def distance_two_interpolation(A,
C,
splitting,
theta=None,
norm='min',
plus_i=True,
cost=[0]):
"""Create prolongator using distance-two AMG interpolation (extended+i interpolaton).
Parameters
----------
A : {csr_matrix}
NxN matrix in CSR format
C : {csr_matrix}
Strength-of-Connection matrix
Must have zero diagonal
splitting : array
C/F splitting stored in an array of length N
theta : float in [0,1), default None
theta value defining strong connections in a classical AMG sense. Provide if
different SOC used for P than for CF-splitting; otherwise, theta = None.
norm : string, default 'abs'
Norm used in redefining classical SOC. Options are 'min' and 'abs' for CSR matrices,
and 'min', 'abs', and 'fro' for BSR matrices. See strength.py for more information.
plus_i : bool, default True
Use "Extended+i" interpolation from [0] as opposed to "Extended" interpolation. Typically
gives better interpolation with minimal added expense.
Returns
-------
P : {csr_matrix}
Prolongator using standard interpolation
References
----------
[0] "Distance-Two Interpolation for Parallel Algebraic Multigrid,"
H. De Sterck, R. Falgout, J. Nolting, U. M. Yang, (2007).
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.classical import standard_interpolation
>>> import numpy as np
>>> A = poisson((5,),format='csr')
>>> splitting = np.array([1,0,1,0,1], dtype='intc')
>>> P = standard_interpolation(A, A, splitting)
>>> print P.todense()
[[ 1. 0. 0. ]
[ 0.5 0.5 0. ]
[ 0. 1. 0. ]
[ 0. 0.5 0.5]
[ 0. 0. 1. ]]
"""
if not isspmatrix_csr(C):
raise TypeError('Expected csr_matrix SOC matrix, C.')
nc = np.sum(splitting)
n = A.shape[0]
# Block BSR format. Transfer A to CSR and the splitting and SOC matrix to have
# DOFs corresponding to CSR A
if isspmatrix_bsr(A):
temp_A = A.tocsr()
splitting0 = splitting * np.ones((A.blocksize[0], 1), dtype='intc')
splitting0 = np.reshape(splitting0, (np.prod(splitting0.shape), ),
order='F')
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
C0 = UnAmal(C0, A.blocksize[0], A.blocksize[1])
else:
C0 = UnAmal(C, A.blocksize[0], A.blocksize[1])
C0 = C0.tocsr()
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(temp_A)
P_indptr = np.empty_like(temp_A.indptr)
amg_core.distance_two_amg_interpolation_pass1(temp_A.shape[0],
C0.indptr, C0.indices,
splitting0, P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=temp_A.dtype)
if plus_i:
amg_core.extended_plusi_interpolation_pass2(
temp_A.shape[0], temp_A.indptr, temp_A.indices, temp_A.data,
C0.indptr, C0.indices, C0.data, splitting0, P_indptr,
P_colinds, P_data)
else:
amg_core.extended_interpolation_pass2(temp_A.shape[0],
temp_A.indptr,
temp_A.indices, temp_A.data,
C0.indptr, C0.indices,
C0.data, splitting0,
P_indptr, P_colinds, P_data)
nc = np.sum(splitting0)
n = A.shape[0]
P = csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
return P.tobsr(blocksize=A.blocksize)
# CSR format
else:
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
else:
C0 = C.copy()
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(A)
P_indptr = np.empty_like(A.indptr)
amg_core.distance_two_amg_interpolation_pass1(A.shape[0], C0.indptr,
C0.indices, splitting,
P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=A.dtype)
if plus_i:
amg_core.extended_plusi_interpolation_pass2(
A.shape[0], A.indptr, A.indices, A.data, C0.indptr, C0.indices,
C0.data, splitting, P_indptr, P_colinds, P_data)
else:
amg_core.extended_interpolation_pass2(A.shape[0], A.indptr,
A.indices, A.data, C0.indptr,
C0.indices, C0.data,
splitting, P_indptr,
P_colinds, P_data)
nc = np.sum(splitting)
n = A.shape[0]
return csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
def standard_interpolation(A,
C,
splitting,
theta=None,
norm='min',
modified=True,
cost=[0]):
"""Create prolongator using standard interpolation
Parameters
----------
A : {csr_matrix}
NxN matrix in CSR format
C : {csr_matrix}
Strength-of-Connection matrix
Must have zero diagonal
splitting : array
C/F splitting stored in an array of length N
theta : float in [0,1), default None
theta value defining strong connections in a classical AMG sense. Provide if
different SOC used for P than for CF-splitting; otherwise, theta = None.
norm : string, default 'abs'
Norm used in redefining classical SOC. Options are 'min' and 'abs' for CSR matrices,
and 'min', 'abs', and 'fro' for BSR matrices. See strength.py for more information.
modified : bool, default True
Use modified classical interpolation. More robust if RS coarsening with second
pass is not used for CF splitting. Ignores interpolating from strong F-connections
without a common C-neighbor.
Returns
-------
P : {csr_matrix}
Prolongator using standard interpolation
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.classical import standard_interpolation
>>> import numpy as np
>>> A = poisson((5,),format='csr')
>>> splitting = np.array([1,0,1,0,1], dtype='intc')
>>> P = standard_interpolation(A, A, splitting)
>>> print P.todense()
[[ 1. 0. 0. ]
[ 0.5 0.5 0. ]
[ 0. 1. 0. ]
[ 0. 0.5 0.5]
[ 0. 0. 1. ]]
"""
if not isspmatrix_csr(C):
raise TypeError('Expected csr_matrix SOC matrix, C.')
nc = np.sum(splitting)
n = A.shape[0]
# Block BSR format. Transfer A to CSR and the splitting and SOC matrix to have
# DOFs corresponding to CSR A
if isspmatrix_bsr(A):
temp_A = A.tocsr()
splitting0 = splitting * np.ones((A.blocksize[0], 1), dtype='intc')
splitting0 = np.reshape(splitting0, (np.prod(splitting0.shape), ),
order='F')
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
C0 = UnAmal(C0, A.blocksize[0], A.blocksize[1])
else:
C0 = UnAmal(C, A.blocksize[0], A.blocksize[1])
C0 = C0.tocsr()
# Use modified standard interpolation by ignoring strong F-connections that do
# not have a common C-point.
if modified:
amg_core.remove_strong_FF_connections(temp_A.shape[0], C0.indptr,
C0.indices, C0.data,
splitting)
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(temp_A)
P_indptr = np.empty_like(temp_A.indptr)
amg_core.rs_standard_interpolation_pass1(temp_A.shape[0], C0.indptr,
C0.indices, splitting0,
P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=temp_A.dtype)
if modified:
amg_core.mod_standard_interpolation_pass2(
temp_A.shape[0], temp_A.indptr, temp_A.indices, temp_A.data,
C0.indptr, C0.indices, C0.data, splitting0, P_indptr,
P_colinds, P_data)
else:
amg_core.rs_standard_interpolation_pass2(
temp_A.shape[0], temp_A.indptr, temp_A.indices, temp_A.data,
C0.indptr, C0.indices, C0.data, splitting0, P_indptr,
P_colinds, P_data)
nc = np.sum(splitting0)
n = A.shape[0]
P = csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
return P.tobsr(blocksize=A.blocksize)
# CSR format
else:
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
else:
C0 = C.copy()
# Use modified standard interpolation by ignoring strong F-connections that do
# not have a common C-point.
if modified:
amg_core.remove_strong_FF_connections(A.shape[0], C0.indptr,
C0.indices, C0.data,
splitting)
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(A)
P_indptr = np.empty_like(A.indptr)
amg_core.rs_standard_interpolation_pass1(A.shape[0], C0.indptr,
C0.indices, splitting,
P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=A.dtype)
if modified:
amg_core.mod_standard_interpolation_pass2(
A.shape[0], A.indptr, A.indices, A.data, C0.indptr, C0.indices,
C0.data, splitting, P_indptr, P_colinds, P_data)
else:
amg_core.rs_standard_interpolation_pass2(
A.shape[0], A.indptr, A.indices, A.data, C0.indptr, C0.indices,
C0.data, splitting, P_indptr, P_colinds, P_data)
nc = np.sum(splitting)
n = A.shape[0]
return csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])