxation are used to inform the initial candidate.

symmetrystr

‘symmetric’ refers to both real and complex symmetric ‘hermitian’ refers to both complex Hermitian and real Hermitian Note that for the strictly real case, these two options are the same. Note that this flag does not denote definiteness of the operator.

pdefbool

True or False, whether A is known to be positive definite.

num_candidatesint

Number of near-nullspace candidates to generate.

candidate_itersint

Number of smoothing passes/multigrid cycles used at each level of the adaptive setup phase.

improvement_itersint

Number of times each candidate is improved.

epsilonfloat

Target convergence factor.

max_levelsint

Maximum number of levels to be used in the multilevel solver.

max_coarseint

Maximum number of variables permitted on the coarse grid.

aggregate[‘standard’, ‘lloyd’, ‘naive’, (‘predefined’, {‘AggOp’: csr_array})]

Method used to aggregate nodes. See smoothed_aggregation_solver documentation.

prepostsmootherstr, dict

Pre- and post-smoother used in the adaptive method.

smooth[‘jacobi’, ‘richardson’, ‘energy’, None]

Method used used to smooth the tentative prolongator. See smoothed_aggregation_solver documentation.

strength[‘symmetric’, ‘classical’, ‘evolution’, None]

Method used to determine the strength of connection between unknowns of the linear system. See smoothed_aggregation_solver(…) documentation. Predefined strength may be used with (‘predefined’, {‘C’: csr_array}).

coarse_solver[‘splu’, ‘lu’, ‘cholesky, ‘pinv’, ‘gauss_seidel’, … ]

Solver used at the coarsest level of the MG hierarchy. Optionally, may be a tuple (fn, args), where fn is a string such as [‘splu’, ‘lu’, …] or a callable function, and args is a dictionary of arguments to be passed to fn.

eliminate_localtuple

Length 2 tuple. If the first entry is True, then eliminate candidates where they aren’t needed locally, using the second entry of the tuple to contain arguments to local elimination routine. Given the rigid sparse data structures, this doesn’t help much, if at all, with complexity. Its more of a diagnostic utility.

keepbool

Flag to indicate keeping extra operators in the hierarchy for diagnostics. For example, if True, then strength of connection (C), tentative prolongation (T), and aggregation (AggOp) are kept.

**kwargsdict

Extra keywords passed to Mulilevel class.

Returns:

MultilevelSolver

Smoothed aggregation solver with adaptively generated candidates.

Notes

• Floating point value representing the ‘work’ required to generate the solver. This value is the total cost of just relaxation, relative to the fine grid. The relaxation method used is assumed to symmetric Gauss-Seidel.

• Unlike the standard Smoothed Aggregation (SA) method, adaptive SA does not require knowledge of near-nullspace candidate vectors. Instead, an adaptive procedure computes one or more candidates ‘from scratch’. This approach is useful when no candidates are known or the candidates have been invalidated due to changes to matrix A.

References

[1]

Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge “Adaptive Smoothed Aggregation (alpha SA) Multigrid” SIAM Review Volume 47, Issue 2 (2005)

Examples

>>> from pyamg.gallery import stencil_grid
>>> from pyamg.aggregation import adaptive_sa_solver
>>> import numpy as np
>>> A=stencil_grid([[-1,-1,-1],[-1,8.0,-1],[-1,-1,-1]], (31,31),format='csr')
>>> [asa,work] = adaptive_sa_solver(A,num_candidates=1)
>>> res=[]
>>> x=asa.solve(b=np.ones((A.shape[0],)), x0=np.ones((A.shape[0],)), residuals=res)

pyamg.aggregation.balanced_lloyd_aggregation(C, ratio=0.1, measure=None, maxiter=5, rebalance_iters=5, pad=None, A=None)

Aggregate nodes using Balanced Lloyd Clustering.

Parameters:

Ccsr_array

Strength of connection matrix with positive weights.

ratioscalar

Fraction of nodes to be aggregate (centers). ratio=0.1 is a coarsening by 10.

naggsint

Number of aggregates or clusters expected (default: C.shape[0] / 10)

measure[‘unit’,’abs’,’inv’,None]

Distance measure to use and assigned to each edge graph.

For each nonzero value C[i,j]:

None	G[i,j] = C[i,j]
‘abs’	G[i,j] = abs(C[i,j])
‘inv’	G[i,j] = 1.0/abs(C[i,j])
‘unit’	G[i,j] = 1
‘sub’	G[i,j] = C[i,j] - min(C)

maxiterint

Maximum number of iterations to perform.

rebalance_itersint

Number of rebalance iterations to perform in balanced Lloyd clustering.

padfloat

A pad for the measure with the sparsity of A.

Acsr_array

Sparse matrix to pad with.

Returns:

csr_array

Aggregation operator, AggOp, which determines the sparsity pattern of the tentative prolongator. Node i is in cluster j if AggOp[i,j] = 1.

array

Array of centers or Cpts, i.e., centers[i] = root node of aggregate i.

See also

amg_core.standard_aggregation

amg_core.balanced_lloyd_cluster

Notes

If pad is not None, then C will be augmented by

C = C + E

where E=A and E.data = pad. The goal is to “fix up” the connectivity of G. As an example, if pad is small and if measure=’inv’ is used, then C + E will have “long” edges for the pad.

References

..[1] Zaman, Tareq, Nicolas Nytko, Ali Taghibakhshi, Scott MacLachlan,

Luke Olson, and Matthew West. “Generalizing lloyd’s algorithm for graph clustering.” SIAM Journal on Scientific Computing 46, no. 5 (2024): A2819-A2847. https://epubs.siam.org/doi/abs/10.1137/23M1556800?journalCode=sjoce3

Examples

>>> import numpy as np
>>> import pyamg
>>> from pyamg.aggregation.aggregate import balanced_lloyd_aggregation
>>> data = pyamg.gallery.load_example('unit_square')
>>> G = data['A'].tocsr()
>>> xy = data['vertices'][:,:2]
>>> G.data[:] = np.ones(len(G.data))
>>> np.random.seed(787888)
>>> AggOp, seeds = balanced_lloyd_aggregation(G)

pyamg.aggregation.energy_prolongation_smoother(A, T, Atilde, B, Bf, Cpt_params, krylov='cg', maxiter=4, tol=1e-08, degree=1, weighting='local', prefilter=None, postfilter=None)

Minimize the energy of the coarse basis functions (columns of T).

Both root-node and non-root-node style prolongation smoothing is available, see Cpt_params description below.

Parameters:

Acsr_array, bsr_array

Sparse NxN matrix.

Tbsr_array

Tentative prolongator, a NxM sparse matrix (M < N).

Atildecsr_array

Strength of connection matrix.

Barray

Near-nullspace modes for coarse grid. Has shape (M,k) where k is the number of coarse candidate vectors.

Bfarray

Near-nullspace modes for fine grid. Has shape (N,k) where k is the number of coarse candidate vectors.

Cpt_paramstuple

Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the standard SA prolongation smoothing is carried out. If True, then root-node style prolongation smoothing is carried out. The dict must be a dictionary of parameters containing, (1) for P_I, P_I.T is the injection matrix for the Cpts, (2) I_F is an identity matrix for only the F-points (i.e. I, but with zero rows and columns for C-points) and I_C is the C-point analogue to I_F. See Notes below for more information.

krylovstr

‘cg’ for SPD systems. Solve A T = 0 in a constraint space with CG ‘cgnr’ for nonsymmetric and/or indefinite systems. Solve A T = 0 in a constraint space with CGNR ‘gmres’ for nonsymmetric and/or indefinite systems. Solve A T = 0 in a constraint space with GMRES.

maxiterint

Number of energy minimization steps to apply to the prolongator.

tolscalar

Minimization tolerance.

degreeint

Generate sparsity pattern for P based on (Atilde^degree T).

weightingstr

‘block’, ‘diagonal’ or ‘local’ construction of the diagonal preconditioning ‘local’ Uses a local row-wise weight based on the Gershgorin estimate. Avoids any potential under-damping due to inaccurate spectral radius estimates. ‘block’ Uses a block diagonal inverse of A if A is BSR. ‘diagonal’ Uses the inverse of the diagonal of A.

prefilterdict

Filter elements by row in sparsity pattern for P to reduce operator and setup complexity. If None or an empty dictionary, then no dropping in P is done. If postfilter has key ‘k’, then the largest ‘k’ entries are kept in each row. If postfilter has key ‘theta’, all entries such that \(P[i,j] < kwargs['theta']*max(abs(P[i,:]))\) are dropped. If postfilter[‘k’] and postfiler[‘theta’] are present, then they are used with the union of their patterns.

postfilterdict

Filters elements by row in smoothed P to reduce operator complexity. Only supported if using the rootnode_solver. If None or an empty dictionary, no dropping in P is done. If postfilter has key ‘k’, then the largest ‘k’ entries are kept in each row. If postfilter has key ‘theta’, all entries such that P[i,j] < kwargs['theta']*max(abs(P[i,:])) are dropped. If postfilter[‘k’] and postfiler[‘theta’] are present, then they are used with the union of their patterns.

Returns:

bsr_array

Smoothed prolongator.

Notes

Only ‘diagonal’ weighting is supported for the CGNR method, because we are working with A^* A and not A.

When Cpt_params[0] == True, root-node style prolongation smoothing is used to minimize the energy of columns of T. Essentially, an identity block is maintained in T, corresponding to injection from the coarse-grid to the fine-grid root-nodes. See [2] for more details, and see util.utils.get_Cpt_params for the helper function to generate Cpt_params.

If Cpt_params[0] == False, the energy of columns of T are still minimized, but without maintaining the identity block.

See [1] for more details on smoothed aggregation.

References

[1]

Jan Mandel, Marian Brezina, and Petr Vanek “Energy Optimization of Algebraic Multigrid Bases” Computing 62, 205-228, 1999 http://dx.doi.org/10.1007/s006070050022

[2]

Olson, L. and Schroder, J. and Tuminaro, R., “A general interpolation strategy for algebraic multigrid using energy minimization”, SIAM Journal on Scientific Computing (SISC), vol. 33, pp. 966–991, 2011.

Examples

>>> from pyamg.aggregation import energy_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_array
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0, 6, dtype=np.int32)
>>> col = np.kron([0, 1], np.ones(3)).astype(np.int32)
>>> T = coo_array((data,(row, col)),shape=(6, 2)).tocsr()
>>> print(T.toarray())
[[1. 0.]
 [1. 0.]
 [1. 0.]
 [0. 1.]
 [0. 1.]
 [0. 1.]]
>>> A = poisson((6,),format='csr')
>>> B = np.ones((2,1),dtype=float)
>>> P = energy_prolongation_smoother(A,T,A,B, None, (False,{}))
>>> print(P.toarray())
[[1.         0.        ]
 [1.         0.        ]
 [0.66666667 0.33333333]
 [0.33333333 0.66666667]
 [0.         1.        ]
 [0.         1.        ]]

pyamg.aggregation.fit_candidates(AggOp, B, tol=1e-10)

Fit near-nullspace candidates to form the tentative prolongator.

Parameters:

AggOpcsr_array

Describes the sparsity pattern of the tentative prolongator. Has dimension (#blocks, #aggregates).

Barray

The near-nullspace candidates stored in column-wise fashion. Has dimension (#blocks * blocksize, #candidates).

tolscalar

Threshold for eliminating local basis functions. If after orthogonalization a local basis function Q[:, j] is small, i.e. ||Q[:, j]|| < tol, then Q[:, j] is set to zero.

Returns:

(bsr_array, array)

The tentative prolongator Q is a sparse block matrix with dimensions (#blocks * blocksize, #aggregates * #candidates) formed by dense blocks of size (blocksize, #candidates). The coarse level candidates are stored in R which has dimensions (#aggregates * #candidates, #candidates).

See also

amg_core.fit_candidates

Notes

Assuming that each row of AggOp contains exactly one non-zero entry, i.e. all unknowns belong to an aggregate, then Q and R satisfy the relationship B = Q@R. In other words, the near-nullspace candidates are represented exactly by the tentative prolongator.

If AggOp contains rows with no non-zero entries, then the range of the tentative prolongator will not include those degrees of freedom. This situation is illustrated in the examples below.

References

[1]

Vanek, P. and Mandel, J. and Brezina, M., “Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems”, Computing, vol. 56, no. 3, pp. 179–196, 1996. http://citeseer.ist.psu.edu/vanek96algebraic.html

Examples

>>> from scipy.sparse import csr_array
>>> from pyamg.aggregation.tentative import fit_candidates
>>> # four nodes divided into two aggregates
... AggOp = csr_array( [[1, 0],
...                      [1, 0],
...                      [0, 1],
...                      [0, 1]] )
>>> # B contains one candidate, the constant vector
... B = [[1],
...      [1],
...      [1],
...      [1]]
>>> Q, R = fit_candidates(AggOp, B)
>>> Q.toarray()
array([[0.70710678, 0.        ],
       [0.70710678, 0.        ],
       [0.        , 0.70710678],
       [0.        , 0.70710678]])
>>> R
array([[1.41421356],
       [1.41421356]])
>>> # Two candidates, the constant vector and a linear function
... B = [[1, 0],
...      [1, 1],
...      [1, 2],
...      [1, 3]]
>>> Q, R = fit_candidates(AggOp, B)
>>> Q.toarray()
array([[ 0.70710678, -0.70710678,  0.        ,  0.        ],
       [ 0.70710678,  0.70710678,  0.        ,  0.        ],
       [ 0.        ,  0.        ,  0.70710678, -0.70710678],
       [ 0.        ,  0.        ,  0.70710678,  0.70710678]])
>>> R
array([[1.41421356, 0.70710678],
       [0.        , 0.70710678],
       [1.41421356, 3.53553391],
       [0.        , 0.70710678]])
>>> # aggregation excludes the third node
... AggOp = csr_array( [[1, 0],
...                      [1, 0],
...                      [0, 0],
...                      [0, 1]] )
>>> B = [[1],
...      [1],
...      [1],
...      [1]]
>>> Q, R = fit_candidates(AggOp, B)
>>> Q.toarray()
array([[0.70710678, 0.        ],
       [0.70710678, 0.        ],
       [0.        , 0.        ],
       [0.        , 1.        ]])
>>> R
array([[1.41421356],
       [1.        ]])

pyamg.aggregation.jacobi_prolongation_smoother(S, T, C, B, omega=1.3333333333333333, degree=1, filter_entries=False, weighting='diagonal')

Jacobi prolongation smoother.

Parameters:

Scsr_array, bsr_array

Sparse NxN matrix used for smoothing. Typically, A.

Tcsr_array, bsr_array

Tentative prolongator.

Ccsr_array, bsr_array

Strength-of-connection matrix.

Barray

Near nullspace modes for the coarse grid such that T@B exactly reproduces the fine grid near nullspace modes.

omegascalar

Damping parameter.

degreeint

Number of passes.

filter_entriesbool

If true, filter S before smoothing T. This option can greatly control complexity.

weightingstr

‘block’, ‘diagonal’ or ‘local’ weighting for constructing the Jacobi D ‘local’ Uses a local row-wise weight based on the Gershgorin estimate. Avoids any potential under-damping due to inaccurate spectral radius estimates. ‘block’ uses a block diagonal inverse of A if A is BSR ‘diagonal’ uses classic Jacobi with D = diagonal(A).

Returns:

csr_array, bsr_array

Smoothed (final) prolongator defined by P = (I - omega/rho(K) K) @ T where K = diag(S)^-1 @ S and rho(K) is an approximation to the spectral radius of K.

Notes

If weighting is not ‘local’, then results using Jacobi prolongation smoother are not precisely reproducible due to a random initial guess used for the spectral radius approximation. For precise reproducibility, set numpy.random.seed(..) to the same value before each test.

Examples

>>> from pyamg.aggregation import jacobi_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_array
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0,6)
>>> col = np.kron([0, 1], np.ones(3, dtype=int))
>>> T = coo_array((data,(row, col)),shape=(6, 2)).tocsr()
>>> T.toarray()
array([[1., 0.],
       [1., 0.],
       [1., 0.],
       [0., 1.],
       [0., 1.],
       [0., 1.]])
>>> A = poisson((6,),format='csr')
>>> P = jacobi_prolongation_smoother(A,T,A,np.ones((2,1)))
>>> P.toarray()
array([[0.64930164, 0.        ],
       [1.        , 0.        ],
       [0.64930164, 0.35069836],
       [0.35069836, 0.64930164],
       [0.        , 1.        ],
       [0.        , 0.64930164]])

pyamg.aggregation.lloyd_aggregation(C, ratio=0.1, measure='unit', maxiter=5)

Aggregate nodes using Lloyd Clustering.

Parameters:

Ccsr_array

Strength of connection matrix.

ratioscalar

Fraction of nodes to be aggregate (centers). ratio=0.1 is a coarsening by 10.

measure[‘unit’,’abs’,’inv’,None]

Distance measure to use and assigned to each edge graph.

For each nonzero value C[i,j]:

=======  ===========================
None     G[i,j] = C[i,j]
'abs'    G[i,j] = abs(C[i,j])
'inv'    G[i,j] = 1.0/abs(C[i,j])
'unit'   G[i,j] = 1
'sub'    G[i,j] = C[i,j] - min(C)
=======  ===========================

maxiterint

Maximum number of iterations to perform.

Returns:

csr_array

Aggregation operator which determines the sparsity pattern of the tentative prolongator. Node i is in cluster j if AggOp[i,j] = 1.

array

Array of centers or Cpts, i.e., Cpts[i] = root node of aggregate i.

See also

amg_core.standard_aggregation

Examples

>>> import pyamg
>>> from pyamg.gallery import poisson
>>> from pyamg.aggregation.aggregate import lloyd_aggregation
>>> A = poisson((4,), format='csr')   # 1D mesh with 4 vertices
>>> A.toarray()
array([[ 2., -1.,  0.,  0.],
       [-1.,  2., -1.,  0.],
       [ 0., -1.,  2., -1.],
       [ 0.,  0., -1.,  2.]])
>>> lloyd_aggregation(A)[0].toarray() # one aggregate
array([[1],
       [1],
       [1],
       [1]], dtype=int32)
>>> # more seeding for two aggregates
>>> Agg = lloyd_aggregation(A,ratio=0.5)[0].toarray()

pyamg.aggregation.metis_aggregation(C, ratio=0.1, measure=None)

Aggregate nodes using a METIS partition.

Parameters:

Ccsr_array

strength of connection matrix

ratioscalar

Fraction of nodes to be aggregated (centers). ratio=0.1 is a coarsening by 10

measure[‘unit’,’abs’,’inv’,None]

Distance measure to use and assigned to each edge graph. METIS requires integer weights. None, simply convert to integer (rounding up). range maps to integers on the range [1,10], and unit gives each unit length.

For each nonzero value C[i,j]: ======= =========================== None G[i,j] = ceil(C[i,j]) ‘range’ G[i,j] = np.round(9 * C[i,j])+1 ‘unit’ G[i,j] = 1 ======= ===========================

maxiterint

Maximum number of iterations to perform

Returns:

AggOpcsr_array

aggregation operator which determines the sparsity pattern of the tentative prolongator. Node i is in cluster j if AggOp[i,j] = 1.

See also

amg_core.standard_aggregation

pyamg.aggregation.naive_aggregation(C)

Compute the sparsity pattern of the tentative prolongator.

Parameters:

Ccsr_array

Strength of connection matrix.

Returns:

csr_array

Aggregation operator which determines the sparsity pattern of the tentative prolongator.

array

Array of Cpts, i.e., Cpts[i] = root node of aggregate i.

See also

amg_core.naive_aggregation

Notes

Differs from standard aggregation. Each dof is considered. If it has been aggregated, skip over. Otherwise, put dof and any unaggregated neighbors in an aggregate. Results in possibly much higher complexities than standard aggregation.

Examples

>>> from scipy.sparse import csr_array
>>> from pyamg.gallery import poisson
>>> from pyamg.aggregation.aggregate import naive_aggregation
>>> A = poisson((4,), format='csr')   # 1D mesh with 4 vertices
>>> A.toarray()
array([[ 2., -1.,  0.,  0.],
       [-1.,  2., -1.,  0.],
       [ 0., -1.,  2., -1.],
       [ 0.,  0., -1.,  2.]])
>>> naive_aggregation(A)[0].toarray() # two aggregates
array([[1, 0],
       [1, 0],
       [0, 1],
       [0, 1]], dtype=int32)
>>> A = csr_array([[1,0,0],[0,1,1],[0,1,1]])
>>> A.toarray()                      # first vertex is isolated
array([[1, 0, 0],
       [0, 1, 1],
       [0, 1, 1]])
>>> naive_aggregation(A)[0].toarray() # two aggregates
array([[1, 0],
       [0, 1],
       [0, 1]], dtype=int32)

pyamg.aggregation.pairwise_solver(A, aggregate=('pairwise', {'matchings': 2, 'norm': 'min', 'theta': 0.25}), presmoother=('block_gauss_seidel', {'sweep': 'symmetric'}), postsmoother=('block_gauss_seidel', {'sweep': 'symmetric'}), max_levels=20, max_coarse=10, **kwargs)

Create a multilevel solver using Pairwise Aggregation.

Parameters:

Acsr_array, bsr_array

Sparse NxN matrix in CSR or BSR format.

aggregatetuple, str, list

Method choice must be ‘pairwise’; inner pairwise options including matchings, theta, and norm can be modified.

presmoothertuple, str, list

Defines the presmoother for the multilevel cycling. The default block Gauss-Seidel option defaults to point-wise Gauss-Seidel, if the matrix is CSR or is a BSR matrix with blocksize of 1. See notes below for varying this parameter on a per level basis.

postsmoothertuple, str, list

Same as presmoother, except defines the postsmoother.

max_levelsint

Maximum number of levels to be used in the multilevel solver.

max_coarseint

Maximum number of variables permitted on the coarse grid.

**kwargsdict

Extra keywords passed to the Multilevel class

cycle_type	[‘V’,’W’,’F’], Structrure of multigrid cycle
coarse_solver	[‘splu’, ‘lu’, ‘cholesky, ‘pinv’, ‘gauss_seidel’, … ] Solver used at the coarsest level of the MG hierarchy. Optionally, may be a tuple (fn, args), where fn is a string such as [‘splu’, ‘lu’, …] or a callable function, and args is a dictionary of arguments to be passed to fn.

See MultiLevel class for more details.

Returns:

MultilevelSolver

Multigrid hierarchy of matrices and prolongation operators.

See also

multilevel_solver

classical.ruge_stuben_solver

aggregation.smoothed_aggregation_solver

Notes

See [1] for more details.

References

[1]

Notay, Y. (2010). An aggregation-based algebraic multigrid method. Electronic transactions on numerical analysis, 37(6), 123-146.

Examples

>>> from pyamg import pairwise_solver
>>> from pyamg.gallery import poisson
>>> from scipy.sparse.linalg import cg
>>> import numpy as np
>>> A = poisson((100, 100), format='csr')       # matrix
>>> b = np.ones((A.shape[0]))                   # RHS
>>> ml = pairwise_solver(A)                     # AMG solver
>>> M = ml.aspreconditioner(cycle='V')          # preconditioner
>>> x, info = cg(A, b, rtol=1e-8, maxiter=30, M=M)   # solve with CG

pyamg.aggregation.richardson_prolongation_smoother(S, T, omega=1.3333333333333333, degree=1)

Richardson prolongation smoother.

Parameters:

Scsr_array, bsr_array

Sparse NxN matrix used for smoothing. Typically, A or the “filtered matrix” obtained from A by lumping weak connections onto the diagonal of A.

Tcsr_array, bsr_array

Tentative prolongator.

omegascalar

Damping parameter.

degreeint

Number of passes.

Returns:

csr_array, bsr_array

Smoothed (final) prolongator defined by P = (I - omega/rho(S) S) @ T where rho(S) is an approximation to the spectral radius of S.

Notes

Results using Richardson prolongation smoother are not precisely reproducible due to a random initial guess used for the spectral radius approximation. For precise reproducibility, set numpy.random.seed(..) to the same value before each test.

Examples

>>> from pyamg.aggregation import richardson_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_array
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0, 6, dtype=np.int32)
>>> col = np.kron([0, 1], np.ones(3, dtype=int))
>>> T = coo_array((data,(row, col)),shape=(6, 2)).tocsr()
>>> T.toarray()
array([[1., 0.],
       [1., 0.],
       [1., 0.],
       [0., 1.],
       [0., 1.],
       [0., 1.]])
>>> A = poisson((6,),format='csr')
>>> P = richardson_prolongation_smoother(A,T)
>>> P.toarray()
array([[0.64930164, 0.        ],
       [1.        , 0.        ],
       [0.64930164, 0.35069836],
       [0.35069836, 0.64930164],
       [0.        , 1.        ],
       [0.        , 0.64930164]])

pyamg.aggregation.rootnode_solver(A, B=None, BH=None, symmetry='hermitian', strength='symmetric', aggregate='standard', smooth='energy', presmoother=('block_gauss_seidel', {'sweep': 'symmetric'}), postsmoother=('block_gauss_seidel', {'sweep': 'symmetric'}), improve_candidates=('block_gauss_seidel', {'iterations': 4, 'sweep': 'symmetric'}), max_levels=10, max_coarse=10, diagonal_dominance=False, keep=False, **kwargs)

Create a multilevel solver using root-node based Smoothed Aggregation (SA).

See the notes below, for the major differences with the classical-style smoothed aggregation solver in aggregation.smoothed_aggregation_solver.

Parameters:

Acsr_array, bsr_array

Sparse NxN matrix in CSR or BSR format.

BNone, array

Right near-nullspace candidates stored in the columns of an NxK array. K must be >= the blocksize of A (see reference [2]). The default value B=None is equivalent to choosing the constant over each block-variable, B=np.kron(np.ones((A.shape[0]/get_blocksize(A), 1)), np.eye(get_blocksize(A))).

BHNone, array

Left near-nullspace candidates stored in the columns of an NxK array. BH is only used if symmetry=’nonsymmetric’. K must be >= the blocksize of A (see reference [2]). The default value B=None is equivalent to choosing the constant over each block-variable, B=np.kron(np.ones((A.shape[0]/get_blocksize(A), 1)), np.eye(get_blocksize(A))).

symmetrystr

• ‘symmetric’ refers to both real and complex symmetric

• ‘hermitian’ refers to both complex Hermitian and real Hermitian

• ‘nonsymmetric’ i.e. nonsymmetric in a hermitian sense

Note that for the strictly real case, symmetric and hermitian are the same. Note that this flag does not denote definiteness of the operator.

strengthlist

Method used to determine the strength of connection between unknowns of the linear system. Method-specific parameters may be passed in using a tuple, e.g. strength=(‘symmetric’,{‘theta’ : 0.25 }). If strength=None, all nonzero entries of the matrix are considered strong.

aggregatelist

Method used to aggregate nodes.

smoothlist

Method used to smooth the tentative prolongator. Method-specific parameters may be passed in using a tuple, e.g. smooth= (‘energy’,{‘krylov’ : ‘gmres’}). Only ‘energy’ and None are valid prolongation smoothing options.

presmoothertuple, str, list

Defines the presmoother for the multilevel cycling. The default block Gauss-Seidel option defaults to point-wise Gauss-Seidel, if the matrix is CSR or is a BSR matrix with blocksize of 1. See notes below for varying this parameter on a per level basis.

postsmoothertuple, str, list

Same as presmoother, except defines the postsmoother.

improve_candidatestuple, str, list

The ith entry defines the method used to improve the candidates B on level i. If the list is shorter than max_levels, then the last entry will define the method for all levels lower. If tuple or string, then this single relaxation descriptor defines improve_candidates on all levels. The list elements are relaxation descriptors of the form used for presmoother and postsmoother. A value of None implies no action on B.

max_levelsint

Maximum number of levels to be used in the multilevel solver.

max_coarseint

Maximum number of variables permitted on the coarse grid.

diagonal_dominancebool, tuple

If True (or the first tuple entry is True), then avoid coarsening diagonally dominant rows. The second tuple entry requires a dictionary, where the key value ‘theta’ is used to tune the diagonal dominance threshold.

keepbool

Flag to indicate keeping extra operators in the hierarchy for diagnostics. For example, if True, then strength of connection (C), tentative prolongation (T), aggregation (AggOp), and arrays storing the C-points (Cpts) and F-points (Fpts) are kept at each level.

**kwargsdict

Extra keywords passed to the Multilevel class

cycle_type	[‘V’,’W’,’F’], Structrure of multigrid cycle
coarse_solver	[‘splu’, ‘lu’, ‘cholesky, ‘pinv’, ‘gauss_seidel’, … ] Solver used at the coarsest level of the MG hierarchy. Optionally, may be a tuple (fn, args), where fn is a string such as [‘splu’, ‘lu’, …] or a callable function, and args is a dictionary of arguments to be passed to fn.

See MultiLevel class for more details.

Returns:

MultilevelSolver

Multigrid hierarchy of matrices and prolongation operators.

See also

MultilevelSolver

aggregation.smoothed_aggregation_solver

classical.ruge_stuben_solver

Notes

• Root-node style SA differs from classical SA primarily by preserving and identity block in the interpolation operator, P. Each aggregate has a ‘root-node’ or ‘center-node’ associated with it, and this root-node is injected from the coarse grid to the fine grid. The injection corresponds to the identity block.

• Only smooth={‘energy’, None} is supported for prolongation smoothing. See reference [2] below for more details on why the ‘energy’ prolongation smoother is the natural counterpart to root-node style SA.

• The additional parameters are passed through as arguments to MultilevelSolver. Refer to pyamg.MultilevelSolver for additional documentation.

• At each level, four steps are executed in order to define the coarser level operator.

  1. Matrix A is given and used to derive a strength matrix, C.

  2. Based on the strength matrix, indices are grouped or aggregated.

  3. The aggregates define coarse nodes and a tentative prolongation operator T is defined by injection

  4. The tentative prolongation operator is smoothed by a relaxation scheme to improve the quality and extent of interpolation from the aggregates to fine nodes.

• The parameters smooth, strength, aggregate, presmoother, postsmoother can be varied on a per level basis. For different methods on different levels, use a list as input so that the i-th entry defines the method at the i-th level. If there are more levels in the hierarchy than list entries, the last entry will define the method for all levels lower.

  Examples are: smooth=[(‘jacobi’, {‘omega’:1.0}), None, ‘jacobi’] presmoother=[(‘block_gauss_seidel’, {‘sweep’:symmetric}), ‘sor’] aggregate=[‘standard’, ‘naive’, ‘lloyd’, ‘pairwise’] strength=[(‘symmetric’, {‘theta’:0.25}), (‘symmetric’, {‘theta’:0.08})]

• Predefined strength of connection and aggregation schemes can be specified. These options are best used together, but aggregation can be predefined while strength of connection is not.

  For predefined strength of connection, use a list consisting of tuples of the form (‘predefined’, {‘C’ : C0}), where C0 is a csr_array and each degree-of-freedom in C0 represents a supernode. For instance to predefine a three-level hierarchy, use [(‘predefined’, {‘C’ : C0}), (‘predefined’, {‘C’ : C1}) ].

  Similarly for predefined aggregation, use a list of tuples. For instance to predefine a three-level hierarchy, use [(‘predefined’, {‘AggOp’ : Agg0}), (‘predefined’, {‘AggOp’ : Agg1}) ], where the dimensions of A, Agg0 and Agg1 are compatible, i.e. Agg0.shape[1] == A.shape[0] and Agg1.shape[1] == Agg0.shape[0]. Each AggOp is a csr_array.

  Because this is a root-nodes solver, if a member of the predefined aggregation list is predefined, it must be of the form (‘predefined’, {‘AggOp’ : Agg, ‘Cnodes’ : Cnodes}).

References

[1996VaMa]

Vanek, P. and Mandel, J. and Brezina, M., “Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems”, Computing, vol. 56, no. 3, pp. 179–196, 1996. http://citeseer.ist.psu.edu/vanek96algebraic.html

[2] (1,2,3)

Olson, L. and Schroder, J. and Tuminaro, R., “A general interpolation strategy for algebraic multigrid using energy minimization”, SIAM Journal on Scientific Computing (SISC), vol. 33, pp. 966–991, 2011.

Examples

>>> from pyamg import rootnode_solver
>>> from pyamg.gallery import poisson
>>> from scipy.sparse.linalg import cg
>>> import numpy as np
>>> A = poisson((100, 100), format='csr')           # matrix
>>> b = np.ones((A.shape[0]))                   # RHS
>>> ml = rootnode_solver(A)                     # AMG solver
>>> M = ml.aspreconditioner(cycle='V')             # preconditioner
>>> x, info = cg(A, b, rtol=1e-8, maxiter=30, M=M)   # solve with CG

pyamg.aggregation.smoothed_aggregation_solver(A, B=None, BH=None, symmetry='hermitian', strength='symmetric', aggregate='standard', smooth=('jacobi', {'omega': 1.3333333333333333}), presmoother=('block_gauss_seidel', {'sweep': 'symmetric'}), postsmoother=('block_gauss_seidel', {'sweep': 'symmetric'}), improve_candidates=(('block_gauss_seidel', {'iterations': 4, 'sweep': 'symmetric'}), None), max_levels=10, max_coarse=10, diagonal_dominance=False, keep=False, **kwargs)

Create a multilevel solver using classical-style Smoothed Aggregation (SA).

Parameters:

Acsr_array, bsr_array

Sparse NxN matrix in CSR or BSR format.

BNone, array

Right near-nullspace candidates stored in the columns of an NxK array. The default value B=None is equivalent to B=ones((N,1)).

BHNone, array

Left near-nullspace candidates stored in the columns of an NxK array. BH is only used if symmetry='nonsymmetric'. The default value B=None is equivalent to BH=B.copy().

symmetrystr

• ‘symmetric’ refers to both real and complex symmetric.

• ‘hermitian’ refers to both complex Hermitian and real Hermitian.

• ‘nonsymmetric’ i.e. nonsymmetric in a hermitian sense.

Note, in the strictly real case, symmetric and hermitian are the same. Note, this flag does not denote definiteness of the operator.

strengthstr, list

Method used to determine the strength of connection between unknowns of the linear system. Method-specific parameters may be passed in using a tuple, e.g. strength=(‘symmetric’,{‘theta’ : 0.25 }). If strength=None, all nonzero entries of the matrix are considered strong. Choose from ‘symmetric’, ‘classical’, ‘evolution’, ‘algebraic_distance’, ‘affinity’, (‘predefined’, {‘C’ : csr_array}), None.

aggregatestr, list

Method used to aggregate nodes. Choose from ‘standard’, ‘lloyd’, ‘naive’, ‘pairwise’, (‘predefined’, {‘AggOp’ : csr_array}).

smoothlist

Method used to smooth the tentative prolongator. Method-specific parameters may be passed in using a tuple, e.g. smooth= (‘jacobi’,{‘filter’ : True }). Choose from ‘jacobi’, ‘richardson’, ‘energy’, None.

presmoothertuple, str, list

Defines the presmoother for the multilevel cycling. The default block Gauss-Seidel option defaults to point-wise Gauss-Seidel, if the matrix is CSR or is a BSR matrix with blocksize of 1.

postsmoothertuple, str, list

Same as presmoother, except defines the postsmoother.

improve_candidatestuple, str, list

The ith entry defines the method used to improve the candidates B on level i. If the list is shorter than max_levels, then the last entry will define the method for all levels lower. If tuple or string, then this single relaxation descriptor defines improve_candidates on all levels. The list elements are relaxation descriptors of the form used for presmoother and postsmoother. A value of None implies no action on B.

max_levelsint

Maximum number of levels to be used in the multilevel solver.

max_coarseint

Maximum number of variables permitted on the coarse grid.

diagonal_dominancebool, tuple

If True (or the first tuple entry is True), then avoid coarsening diagonally dominant rows. The second tuple entry requires a dictionary, where the key value ‘theta’ is used to tune the diagonal dominance threshold.

keepbool

Flag to indicate keeping extra operators in the hierarchy for diagnostics. For example, if True, then strength of connection (C), tentative prolongation (T), and aggregation (AggOp) are kept.

**kwargsdict

Extra keywords passed to the Multilevel class

cycle_type	[‘V’,’W’,’F’], Structrure of multigrid cycle
coarse_solver	[‘splu’, ‘lu’, ‘cholesky, ‘pinv’, ‘gauss_seidel’, … ] Solver used at the coarsest level of the MG hierarchy. Optionally, may be a tuple (fn, args), where fn is a string such as [‘splu’, ‘lu’, …] or a callable function, and args is a dictionary of arguments to be passed to fn.

See MultiLevel class for more details.

Returns:

MultilevelSolver

Multigrid hierarchy of matrices and prolongation operators.

See also

MultilevelSolver

classical.ruge_stuben_solver

aggregation.root_node_solver

Notes

• This method implements classical-style SA, not root-node style SA (see aggregation.rootnode_solver). See [1].

• The additional parameters are passed through as arguments to MultilevelSolver. Refer to pyamg.MultilevelSolver for additional documentation.

• At each level, four steps are executed in order to define the coarser level operator.

  1. Matrix A is given and used to derive a strength matrix, C.

  2. Based on the strength matrix, indices are grouped or aggregated.

  3. The aggregates define coarse nodes and a tentative prolongation operator T is defined by injection

  4. The tentative prolongation operator is smoothed by a relaxation scheme to improve the quality and extent of interpolation from the aggregates to fine nodes.

• The parameters smooth, strength, aggregate, presmoother, postsmoother can be varied on a per level basis. For different methods on different levels, use a list as input so that the i-th entry defines the method at the i-th level. If there are more levels in the hierarchy than list entries, the last entry will define the method for all levels lower.

  Examples are: smooth=[(‘jacobi’, {‘omega’:1.0}), None, ‘jacobi’] presmoother=[(‘block_gauss_seidel’, {‘sweep’:symmetric}), ‘sor’] aggregate=[‘standard’, ‘naive’] strength=[(‘symmetric’, {‘theta’:0.25}), (‘symmetric’, {‘theta’:0.08})]

• Predefined strength of connection and aggregation schemes can be specified. These options are best used together, but aggregation can be predefined while strength of connection is not.

  For predefined strength of connection, use a list consisting of tuples of the form (‘predefined’, {‘C’ : C0}), where C0 is a csr_array and each degree-of-freedom in C0 represents a supernode. For instance to predefine a three-level hierarchy, use [(‘predefined’, {‘C’ : C0}), (‘predefined’, {‘C’ : C1}) ].

  Similarly for predefined aggregation, use a list of tuples. For instance to predefine a three-level hierarchy, use [(‘predefined’, {‘AggOp’ : Agg0}), (‘predefined’, {‘AggOp’ : Agg1}) ], where the dimensions of A, Agg0 and Agg1 are compatible, i.e. Agg0.shape[1] == A.shape[0] and Agg1.shape[1] == Agg0.shape[0]. Each AggOp is a csr_array.

References

[1]

Vanek, P. and Mandel, J. and Brezina, M., “Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems”, Computing, vol. 56, no. 3, pp. 179–196, 1996. http://citeseer.ist.psu.edu/vanek96algebraic.html

Examples

>>> from pyamg import smoothed_aggregation_solver
>>> from pyamg.gallery import poisson
>>> from scipy.sparse.linalg import cg
>>> import numpy as np
>>> A = poisson((100,100), format='csr')           # matrix
>>> b = np.ones((A.shape[0]))                      # RHS
>>> ml = smoothed_aggregation_solver(A)            # AMG solver
>>> M = ml.aspreconditioner(cycle='V')             # preconditioner
>>> x,info = cg(A, b, rtol=1e-8, maxiter=30, M=M)   # solve with CG

pyamg.aggregation.standard_aggregation(C)

Compute the sparsity pattern of the tentative prolongator.

Parameters:

Ccsr_array

Strength of connection matrix.

Returns:

csr_array

Aggregation operator which determines the sparsity pattern of the tentative prolongator.

array

Array of Cpts, i.e., Cpts[i] = root node of aggregate i.

See also

amg_core.standard_aggregation

Examples

>>> from scipy.sparse import csr_array
>>> from pyamg.gallery import poisson
>>> from pyamg.aggregation.aggregate import standard_aggregation
>>> A = poisson((4,), format='csr')   # 1D mesh with 4 vertices
>>> A.toarray()
array([[ 2., -1.,  0.,  0.],
       [-1.,  2., -1.,  0.],
       [ 0., -1.,  2., -1.],
       [ 0.,  0., -1.,  2.]])
>>> standard_aggregation(A)[0].toarray() # two aggregates
array([[1, 0],
       [1, 0],
       [0, 1],
       [0, 1]], dtype=int32)
>>> A = csr_array([[1,0,0],[0,1,1],[0,1,1]])
>>> A.toarray()                      # first vertex is isolated
array([[1, 0, 0],
       [0, 1, 1],
       [0, 1, 1]])
>>> standard_aggregation(A)[0].toarray() # one aggregate
array([[0],
       [1],
       [1]], dtype=int32)