def energy_prolongation_smoother(A,
T,
Atilde,
B,
Bf,
Cpt_params,
krylov='cg',
maxiter=4,
tol=1e-8,
degree=1,
weighting='local',
prefilter={},
postfilter={},
cost=[0.0]):
"""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
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N)
Atilde : {csr_matrix}
Strength of connection matrix
B : {array}
Near-nullspace modes for coarse grid. Has shape (M,k) where
k is the number of coarse candidate vectors.
Bf : {array}
Near-nullspace modes for fine grid. Has shape (N,k) where
k is the number of coarse candidate vectors.
Cpt_params : {tuple}
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) P_I: P_I.T is the
injection matrix for the Cpts, (2) I_F: an identity matrix for only the
F-points (i.e. I, but with zero rows and columns for C-points) and I_C:
the C-point analogue to I_F. See Notes below for more information.
krylov : {string}
'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
maxiter : integer
Number of energy minimization steps to apply to the prolongator
tol : {scalar}
Minimization tolerance
degree : {int}
Generate sparsity pattern for P based on (Atilde^degree T)
weighting : {string}
'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': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Use inverse of the diagonal of A
prefilter : {dictionary} : Default {}
Filters elements by row in sparsity pattern for P to reduce operator and
setup complexity. If None or 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 in conjunction, with the union of their patterns used.
postfilter : {dictionary} : Default {}
Filters elements by row in smoothed P to reduce operator complexity.
Only supported if using the rootnode_solver. If None or 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 in conjunction, with the union of their patterns used.
Returns
-------
T : {bsr_matrix}
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.
Examples
--------
>>> from pyamg.aggregation import energy_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_matrix
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0,6)
>>> col = np.kron([0,1],np.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> print T.todense()
[[ 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.todense()
[[ 1. 0. ]
[ 1. 0. ]
[ 0.66666667 0.33333333]
[ 0.33333333 0.66666667]
[ 0. 1. ]
[ 0. 1. ]]
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.
"""
# Test Inputs
if maxiter < 0:
raise ValueError('maxiter must be > 0')
if tol > 1:
raise ValueError('tol must be <= 1')
if sparse.isspmatrix_csr(A):
A = A.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(A):
pass
else:
raise TypeError("A must be csr_matrix or bsr_matrix")
if sparse.isspmatrix_csr(T):
T = T.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(T):
pass
else:
raise TypeError("T must be csr_matrix or bsr_matrix")
if T.blocksize[0] != A.blocksize[0]:
raise ValueError("T row-blocksize should be the same as A blocksize")
if B.shape[0] != T.shape[1]:
raise ValueError("B is the candidates for the coarse grid. \
num_rows(b) = num_cols(T)")
if min(T.nnz, A.nnz) == 0:
return T
if not sparse.isspmatrix_csr(Atilde):
raise TypeError("Atilde must be csr_matrix")
if ('theta' in prefilter) and (prefilter['theta'] == 0):
prefilter.pop('theta', None)
if ('theta' in postfilter) and (postfilter['theta'] == 0):
postfilter.pop('theta', None)
# Prepocess Atilde, the strength matrix
if Atilde is None:
Atilde = sparse.csr_matrix(
(np.ones(len(A.indices)), A.indices.copy(), A.indptr.copy()),
shape=(A.shape[0] / A.blocksize[0], A.shape[1] / A.blocksize[1]))
# If Atilde has no nonzeros, then return T
if min(T.nnz, A.nnz) == 0:
return T
# Expand allowed sparsity pattern for P through multiplication by Atilde
if degree > 0:
# Construct Sparsity_Pattern by multiplying with Atilde
T.sort_indices()
shape = (int(T.shape[0] / T.blocksize[0]),
int(T.shape[1] / T.blocksize[1]))
Sparsity_Pattern = sparse.csr_matrix(
(np.ones(T.indices.shape), T.indices, T.indptr), shape=shape)
AtildeCopy = Atilde.copy()
for i in range(degree):
cost[0] += mat_mat_complexity(AtildeCopy,
Sparsity_Pattern) / float(A.nnz)
Sparsity_Pattern = AtildeCopy * Sparsity_Pattern
# Optional filtering of sparsity pattern before smoothing
# - Complexity: two passes through T for theta-filter, a sort on
# each row for k-filter, adding matrices if both.
if 'theta' in prefilter and 'k' in prefilter:
temp_cost = [0.0]
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
cost[0] += Sparsity_Pattern.nnz / float(A.nnz)
elif 'k' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif 'theta' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
# UnAmal returns a BSR matrix with 1's in the nonzero locations
Sparsity_Pattern = UnAmal(Sparsity_Pattern, T.blocksize[0],
T.blocksize[1])
Sparsity_Pattern.sort_indices()
else:
# If degree is 0, just copy T for the sparsity pattern
Sparsity_Pattern = T.copy()
if 'theta' in prefilter and 'k' in prefilter:
temp_cost = [0.0]
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
cost[0] += Sparsity_Pattern.nnz / float(A.nnz)
elif 'k' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = truncate_rows(Sparsity_Pattern,
prefilter['k'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif 'theta' in prefilter:
temp_cost = [0.0]
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern,
prefilter['theta'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
Sparsity_Pattern.data[:] = 1.0
Sparsity_Pattern.sort_indices()
# If using root nodes, enforce identity at C-points
if Cpt_params[0]:
Sparsity_Pattern = Cpt_params[1]['I_F'] * Sparsity_Pattern
Sparsity_Pattern = Cpt_params[1]['P_I'] + Sparsity_Pattern
cost[0] += Sparsity_Pattern.nnz / float(A.nnz)
# Construct array of inv(Bi'Bi), where Bi is B restricted to row i's
# sparsity pattern in Sparsity Pattern. This array is used multiple times
# in Satisfy_Constraints(...).
temp_cost = [0.0]
BtBinv = compute_BtBinv(B, Sparsity_Pattern, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# If using root nodes and B has more columns that A's blocksize, then
# T must be updated so that T*B = Bfine. Note, if this is a 'secondpass'
# after dropping entries in P, then we must re-enforce the constraints
if (Cpt_params[0] and
(B.shape[1] > A.blocksize[0])) or ('secondpass' in postfilter):
temp_cost = [0.0]
T = filter_operator(T, Sparsity_Pattern, B, Bf, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
# Iteratively minimize the energy of T subject to the constraints of
# Sparsity_Pattern and maintaining T's effect on B, i.e. T*B =
# (T+Update)*B, i.e. Update*B = 0
if krylov == 'cg':
T = cg_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params,
cost)
elif krylov == 'cgnr':
T = cgnr_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params,
cost)
elif krylov == 'gmres':
T = gmres_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params,
cost)
T.eliminate_zeros()
# Filter entries in P, only in the rootnode case, i.e., Cpt_params[0] == True
if (len(postfilter) == 0) or ('secondpass'
in postfilter) or (Cpt_params[0] is False):
return T
else:
if 'theta' in postfilter and 'k' in postfilter:
temp_cost = [0.0]
T_theta = filter_matrix_rows(T,
postfilter['theta'],
cost=temp_cost)
T_k = truncate_rows(T, postfilter['k'], cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Union two sparsity patterns
T_theta.data[:] = 1.0
T_k.data[:] = 1.0
T_filter = T_theta + T_k
T_filter.data[:] = 1.0
T_filter = T.multiply(T_filter)
elif 'k' in postfilter:
temp_cost = [0.0]
T_filter = truncate_rows(T, postfilter['k'], cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
elif 'theta' in postfilter:
temp_cost = [0.0]
T_filter = filter_matrix_rows(T,
postfilter['theta'],
cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
else:
raise ValueError("Unrecognized postfilter option")
# Re-smooth T_filter and re-fit the modes B into the span.
# Note, we set 'secondpass', because this is the second
# filtering pass
T = energy_prolongation_smoother(A,
T_filter,
Atilde,
B,
Bf,
Cpt_params,
krylov=krylov,
maxiter=1,
tol=1e-8,
degree=0,
weighting=weighting,
prefilter={},
postfilter={'secondpass': True},
cost=cost)
return T
def energy_prolongation_smoother(A, T, Atilde, B, Bf, Cpt_params,
krylov='cg', maxiter=4, tol=1e-8,
degree=1, weighting='local',
prefilter={}, postfilter={}):
"""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
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N)
Atilde : {csr_matrix}
Strength of connection matrix
B : {array}
Near-nullspace modes for coarse grid. Has shape (M,k) where
k is the number of coarse candidate vectors.
Bf : {array}
Near-nullspace modes for fine grid. Has shape (N,k) where
k is the number of coarse candidate vectors.
Cpt_params : {tuple}
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) P_I: P_I.T is the
injection matrix for the Cpts, (2) I_F: an identity matrix for only the
F-points (i.e. I, but with zero rows and columns for C-points) and I_C:
the C-point analogue to I_F. See Notes below for more information.
krylov : {string}
'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
maxiter : integer
Number of energy minimization steps to apply to the prolongator
tol : {scalar}
Minimization tolerance
degree : {int}
Generate sparsity pattern for P based on (Atilde^degree T)
weighting : {string}
'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': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Use inverse of the diagonal of A
prefilter : {dictionary} : Default {}
Filters elements by row in sparsity pattern for P to reduce operator and
setup complexity. If None or 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 in conjunction, with the union of their patterns used.
postfilter : {dictionary} : Default {}
Filters elements by row in smoothed P to reduce operator complexity.
Only supported if using the rootnode_solver. If None or 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 in conjunction, with the union of their patterns used.
Returns
-------
T : {bsr_matrix}
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.
Examples
--------
>>> from pyamg.aggregation import energy_prolongation_smoother
>>> from pyamg.gallery import poisson
>>> from scipy.sparse import coo_matrix
>>> import numpy as np
>>> data = np.ones((6,))
>>> row = np.arange(0,6)
>>> col = np.kron([0,1],np.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> print T.todense()
[[ 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.todense()
[[ 1. 0. ]
[ 1. 0. ]
[ 0.66666667 0.33333333]
[ 0.33333333 0.66666667]
[ 0. 1. ]
[ 0. 1. ]]
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.
"""
# Test Inputs
if maxiter < 0:
raise ValueError('maxiter must be > 0')
if tol > 1:
raise ValueError('tol must be <= 1')
if sparse.isspmatrix_csr(A):
A = A.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(A):
pass
else:
raise TypeError("A must be csr_matrix or bsr_matrix")
if sparse.isspmatrix_csr(T):
T = T.tobsr(blocksize=(1, 1), copy=False)
elif sparse.isspmatrix_bsr(T):
pass
else:
raise TypeError("T must be csr_matrix or bsr_matrix")
if T.blocksize[0] != A.blocksize[0]:
raise ValueError("T row-blocksize should be the same as A blocksize")
if B.shape[0] != T.shape[1]:
raise ValueError("B is the candidates for the coarse grid. \
num_rows(b) = num_cols(T)")
if min(T.nnz, A.nnz) == 0:
return T
if not sparse.isspmatrix_csr(Atilde):
raise TypeError("Atilde must be csr_matrix")
if ('theta' in prefilter) and (prefilter['theta'] == 0):
prefilter.pop('theta', None)
if ('theta' in postfilter) and (postfilter['theta'] == 0):
postfilter.pop('theta', None)
# Prepocess Atilde, the strength matrix
if Atilde is None:
Atilde = sparse.csr_matrix((np.ones(len(A.indices)),
A.indices.copy(), A.indptr.copy()),
shape=(A.shape[0]/A.blocksize[0],
A.shape[1]/A.blocksize[1]))
# If Atilde has no nonzeros, then return T
if min(T.nnz, A.nnz) == 0:
return T
# Expand allowed sparsity pattern for P through multiplication by Atilde
if degree > 0:
# Construct Sparsity_Pattern by multiplying with Atilde
T.sort_indices()
shape = (int(T.shape[0]/T.blocksize[0]), int(T.shape[1]/T.blocksize[1]))
Sparsity_Pattern = sparse.csr_matrix((np.ones(T.indices.shape),
T.indices, T.indptr),
shape=shape)
AtildeCopy = Atilde.copy()
AtildeCopy.data[:] = 1.0
for i in range(degree):
Sparsity_Pattern = AtildeCopy*Sparsity_Pattern
# Optional filtering of sparsity pattern before smoothing
if 'theta' in prefilter and 'k' in prefilter:
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
elif 'k' in prefilter:
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
elif 'theta' in prefilter:
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
# UnAmal returns a BSR matrix with 1's in the nonzero locations
Sparsity_Pattern = UnAmal(Sparsity_Pattern,
T.blocksize[0], T.blocksize[1])
Sparsity_Pattern.sort_indices()
else:
# If degree is 0, just copy T for the sparsity pattern
Sparsity_Pattern = T.copy()
if 'theta' in prefilter and 'k' in prefilter:
Sparsity_theta = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
# Union two sparsity patterns
Sparsity_Pattern += Sparsity_theta
elif 'k' in prefilter:
Sparsity_Pattern = truncate_rows(Sparsity_Pattern, prefilter['k'])
elif 'theta' in prefilter:
Sparsity_Pattern = filter_matrix_rows(Sparsity_Pattern, prefilter['theta'])
elif len(prefilter) > 0:
raise ValueError("Unrecognized prefilter option")
Sparsity_Pattern.data[:] = 1.0
Sparsity_Pattern.sort_indices()
# If using root nodes, enforce identity at C-points
if Cpt_params[0]:
Sparsity_Pattern = Cpt_params[1]['I_F'] * Sparsity_Pattern
Sparsity_Pattern = Cpt_params[1]['P_I'] + Sparsity_Pattern
# Construct array of inv(Bi'Bi), where Bi is B restricted to row i's
# sparsity pattern in Sparsity Pattern. This array is used multiple times
# in Satisfy_Constraints(...).
BtBinv = compute_BtBinv(B, Sparsity_Pattern)
# If using root nodes and B has more columns that A's blocksize, then
# T must be updated so that T*B = Bfine. Note, if this is a 'secondpass'
# after dropping entries in P, then we must re-enforce the constraints
if (Cpt_params[0] and (B.shape[1] > A.blocksize[0])) or ('secondpass' in postfilter):
T = filter_operator(T, Sparsity_Pattern, B, Bf, BtBinv)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F']*T + Cpt_params[1]['P_I']
# Iteratively minimize the energy of T subject to the constraints of
# Sparsity_Pattern and maintaining T's effect on B, i.e. T*B =
# (T+Update)*B, i.e. Update*B = 0
if krylov == 'cg':
T = cg_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params)
elif krylov == 'cgnr':
T = cgnr_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params)
elif krylov == 'gmres':
T = gmres_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern,
maxiter, tol, weighting, Cpt_params)
T.eliminate_zeros()
# Filter entries in P, only in the rootnode case, i.e., Cpt_params[0] == True
if (len(postfilter) == 0) or ('secondpass' in postfilter) or (Cpt_params[0] is False):
return T
else:
if 'theta' in postfilter and 'k' in postfilter:
T_theta = filter_matrix_rows(T, postfilter['theta'])
T_k = truncate_rows(T, postfilter['k'])
# Union two sparsity patterns
T_theta.data[:] = 1.0
T_k.data[:] = 1.0
T_filter = T_theta + T_k
T_filter.data[:] = 1.0
T_filter = T.multiply(T_filter)
elif 'k' in postfilter:
T_filter = truncate_rows(T, postfilter['k'])
elif 'theta' in postfilter:
T_filter = filter_matrix_rows(T, postfilter['theta'])
else:
raise ValueError("Unrecognized postfilter option")
# Re-smooth T_filter and re-fit the modes B into the span.
# Note, we set 'secondpass', because this is the second
# filtering pass
T = energy_prolongation_smoother(A, T_filter,
Atilde, B, Bf, Cpt_params,
krylov=krylov, maxiter=1,
tol=1e-8, degree=0,
weighting=weighting,
prefilter={},
postfilter={'secondpass' : True} )
return T