"""Support for aggregation-based AMG."""
from warnings import warn
import numpy as np
from scipy.sparse import csr_matrix, isspmatrix_csr, isspmatrix_bsr, \
SparseEfficiencyWarning
from ..multilevel import MultilevelSolver
from ..relaxation.smoothing import change_smoothers
from ..relaxation.utils import relaxation_as_linear_operator
from ..util.utils import scale_T, get_Cpt_params, \
eliminate_diag_dom_nodes, get_blocksize, \
levelize_strength_or_aggregation, \
levelize_smooth_or_improve_candidates
from ..strength import classical_strength_of_connection, \
symmetric_strength_of_connection, evolution_strength_of_connection, \
energy_based_strength_of_connection, distance_strength_of_connection, \
algebraic_distance, affinity_distance
from .aggregate import standard_aggregation, naive_aggregation, \
lloyd_aggregation, pairwise_aggregation
from .tentative import fit_candidates
from .smooth import energy_prolongation_smoother
[docs]
def 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',
{'sweep': 'symmetric',
'iterations': 4}),
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
----------
A : csr_matrix, bsr_matrix
Sparse NxN matrix in CSR or BSR format
B : None, array_like
Right near-nullspace candidates stored in the columns of an NxK array.
K must be >= the blocksize of A (see reference [2011OlScTu]_). 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)))
BH : None, array_like
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 [2011OlScTu]_). 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)))
symmetry : string
'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.
strength : 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.
aggregate : list
Method used to aggregate nodes.
smooth : list
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.
presmoother : tuple, string, 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.
postsmoother : tuple, string, list
Same as presmoother, except defines the postsmoother.
improve_candidates : tuple, string, 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_levels : integer
Maximum number of levels to be used in the multilevel solver.
max_coarse : integer
Maximum number of variables permitted on the coarse grid.
diagonal_dominance : bool, 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.
keep : bool
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.
Other Parameters
----------------
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.
Returns
-------
ml : 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 [2011OlScTu]_ 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_matrix 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_matrix.
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}).
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, tol=1e-8, maxiter=30, M=M) # solve with CG
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
.. [2011OlScTu] 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.
"""
if not (isspmatrix_csr(A) or isspmatrix_bsr(A)):
try:
A = csr_matrix(A)
warn('Implicit conversion of A to CSR',
SparseEfficiencyWarning)
except Exception as e:
raise TypeError('Argument A must have type csr_matrix, '
'bsr_matrix, or be convertible to csr_matrix') from e
A = A.asfptype()
if symmetry not in ('symmetric', 'hermitian', 'nonsymmetric'):
raise ValueError('Expected "symmetric", "nonsymmetric" '
'or "hermitian" for the symmetry parameter.')
A.symmetry = symmetry
if A.shape[0] != A.shape[1]:
raise ValueError('expected square matrix')
# Right near nullspace candidates use constant for each variable as default
if B is None:
B = np.kron(np.ones((int(A.shape[0]/get_blocksize(A)), 1), dtype=A.dtype),
np.eye(get_blocksize(A)))
else:
B = np.asarray(B, dtype=A.dtype)
if len(B.shape) == 1:
B = B.reshape(-1, 1)
if B.shape[0] != A.shape[0]:
raise ValueError('The near null-space modes B have incorrect \
dimensions for matrix A')
if B.shape[1] < get_blocksize(A):
raise ValueError('B.shape[1] must be >= the blocksize of A')
# Left near nullspace candidates
if A.symmetry == 'nonsymmetric':
if BH is None:
BH = B.copy()
else:
BH = np.asarray(BH, dtype=A.dtype)
if len(BH.shape) == 1:
BH = BH.reshape(-1, 1)
if BH.shape[1] != B.shape[1]:
raise ValueError('The number of left and right near \
null-space modes B and BH, must be equal')
if BH.shape[0] != A.shape[0]:
raise ValueError('The near null-space modes BH have \
incorrect dimensions for matrix A')
# Levelize the user parameters, so that they become lists describing the
# desired user option on each level.
max_levels, max_coarse, strength =\
levelize_strength_or_aggregation(strength, max_levels, max_coarse)
max_levels, max_coarse, aggregate =\
levelize_strength_or_aggregation(aggregate, max_levels, max_coarse)
improve_candidates =\
levelize_smooth_or_improve_candidates(improve_candidates, max_levels)
smooth = levelize_smooth_or_improve_candidates(smooth, max_levels)
# Construct multilevel structure
levels = []
levels.append(MultilevelSolver.Level())
levels[-1].A = A # matrix
# Append near nullspace candidates
levels[-1].B = B # right candidates
if A.symmetry == 'nonsymmetric':
levels[-1].BH = BH # left candidates
while len(levels) < max_levels and \
int(levels[-1].A.shape[0]/get_blocksize(levels[-1].A)) > max_coarse:
_extend_hierarchy(levels, strength, aggregate, smooth,
improve_candidates, diagonal_dominance, keep)
ml = MultilevelSolver(levels, **kwargs)
change_smoothers(ml, presmoother, postsmoother)
return ml
def _extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates,
diagonal_dominance=False, keep=True):
"""Extend the multigrid hierarchy.
Service routine to implement the strength of connection, aggregation,
tentative prolongation construction, and prolongation smoothing. Called by
smoothed_aggregation_solver.
"""
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
return v, {}
A = levels[-1].A
B = levels[-1].B
if A.symmetry == 'nonsymmetric':
AH = A.H.asformat(A.format)
BH = levels[-1].BH
# Compute the strength-of-connection matrix C, where larger
# C[i, j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength[len(levels)-1])
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif fn in ('ode', 'evolution'):
if 'B' in kwargs:
C = evolution_strength_of_connection(A, **kwargs)
else:
C = evolution_strength_of_connection(A, B, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'predefined':
C = kwargs['C'].tocsr()
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A.tocsr()
else:
raise ValueError(f'Unrecognized strength of connection method: {str(fn)}')
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate[len(levels)-1])
if fn == 'standard':
AggOp, Cnodes = standard_aggregation(C, **kwargs)
elif fn == 'naive':
AggOp, Cnodes = naive_aggregation(C, **kwargs)
elif fn == 'lloyd':
AggOp, Cnodes = lloyd_aggregation(C, **kwargs)
elif fn == 'pairwise':
AggOp, Cnodes = pairwise_aggregation(A, **kwargs)
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
Cnodes = kwargs['Cnodes']
else:
raise ValueError(f'Unrecognized aggregation method: {str(fn)}')
# Improve near nullspace candidates by relaxing on A B = 0
fn, kwargs = unpack_arg(improve_candidates[len(levels)-1])
if fn is not None:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
B = relaxation_as_linear_operator((fn, kwargs), A, b) * B
levels[-1].B = B
if A.symmetry == 'nonsymmetric':
BH = relaxation_as_linear_operator((fn, kwargs), AH, b) * BH
levels[-1].BH = BH
# Compute the tentative prolongator, T, which is a tentative interpolation
# matrix from the coarse-grid to the fine-grid. T exactly interpolates
# B_fine[:, 0:get_blocksize(A)] = T B_coarse[:, 0:get_blocksize(A)].
T, dummy = fit_candidates(AggOp, B[:, 0:get_blocksize(A)])
del dummy
if A.symmetry == 'nonsymmetric':
TH, dummyH = fit_candidates(AggOp, BH[:, 0:get_blocksize(A)])
del dummyH
# Create necessary root node matrices
Cpt_params = (True, get_Cpt_params(A, Cnodes, AggOp, T))
T = scale_T(T, Cpt_params[1]['P_I'], Cpt_params[1]['I_F'])
if A.symmetry == 'nonsymmetric':
TH = scale_T(TH, Cpt_params[1]['P_I'], Cpt_params[1]['I_F'])
# Set coarse grid near nullspace modes as injected fine grid near
# null-space modes
B = Cpt_params[1]['P_I'].T*levels[-1].B
if A.symmetry == 'nonsymmetric':
BH = Cpt_params[1]['P_I'].T*levels[-1].BH
# Smooth the tentative prolongator, so that it's accuracy is greatly
# improved for algebraically smooth error.
fn, kwargs = unpack_arg(smooth[len(levels)-1])
if fn == 'energy':
P = energy_prolongation_smoother(A, T, C, B, levels[-1].B,
Cpt_params=Cpt_params, **kwargs)
elif fn is None:
P = T
else:
raise ValueError(f'Unrecognized prolongation smoother method: {str(fn)}')
# Compute the restriction matrix R, which interpolates from the fine-grid
# to the coarse-grid. If A is nonsymmetric, then R must be constructed
# based on A.H. Otherwise R = P.H or P.T.
symmetry = A.symmetry
if symmetry == 'hermitian':
R = P.H
elif symmetry == 'symmetric':
R = P.T
elif symmetry == 'nonsymmetric':
fn, kwargs = unpack_arg(smooth[len(levels)-1])
if fn == 'energy':
R = energy_prolongation_smoother(AH, TH, C, BH, levels[-1].BH,
Cpt_params=Cpt_params, **kwargs)
R = R.H
elif fn is None:
R = T.H
else:
raise ValueError(f'Unrecognized prolongation smoother method: {str(fn)}')
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].AggOp = AggOp # aggregation operator
levels[-1].T = T # tentative prolongator
levels[-1].Fpts = Cpt_params[1]['Fpts'] # Fpts
levels[-1].P_I = Cpt_params[1]['P_I'] # Injection operator
levels[-1].I_F = Cpt_params[1]['I_F'] # Identity on F-pts
levels[-1].I_C = Cpt_params[1]['I_C'] # Identity on C-pts
levels[-1].P = P # smoothed prolongator
levels[-1].R = R # restriction operator
levels[-1].Cpts = Cpt_params[1]['Cpts'] # Cpts (i.e., rootnodes)
levels.append(MultilevelSolver.Level())
A = R * A * P # Galerkin operator
A.symmetry = symmetry
levels[-1].A = A
levels[-1].B = B # right near nullspace candidates
if A.symmetry == 'nonsymmetric':
levels[-1].BH = BH # left near nullspace candidates
