def setup_block_jacobi(lvl, iterations=1, omega=1.0, Dinv=None, blocksize=None,
withrho=True):
# Determine Blocksize
if blocksize is None and Dinv is None:
if sp.sparse.isspmatrix_csr(lvl.A):
blocksize = 1
elif sp.sparse.isspmatrix_bsr(lvl.A):
blocksize = lvl.A.blocksize[0]
elif blocksize is None:
blocksize = Dinv.shape[1]
if blocksize == 1:
# Block Jacobi is equivalent to normal Jacobi
return setup_jacobi(lvl, iterations=iterations, omega=omega,
withrho=withrho)
else:
# Use Block Jacobi
if Dinv is None:
Dinv = get_block_diag(lvl.A, blocksize=blocksize, inv_flag=True)
if withrho:
omega = omega/rho_block_D_inv_A(lvl.A, Dinv)
def smoother(A, x, b):
relaxation.block_jacobi(A, x, b, iterations=iterations,
omega=omega, Dinv=Dinv,
blocksize=blocksize)
return smoother
def setup_block_jacobi(lvl, iterations=DEFAULT_NITER, omega=1.0, Dinv=None,
blocksize=None, withrho=True):
# Determine Blocksize
if blocksize is None and Dinv is None:
if sp.sparse.isspmatrix_csr(lvl.A):
blocksize = 1
elif sp.sparse.isspmatrix_bsr(lvl.A):
blocksize = lvl.A.blocksize[0]
elif blocksize is None:
blocksize = Dinv.shape[1]
if blocksize == 1:
# Block Jacobi is equivalent to normal Jacobi
return setup_jacobi(lvl, iterations=iterations, omega=omega,
withrho=withrho)
else:
# Use Block Jacobi
if Dinv is None:
Dinv = get_block_diag(lvl.A, blocksize=blocksize, inv_flag=True)
if withrho:
omega = omega/rho_block_D_inv_A(lvl.A, Dinv)
def smoother(A, x, b):
relaxation.block_jacobi(A, x, b, iterations=iterations,
omega=omega, Dinv=Dinv,
blocksize=blocksize)
return smoother
def setup_block_gauss_seidel(lvl, iterations=DEFAULT_NITER,
sweep=DEFAULT_SWEEP,
Dinv=None, blocksize=None):
# Determine Blocksize
if blocksize is None and Dinv is None:
if sp.sparse.isspmatrix_csr(lvl.A):
blocksize = 1
elif sp.sparse.isspmatrix_bsr(lvl.A):
blocksize = lvl.A.blocksize[0]
elif blocksize is None:
blocksize = Dinv.shape[1]
if blocksize == 1:
# Block GS is equivalent to normal GS
return setup_gauss_seidel(lvl, iterations=iterations, sweep=sweep)
else:
# Use Block GS
if Dinv is None:
Dinv = get_block_diag(lvl.A, blocksize=blocksize, inv_flag=True)
def smoother(A, x, b):
relaxation.block_gauss_seidel(A, x, b, iterations=iterations,
Dinv=Dinv, blocksize=blocksize,
sweep=sweep)
return smoother
def setup_block_gauss_seidel(lvl,
iterations=DEFAULT_NITER,
sweep=DEFAULT_SWEEP,
Dinv=None,
blocksize=None):
# Determine Blocksize
if blocksize is None and Dinv is None:
if sp.sparse.isspmatrix_csr(lvl.A):
blocksize = 1
elif sp.sparse.isspmatrix_bsr(lvl.A):
blocksize = lvl.A.blocksize[0]
elif blocksize is None:
blocksize = Dinv.shape[1]
if blocksize == 1:
# Block GS is equivalent to normal GS
return setup_gauss_seidel(lvl, iterations=iterations, sweep=sweep)
else:
# Use Block GS
if Dinv is None:
Dinv = get_block_diag(lvl.A, blocksize=blocksize, inv_flag=True)
def smoother(A, x, b):
relaxation.block_gauss_seidel(A,
x,
b,
iterations=iterations,
Dinv=Dinv,
blocksize=blocksize,
sweep=sweep)
return smoother
def setup_FC_block_jacobi(lvl, F_iterations=DEFAULT_NITER, C_iterations=DEFAULT_NITER,
iterations=DEFAULT_NITER, omega=1.0, Dinv=None, blocksize=None,
withrho=False):
# Determine Blocksize
if blocksize is None and Dinv is None:
if sp.sparse.isspmatrix_csr(lvl.A):
blocksize = 1
elif sp.sparse.isspmatrix_bsr(lvl.A):
blocksize = lvl.A.blocksize[0]
elif blocksize is None:
if sp.sparse.isspmatrix_bsr(Dinv):
blocksize = Dinv.blocksize[1]
else:
blocksize = 1
# Check for compatible dimensions
if (lvl.A.shape[0] % blocksize) != 0:
raise ValueError("Blocksize does not divide size of matrix.")
elif (len(lvl.splitting)*blocksize != lvl.A.shape[0]):
raise ValueError("Blocksize not compatible with CF-splitting and matrix size.")
if blocksize == 1:
# Block Jacobi is equivalent to normal Jacobi
return setup_FC_jacobi(lvl, iterations=iterations, omega=omega,
withrho=withrho)
else:
# Get C-points and F-points from splitting
try:
Fpts = np.array(np.where(lvl.splitting == 0)[0], dtype='int32')
Cpts = np.array(np.where(lvl.splitting == 1)[0], dtype='int32')
lvl.nf = len(Fpts)
lvl.nc = len(Cpts)
except:
raise ValueError("CF-splitting array needs to be stored in hierarchy.")
# Use Block Jacobi
if Dinv is None:
Dinv = get_block_diag(lvl.A, blocksize=blocksize, inv_flag=True)
if withrho:
omega = omega/rho_block_D_inv_A(lvl.A, Dinv)
def smoother(A, x, b):
relaxation.FC_block_jacobi(A, x, b, Cpts=Cpts, Fpts=Fpts, iterations=iterations,
F_iterations=F_iterations, C_iterations=C_iterations,
omega=omega, Dinv=Dinv, blocksize=blocksize)
return smoother
def jacobi_prolongation_smoother(S, T, C, B, omega=4.0/3.0, degree=1, filter=False, weighting='diagonal'):
"""Jacobi prolongation smoother
Parameters
----------
S : {csr_matrix, bsr_matrix}
Sparse NxN matrix used for smoothing. Typically, A.
T : {csr_matrix, bsr_matrix}
Tentative prolongator
C : {csr_matrix, bsr_matrix}
Strength-of-connection matrix
B : {array}
Near nullspace modes for the coarse grid such that T*B
exactly reproduces the fine grid near nullspace modes
omega : {scalar}
Damping parameter
filter : {boolean}
If true, filter S before smoothing T. This option can greatly control
complexity.
weighting : {string}
'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': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Classic Jacobi D = diagonal(A)
Returns
-------
P : {csr_matrix, bsr_matrix}
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_matrix
>>> import numpy
>>> data = numpy.ones((6,))
>>> row = numpy.arange(0,6)
>>> col = numpy.kron([0,1],numpy.ones((3,)))
>>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr()
>>> T.todense()
matrix([[ 1., 0.],
[ 1., 0.],
[ 1., 0.],
[ 0., 1.],
[ 0., 1.],
[ 0., 1.]])
>>> A = poisson((6,),format='csr')
>>> P = jacobi_prolongation_smoother(A,T,A,numpy.ones((2,1)))
>>> P.todense()
matrix([[ 0.64930164, 0. ],
[ 1. , 0. ],
[ 0.64930164, 0.35069836],
[ 0.35069836, 0.64930164],
[ 0. , 1. ],
[ 0. , 0.64930164]])
"""
# preprocess weighting
if weighting == 'block':
if isspmatrix_csr(S):
weighting = 'diagonal'
elif isspmatrix_bsr(S):
if S.blocksize[0] == 1:
weighting = 'diagonal'
if filter:
##
# Implement filtered prolongation smoothing for the general case by
# utilizing satisfy constraints
if isspmatrix_bsr(S):
numPDEs = S.blocksize[0]
else:
numPDEs = 1
# Create a filtered S with entries dropped that aren't in C
C = UnAmal(C, numPDEs, numPDEs)
S = S.multiply(C)
S.eliminate_zeros()
if weighting == 'diagonal':
# Use diagonal of S
D_inv = get_diagonal(S, inv=True)
D_inv_S = scale_rows(S, D_inv, copy=True)
D_inv_S = (omega/approximate_spectral_radius(D_inv_S))*D_inv_S
elif weighting == 'block':
# Use block diagonal of S
D_inv = get_block_diag(S, blocksize=S.blocksize[0], inv_flag=True)
D_inv = bsr_matrix( (D_inv, numpy.arange(D_inv.shape[0]), \
numpy.arange(D_inv.shape[0]+1)), shape = S.shape)
D_inv_S = D_inv*S
D_inv_S = (omega/approximate_spectral_radius(D_inv_S))*D_inv_S
elif weighting == 'local':
# Use the Gershgorin estimate as each row's weight, instead of a global
# spectral radius estimate
D = numpy.abs(S)*numpy.ones((S.shape[0],1), dtype=S.dtype)
D_inv = numpy.zeros_like(D)
D_inv[D != 0] = 1.0 / numpy.abs(D[D != 0])
D_inv_S = scale_rows(S, D_inv, copy=True)
D_inv_S = omega*D_inv_S
else:
raise ValueError('Incorrect weighting option')
if filter:
##
# Carry out Jacobi, but after calculating the prolongator update, U,
# apply satisfy constraints so that U*B = 0
P = T
for i in range(degree):
U = (D_inv_S*P).tobsr(blocksize=P.blocksize)
##
# Enforce U*B = 0
# (1) 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, U)
# (2) Apply satisfy constraints
Satisfy_Constraints(U, B, BtBinv)
##
# Update P
P = P - U
else:
##
# Carry out Jacobi as normal
P = T
for i in range(degree):
P = P - (D_inv_S*P)
return P
def cg_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern, maxiter, tol, weighting='local', Cpt_params = None):
'''
Helper function for energy_prolongation_smoother(...)
Use CG to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal preconditioning
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
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.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using conjugate gradients to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
# Preallocate
AP = bsr_matrix((numpy.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype), Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape) )
# CG will be run with diagonal preconditioning
if weighting == 'diagonal':
Dinv = get_diagonal(A, norm_eq=False, inv=True)
elif weighting == 'block':
Dinv = get_block_diag(A, blocksize=A.blocksize[0], inv_flag=True)
Dinv = bsr_matrix( (Dinv, numpy.arange(Dinv.shape[0]), numpy.arange(Dinv.shape[0]+1)), shape = A.shape)
elif weighting == 'local':
# Based on Gershgorin estimate
D = numpy.abs(A)*numpy.ones((A.shape[0],1), dtype=A.dtype)
Dinv = numpy.zeros_like(D)
Dinv[D != 0] = 1.0 / numpy.abs(D[D != 0])
else:
raise ValueError('weighting value is invalid')
# Calculate initial residual
# Equivalent to R = -A*T; R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
R = bsr_matrix((numpy.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype), Sparsity_Pattern.indices,
Sparsity_Pattern.indptr), shape=(Sparsity_Pattern.shape) )
pyamg.amg_core.incomplete_mat_mult_bsr(A.indptr, A.indices, numpy.ravel(A.data),
T.indptr, T.indices, numpy.ravel(T.data),
R.indptr, R.indices, numpy.ravel(R.data),
T.shape[0]/T.blocksize[0], T.shape[1]/T.blocksize[1],
A.blocksize[0], A.blocksize[1], T.blocksize[1])
R.data *= -1.0
# Enforce R*B = 0
Satisfy_Constraints(R, B, BtBinv)
if R.nnz == 0:
print "Error in sa_energy_min(..). Initial R no nonzeros on a level. Returning tentative prolongator\n"
return T
#Calculate Frobenius norm of the residual
resid = R.nnz ##numpy.sqrt((R.data.conjugate()*R.data).sum())
#print "Energy Minimization of Prolongator --- Iteration 0 --- r = " + str(resid)
i = 0
while i < maxiter and resid > tol:
#Apply diagonal preconditioner
if weighting == 'local' or weighting == 'diagonal':
Z = scale_rows(R, Dinv)
else:
Z = Dinv*R
#Frobenius inner-product of (R,Z) = sum( numpy.conjugate(rk).*zk)
newsum = (R.conjugate().multiply(Z)).sum()
if newsum < tol:
# met tolerance, so halt
break
#P is the search direction, not the prolongator, which is T.
if(i == 0):
P = Z
else:
beta = newsum/oldsum
P = Z + beta*P
oldsum = newsum
# Calculate new direction and enforce constraints
# Equivalent to: AP = A*P; AP = AP.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not !!!!
AP.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(A.indptr, A.indices, numpy.ravel(A.data),
P.indptr, P.indices, numpy.ravel(P.data),
AP.indptr, AP.indices, numpy.ravel(AP.data),
T.shape[0]/T.blocksize[0], T.shape[1]/T.blocksize[1],
A.blocksize[0], A.blocksize[1], P.blocksize[1])
# Enforce AP*B = 0
Satisfy_Constraints(AP, B, BtBinv)
#Frobenius inner-product of (P, AP)
alpha = newsum/(P.conjugate().multiply(AP)).sum()
#Update the prolongator, T
T = T + alpha*P
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F']*T + Cpt_params[1]['P_I']
#Update residual
R = R - alpha*AP
i += 1
#Calculate Frobenius norm of the residual
resid = R.nnz #numpy.sqrt((R.data.conjugate()*R.data).sum())
#print "Energy Minimization of Prolongator --- Iteration " + str(i) + " --- r = " + str(resid)
return T
def jacobi_prolongation_smoother(S,
T,
C,
B,
omega=4.0 / 3.0,
degree=1,
filter=False,
weighting='diagonal',
cost=[0.0]):
"""Jacobi prolongation smoother
Parameters
----------
S : {csr_matrix, bsr_matrix}
Sparse NxN matrix used for smoothing. Typically, A.
T : {csr_matrix, bsr_matrix}
Tentative prolongator
C : {csr_matrix, bsr_matrix}
Strength-of-connection matrix
B : {array}
Near nullspace modes for the coarse grid such that T*B
exactly reproduces the fine grid near nullspace modes
omega : {scalar}
Damping parameter
filter : {boolean}
If true, filter S before smoothing T. This option can greatly control
complexity.
weighting : {string}
'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': If A is a BSR matrix, use a block diagonal inverse of A
'diagonal': Classic Jacobi D = diagonal(A)
Returns
-------
P : {csr_matrix, bsr_matrix}
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_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()
>>> T.todense()
matrix([[ 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.todense()
matrix([[ 0.64930164, 0. ],
[ 1. , 0. ],
[ 0.64930164, 0.35069836],
[ 0.35069836, 0.64930164],
[ 0. , 1. ],
[ 0. , 0.64930164]])
"""
# preprocess weighting
if weighting == 'block':
if sparse.isspmatrix_csr(S):
weighting = 'diagonal'
elif sparse.isspmatrix_bsr(S):
if S.blocksize[0] == 1:
weighting = 'diagonal'
if filter:
# Implement filtered prolongation smoothing for the general case by
# utilizing satisfy constraints
if sparse.isspmatrix_bsr(S):
numPDEs = S.blocksize[0]
else:
numPDEs = 1
# Create a filtered S with entries dropped that aren't in C
C = UnAmal(C, numPDEs, numPDEs)
S = S.multiply(C)
S.eliminate_zeros()
cost[0] += 1.0
if weighting == 'diagonal':
# Use diagonal of S
D_inv = get_diagonal(S, inv=True)
D_inv_S = scale_rows(S, D_inv, copy=True)
D_inv_S = (omega / approximate_spectral_radius(D_inv_S)) * D_inv_S
# 15 WU to find spectral radius, 2 to scale D_inv_S twice
cost[0] += 17
elif weighting == 'block':
# Use block diagonal of S
D_inv = get_block_diag(S, blocksize=S.blocksize[0], inv_flag=True)
D_inv = sparse.bsr_matrix(
(D_inv, np.arange(D_inv.shape[0]), np.arange(D_inv.shape[0] + 1)),
shape=S.shape)
D_inv_S = D_inv * S
# 15 WU to find spectral radius, 2 to scale D_inv_S twice
D_inv_S = (omega / approximate_spectral_radius(D_inv_S)) * D_inv_S
cost[0] += 17
elif weighting == 'local':
# Use the Gershgorin estimate as each row's weight, instead of a global
# spectral radius estimate
D = np.abs(S) * np.ones((S.shape[0], 1), dtype=S.dtype)
D_inv = np.zeros_like(D)
D_inv[D != 0] = 1.0 / np.abs(D[D != 0])
D_inv_S = scale_rows(S, D_inv, copy=True)
D_inv_S = omega * D_inv_S
cost[0] += 3
else:
raise ValueError('Incorrect weighting option')
if filter:
# Carry out Jacobi, but after calculating the prolongator update, U,
# apply satisfy constraints so that U*B = 0
P = T
for i in range(degree):
if sparse.isspmatrix_bsr(P):
U = (D_inv_S * P).tobsr(blocksize=P.blocksize)
else:
U = D_inv_S * P
cost[0] += P.nnz / float(S.nnz)
# (1) Enforce U*B = 0. 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, U, cost=temp_cost)
cost[0] += temp_cost[0] / float(S.nnz)
# (2) Apply satisfy constraints
temp_cost = [0.0]
Satisfy_Constraints(U, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(S.nnz)
# Update P
P = P - U
cost[0] += max(P.nnz, U.nnz) / float(S.nnz)
else:
# Carry out Jacobi as normal
P = T
for i in range(degree):
P = P - (D_inv_S * P)
cost[0] += P.nnz / float(S.nnz)
return P
def gmres_prolongation_smoothing(A,
T,
B,
BtBinv,
Sparsity_Pattern,
maxiter,
tol,
weighting='local',
Cpt_params=None,
cost=[0.0]):
'''
Helper function for energy_prolongation_smoother(...).
Use GMRES to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
Should be at least nonsymmetric or indefinite
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal
preconditioning
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
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.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using GMRES to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
# For non-SPD system, apply GMRES with Diagonal Preconditioning
# Preallocate space for new search directions
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
AV = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
# Preallocate for Givens Rotations, Hessenberg matrix and Krylov Space
xtype = sparse.sputils.upcast(A.dtype, T.dtype, B.dtype)
Q = [] # Givens Rotations
V = [] # Krylov Space
# vs = [] # vs store the pointers to each column of V for speed
# Upper Hessenberg matrix, converted to upper tri with Givens Rots
H = np.zeros((maxiter + 1, maxiter + 1), dtype=xtype)
# GMRES will be run with diagonal preconditioning
if weighting == 'diagonal':
Dinv = get_diagonal(A, norm_eq=False, inv=True)
elif weighting == 'block':
Dinv = get_block_diag(A, blocksize=A.blocksize[0], inv_flag=True)
Dinv = sparse.bsr_matrix(
(Dinv, np.arange(Dinv.shape[0]), np.arange(Dinv.shape[0] + 1)),
shape=A.shape)
elif weighting == 'local':
# Based on Gershgorin estimate
D = np.abs(A) * np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
cost[0] += 1.0
else:
raise ValueError('weighting value is invalid')
# Calculate initial residual
# Equivalent to R = -A*T; R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
R = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), T.indptr, T.indices,
np.ravel(T.data), R.indptr, R.indices, np.ravel(R.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], T.blocksize[1])
R.data *= -1.0
# T is block diagonal, using sparsity pattern of R with
# incomplete=True significantly overestimates complexity.
# More accurate to use full mat-mat with block diagonal T.
cost[0] += mat_mat_complexity(A, T, incomplete=False) / float(A.nnz)
# Apply diagonal preconditioner
if weighting == 'local' or weighting == 'diagonal':
R = scale_rows(R, Dinv)
else:
R = Dinv * R
cost[0] += R.nnz / float(A.nnz)
# Enforce R*B = 0
temp_cost = [0.0]
Satisfy_Constraints(R, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
if R.nnz == 0:
print("Error in sa_energy_min(..). Initial R no nonzeros on a level. \
Returning tentative prolongator\n")
return T
# This is the RHS vector for the problem in the Krylov Space
normr = np.sqrt((R.data.conjugate() * R.data).sum())
g = np.zeros((maxiter + 1, ), dtype=xtype)
g[0] = normr
# First Krylov vector
# V[0] = r/normr
if normr > 0.0:
V.append((1.0 / normr) * R)
i = -1
while i < maxiter - 1 and normr > tol:
i = i + 1
# Calculate new search direction
# Equivalent to: AV = A*V; AV = AV.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not
AV.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), V[i].indptr, V[i].indices,
np.ravel(V[i].data), AV.indptr, AV.indices, np.ravel(AV.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], T.blocksize[1])
cost[0] += mat_mat_complexity(A, AV, incomplete=True) / float(A.nnz)
if weighting == 'local' or weighting == 'diagonal':
AV = scale_rows(AV, Dinv)
else:
AV = Dinv * AV
cost[0] += AV.nnz / float(A.nnz)
# Enforce AV*B = 0
temp_cost = [0.0]
Satisfy_Constraints(AV, B, BtBinv, cost=temp_cost)
V.append(AV.copy())
cost[0] += temp_cost[0] / float(A.nnz)
# Modified Gram-Schmidt
for j in range(i + 1):
# Frobenius inner-product
H[j, i] = (V[j].conjugate().multiply(V[i + 1])).sum()
V[i + 1] = V[i + 1] - H[j, i] * V[j]
cost[0] += 2.0 * max(V[i + 1].nnz, V[j].nnz) / float(A.nnz)
# Frobenius Norm
H[i + 1, i] = np.sqrt(
(V[i + 1].data.conjugate() * V[i + 1].data).sum())
cost[0] += V[i + 1].nnz / float(A.nnz)
# Check for breakdown
if H[i + 1, i] != 0.0:
V[i + 1] = (1.0 / H[i + 1, i]) * V[i + 1]
cost[0] += V[i + 1].nnz / float(A.nnz)
# Apply previous Givens rotations to H
if i > 0:
apply_givens(Q, H[:, i], i)
# Calculate and apply next complex-valued Givens Rotation
if H[i + 1, i] != 0:
h1 = H[i, i]
h2 = H[i + 1, i]
h1_mag = np.abs(h1)
h2_mag = np.abs(h2)
if h1_mag < h2_mag:
mu = h1 / h2
tau = np.conjugate(mu) / np.abs(mu)
else:
mu = h2 / h1
tau = mu / np.abs(mu)
denom = np.sqrt(h1_mag**2 + h2_mag**2)
c = h1_mag / denom
s = h2_mag * tau / denom
Qblock = np.array([[c, np.conjugate(s)], [-s, c]], dtype=xtype)
Q.append(Qblock)
# Apply Givens Rotation to g,
# the RHS for the linear system in the Krylov Subspace.
g[i:i + 2] = sp.dot(Qblock, g[i:i + 2])
# Apply effect of Givens Rotation to H
H[i, i] = sp.dot(Qblock[0, :], H[i:i + 2, i])
H[i + 1, i] = 0.0
normr = np.abs(g[i + 1])
# End while loop
# Find best update to x in Krylov Space, V. Solve (i x i) system.
if i != -1:
y = la.solve(H[0:i + 1, 0:i + 1], g[0:i + 1])
for j in range(i + 1):
T = T + y[j] * V[j]
cost[0] += max(T.nnz, V[j].nnz) / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
return T
def cg_prolongation_smoothing(A,
T,
B,
BtBinv,
Sparsity_Pattern,
maxiter,
tol,
weighting='local',
Cpt_params=None,
cost=[0.0]):
'''
Helper function for energy_prolongation_smoother(...)
Use CG to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal
preconditioning
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
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.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using conjugate gradients to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
# Preallocate
AP = sparse.bsr_matrix(
(np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype),
Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
# CG will be run with diagonal preconditioning
if weighting == 'diagonal':
Dinv = get_diagonal(A, norm_eq=False, inv=True)
elif weighting == 'block':
Dinv = get_block_diag(A, blocksize=A.blocksize[0], inv_flag=True)
Dinv = sparse.bsr_matrix(
(Dinv, np.arange(Dinv.shape[0]), np.arange(Dinv.shape[0] + 1)),
shape=A.shape)
elif weighting == 'local':
# Based on Gershgorin estimate
D = np.abs(A) * np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
cost[0] += 1
else:
raise ValueError('weighting value is invalid')
# Calculate initial residual
# Equivalent to R = -A*T; R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
uones = np.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype)
R = sparse.bsr_matrix(
(uones, Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape))
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), T.indptr, T.indices,
np.ravel(T.data), R.indptr, R.indices, np.ravel(R.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], T.blocksize[1])
R.data *= -1.0
# T is block diagonal, using sparsity pattern of R with
# incomplete=True significantly overestimates complexity.
# More accurate to use full mat-mat with block diagonal T.
cost[0] += mat_mat_complexity(A, T, incomplete=False) / float(A.nnz)
# Enforce R*B = 0
temp_cost = [0.0]
Satisfy_Constraints(R, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
if R.nnz == 0:
print("Error in sa_energy_min(..). Initial R no nonzeros on a level. \
Returning tentative prolongator\n")
return T
# Calculate Frobenius norm of the residual
resid = R.nnz # np.sqrt((R.data.conjugate()*R.data).sum())
i = 0
while i < maxiter and resid > tol:
# Apply diagonal preconditioner
if weighting == 'local' or weighting == 'diagonal':
Z = scale_rows(R, Dinv)
else:
Z = Dinv * R
cost[0] += R.nnz / float(A.nnz)
# Frobenius inner-product of (R,Z) = sum( np.conjugate(rk).*zk)
newsum = (R.conjugate().multiply(Z)).sum()
cost[0] += Z.nnz / float(A.nnz)
if newsum < tol:
# met tolerance, so halt
break
# P is the search direction, not the prolongator, which is T.
if (i == 0):
P = Z
oldsum = newsum
else:
beta = newsum / oldsum
P = Z + beta * P
cost[0] += max(Z.nnz, P.nnz) / float(A.nnz)
oldsum = newsum
# Calculate new direction and enforce constraints
# Equivalent to: AP = A*P; AP = AP.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not !!!!
AP.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(
A.indptr, A.indices, np.ravel(A.data), P.indptr, P.indices,
np.ravel(P.data), AP.indptr, AP.indices, np.ravel(AP.data),
int(T.shape[0] / T.blocksize[0]), int(T.shape[1] / T.blocksize[1]),
A.blocksize[0], A.blocksize[1], P.blocksize[1])
cost[0] += mat_mat_complexity(A, AP, incomplete=True) / float(A.nnz)
# Enforce AP*B = 0
temp_cost = [0.0]
Satisfy_Constraints(AP, B, BtBinv, cost=temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
# Frobenius inner-product of (P, AP)
alpha = newsum / (P.conjugate().multiply(AP)).sum()
cost[0] += max(P.nnz, AP.nnz) / float(A.nnz)
# Update the prolongator, T
T = T + alpha * P
cost[0] += max(P.nnz, T.nnz) / float(A.nnz)
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F'] * T + Cpt_params[1]['P_I']
# Update residual
R = R - alpha * AP
cost[0] += max(R.nnz, AP.nnz) / float(A.nnz)
# Calculate Frobenius norm of the residual
resid = R.nnz # np.sqrt((R.data.conjugate()*R.data).sum())
i += 1
return T
def block_gauss_seidel(A, x, b, iterations=1, sweep='forward', blocksize=1, Dinv=None):
"""Perform block Gauss-Seidel iteration on the linear system Ax=b
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
iterations : int
Number of iterations to perform
sweep : {'forward','backward','symmetric'}
Direction of sweep
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use Gauss-Seidel as a Stand-Alone Solver
>>> from pyamg.relaxation import *
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((10,10), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> block_gauss_seidel(A, x0, b, iterations=10, blocksize=4, sweep='symmetric')
>>> print norm(b-A*x0)
0.958333817624
>>> #
>>> ## Use Gauss-Seidel as the Multigrid Smoother
>>> from pyamg import smoothed_aggregation_solver
>>> sa = smoothed_aggregation_solver(A, B=numpy.ones((A.shape[0],1)),
... coarse_solver='pinv2', max_coarse=50,
... presmoother=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}),
... postsmoother=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}))
>>> x0=numpy.zeros((A.shape[0],1))
>>> residuals=[]
>>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals)
"""
A,x,b = make_system(A, x, b, formats=['csr','bsr'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv == None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != A.shape[0]/blocksize:
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
if sweep == 'forward':
row_start,row_stop,row_step = 0,len(x)/blocksize,1
elif sweep == 'backward':
row_start,row_stop,row_step = len(x)/blocksize-1,-1,-1
elif sweep == 'symmetric':
for iter in xrange(iterations):
block_gauss_seidel(A, x, b, iterations=1, sweep='forward', blocksize=blocksize, Dinv=Dinv)
block_gauss_seidel(A, x, b, iterations=1, sweep='backward',blocksize=blocksize, Dinv=Dinv)
return
else:
raise ValueError("valid sweep directions are 'forward', 'backward', and 'symmetric'")
for iter in xrange(iterations):
amg_core.block_gauss_seidel(A.indptr, A.indices, numpy.ravel(A.data),
x, b, numpy.ravel(Dinv),
row_start, row_stop, row_step, blocksize)
def evolution_strength_of_connection(A, B='ones', epsilon=4.0, k=2,
proj_type="l2", block_flag=False,
symmetrize_measure=True):
"""
Construct strength of connection matrix using an Evolution-based measure
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
B : {string, array}
If B='ones', then the near nullspace vector used is all ones. If B is
an (NxK) array, then B is taken to be the near nullspace vectors.
epsilon : scalar
Drop tolerance
k : integer
ODE num time steps, step size is assumed to be 1/rho(DinvA)
proj_type : {'l2','D_A'}
Define norm for constrained min prob, i.e. define projection
block_flag : {boolean}
If True, use a block D inverse as preconditioner for A during
weighted-Jacobi
Returns
-------
Atilde : {csr_matrix}
Sparse matrix of strength values
References
----------
.. [1] Olson, L. N., Schroder, J., Tuminaro, R. S.,
"A New Perspective on Strength Measures in Algebraic Multigrid",
submitted, June, 2008.
Examples
--------
>>> import numpy as np
>>> from pyamg.gallery import stencil_grid
>>> from pyamg.strength import evolution_strength_of_connection
>>> n=3
>>> stencil = np.array([[-1.0,-1.0,-1.0],
... [-1.0, 8.0,-1.0],
... [-1.0,-1.0,-1.0]])
>>> A = stencil_grid(stencil, (n,n), format='csr')
>>> S = evolution_strength_of_connection(A, np.ones((A.shape[0],1)))
"""
# local imports for evolution_strength_of_connection
from pyamg.util.utils import scale_rows, get_block_diag, scale_columns
from pyamg.util.linalg import approximate_spectral_radius
# ====================================================================
# Check inputs
if epsilon < 1.0:
raise ValueError("expected epsilon > 1.0")
if k <= 0:
raise ValueError("number of time steps must be > 0")
if proj_type not in ['l2', 'D_A']:
raise ValueError("proj_type must be 'l2' or 'D_A'")
if (not sparse.isspmatrix_csr(A)) and (not sparse.isspmatrix_bsr(A)):
raise TypeError("expected csr_matrix or bsr_matrix")
# ====================================================================
# Format A and B correctly.
# B must be in mat format, this isn't a deep copy
if B == 'ones':
Bmat = np.mat(np.ones((A.shape[0], 1), dtype=A.dtype))
else:
Bmat = np.mat(B)
# Pre-process A. We need A in CSR, to be devoid of explicit 0's and have
# sorted indices
if (not sparse.isspmatrix_csr(A)):
csrflag = False
numPDEs = A.blocksize[0]
D = A.diagonal()
# Calculate Dinv*A
if block_flag:
Dinv = get_block_diag(A, blocksize=numPDEs, inv_flag=True)
Dinv = sparse.bsr_matrix((Dinv, np.arange(Dinv.shape[0]),
np.arange(Dinv.shape[0] + 1)),
shape=A.shape)
Dinv_A = (Dinv * A).tocsr()
else:
Dinv = np.zeros_like(D)
mask = (D != 0.0)
Dinv[mask] = 1.0 / D[mask]
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
A = A.tocsr()
else:
csrflag = True
numPDEs = 1
D = A.diagonal()
Dinv = np.zeros_like(D)
mask = (D != 0.0)
Dinv[mask] = 1.0 / D[mask]
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
A.eliminate_zeros()
A.sort_indices()
# Handle preliminaries for the algorithm
dimen = A.shape[1]
NullDim = Bmat.shape[1]
# Get spectral radius of Dinv*A, this will be used to scale the time step
# size for the ODE
rho_DinvA = approximate_spectral_radius(Dinv_A)
# Calculate D_A for later use in the minimization problem
if proj_type == "D_A":
D_A = sparse.spdiags([D], [0], dimen, dimen, format='csr')
else:
D_A = sparse.eye(dimen, dimen, format="csr", dtype=A.dtype)
# Calculate (I - delta_t Dinv A)^k
# In order to later access columns, we calculate the transpose in
# CSR format so that columns will be accessed efficiently
# Calculate the number of time steps that can be done by squaring, and
# the number of time steps that must be done incrementally
nsquare = int(np.log2(k))
ninc = k - 2**nsquare
# Calculate one time step
I = sparse.eye(dimen, dimen, format="csr", dtype=A.dtype)
Atilde = (I - (1.0/rho_DinvA)*Dinv_A)
Atilde = Atilde.T.tocsr()
# Construct a sparsity mask for Atilde that will restrict Atilde^T to the
# nonzero pattern of A, with the added constraint that row i of Atilde^T
# retains only the nonzeros that are also in the same PDE as i.
mask = A.copy()
# Restrict to same PDE
if numPDEs > 1:
row_length = np.diff(mask.indptr)
my_pde = np.mod(range(dimen), numPDEs)
my_pde = np.repeat(my_pde, row_length)
mask.data[np.mod(mask.indices, numPDEs) != my_pde] = 0.0
del row_length, my_pde
mask.eliminate_zeros()
# If the total number of time steps is a power of two, then there is
# a very efficient computational short-cut. Otherwise, we support
# other numbers of time steps, through an inefficient algorithm.
if ninc > 0:
warn("The most efficient time stepping for the Evolution Strength\
Method is done in powers of two.\nYou have chosen " + str(k) +
" time steps.")
# Calculate (Atilde^nsquare)^T = (Atilde^T)^nsquare
for i in range(nsquare):
Atilde = Atilde*Atilde
JacobiStep = (I - (1.0/rho_DinvA)*Dinv_A).T.tocsr()
for i in range(ninc):
Atilde = Atilde*JacobiStep
del JacobiStep
# Apply mask to Atilde, zeros in mask have already been eliminated at
# start of routine.
mask.data[:] = 1.0
Atilde = Atilde.multiply(mask)
Atilde.eliminate_zeros()
Atilde.sort_indices()
elif nsquare == 0:
if numPDEs > 1:
# Apply mask to Atilde, zeros in mask have already been eliminated
# at start of routine.
mask.data[:] = 1.0
Atilde = Atilde.multiply(mask)
Atilde.eliminate_zeros()
Atilde.sort_indices()
else:
# Use computational short-cut for case (ninc == 0) and (nsquare > 0)
# Calculate Atilde^k only at the sparsity pattern of mask.
for i in range(nsquare-1):
Atilde = Atilde*Atilde
# Call incomplete mat-mat mult
AtildeCSC = Atilde.tocsc()
AtildeCSC.sort_indices()
mask.sort_indices()
Atilde.sort_indices()
amg_core.incomplete_mat_mult_csr(Atilde.indptr, Atilde.indices,
Atilde.data, AtildeCSC.indptr,
AtildeCSC.indices, AtildeCSC.data,
mask.indptr, mask.indices, mask.data,
dimen)
del AtildeCSC, Atilde
Atilde = mask
Atilde.eliminate_zeros()
Atilde.sort_indices()
del Dinv, Dinv_A, mask
# Calculate strength based on constrained min problem of
# min( z - B*x ), such that
# (B*x)|_i = z|_i, i.e. they are equal at point i
# z = (I - (t/k) Dinv A)^k delta_i
#
# Strength is defined as the relative point-wise approx. error between
# B*x and z. We don't use the full z in this problem, only that part of
# z that is in the sparsity pattern of A.
#
# Can use either the D-norm, and inner product, or l2-norm and inner-prod
# to solve the constrained min problem. Using D gives scale invariance.
#
# This is a quadratic minimization problem with a linear constraint, so
# we can build a linear system and solve it to find the critical point,
# i.e. minimum.
#
# We exploit a known shortcut for the case of NullDim = 1. The shortcut is
# mathematically equivalent to the longer constrained min. problem
if NullDim == 1:
# Use shortcut to solve constrained min problem if B is only a vector
# Strength(i,j) = | 1 - (z(i)/b(j))/(z(j)/b(i)) |
# These ratios can be calculated by diagonal row and column scalings
# Create necessary vectors for scaling Atilde
# Its not clear what to do where B == 0. This is an
# an easy programming solution, that may make sense.
Bmat_forscaling = np.ravel(Bmat)
Bmat_forscaling[Bmat_forscaling == 0] = 1.0
DAtilde = Atilde.diagonal()
DAtildeDivB = np.ravel(DAtilde) / Bmat_forscaling
# Calculate best approximation, z_tilde, in span(B)
# Importantly, scale_rows and scale_columns leave zero entries
# in the matrix. For previous implementations this was useful
# because we assume data and Atilde.data are the same length below
data = Atilde.data.copy()
Atilde.data[:] = 1.0
Atilde = scale_rows(Atilde, DAtildeDivB)
Atilde = scale_columns(Atilde, np.ravel(Bmat_forscaling))
# If angle in the complex plane between z and z_tilde is
# greater than 90 degrees, then weak. We can just look at the
# dot product to determine if angle is greater than 90 degrees.
angle = np.real(Atilde.data) * np.real(data) +\
np.imag(Atilde.data) * np.imag(data)
angle = angle < 0.0
angle = np.array(angle, dtype=bool)
# Calculate Approximation ratio
Atilde.data = Atilde.data/data
# If approximation ratio is less than tol, then weak connection
weak_ratio = (np.abs(Atilde.data) < 1e-4)
# Calculate Approximation error
Atilde.data = abs(1.0 - Atilde.data)
# Set small ratios and large angles to weak
Atilde.data[weak_ratio] = 0.0
Atilde.data[angle] = 0.0
# Set near perfect connections to 1e-4
Atilde.eliminate_zeros()
Atilde.data[Atilde.data < np.sqrt(np.finfo(float).eps)] = 1e-4
del data, weak_ratio, angle
else:
# For use in computing local B_i^H*B, precompute the element-wise
# multiply of each column of B with each other column. We also scale
# by 2.0 to account for BDB's eventual use in a constrained
# minimization problem
BDBCols = int(np.sum(range(NullDim + 1)))
BDB = np.zeros((dimen, BDBCols), dtype=A.dtype)
counter = 0
for i in range(NullDim):
for j in range(i, NullDim):
BDB[:, counter] = 2.0 *\
(np.conjugate(np.ravel(np.asarray(B[:, i]))) *
np.ravel(np.asarray(D_A * B[:, j])))
counter = counter + 1
# Choose tolerance for dropping "numerically zero" values later
t = Atilde.dtype.char
eps = np.finfo(np.float).eps
feps = np.finfo(np.single).eps
geps = np.finfo(np.longfloat).eps
_array_precision = {'f': 0, 'd': 1, 'g': 2, 'F': 0, 'D': 1, 'G': 2}
tol = {0: feps*1e3, 1: eps*1e6, 2: geps*1e6}[_array_precision[t]]
# Use constrained min problem to define strength
amg_core.evolution_strength_helper(Atilde.data,
Atilde.indptr,
Atilde.indices,
Atilde.shape[0],
np.ravel(np.asarray(B)),
np.ravel(np.asarray(
(D_A * np.conjugate(B)).T)),
np.ravel(np.asarray(BDB)),
BDBCols, NullDim, tol)
Atilde.eliminate_zeros()
# All of the strength values are real by this point, so ditch the complex
# part
Atilde.data = np.array(np.real(Atilde.data), dtype=float)
# Apply drop tolerance
if symmetrize_measure:
Atilde = 0.5*(Atilde + Atilde.T)
if epsilon != np.inf:
amg_core.apply_distance_filter(dimen, epsilon, Atilde.indptr,
Atilde.indices, Atilde.data)
Atilde.eliminate_zeros()
# Set diagonal to 1.0, as each point is strongly connected to itself.
I = sparse.eye(dimen, dimen, format="csr")
I.data -= Atilde.diagonal()
Atilde = Atilde + I
# If converted BSR to CSR, convert back and return amalgamated matrix,
# i.e. the sparsity structure of the blocks of Atilde
if not csrflag:
Atilde = Atilde.tobsr(blocksize=(numPDEs, numPDEs))
n_blocks = Atilde.indices.shape[0]
blocksize = Atilde.blocksize[0]*Atilde.blocksize[1]
CSRdata = np.zeros((n_blocks,))
amg_core.min_blocks(n_blocks, blocksize,
np.ravel(np.asarray(Atilde.data)), CSRdata)
# Atilde = sparse.csr_matrix((data, row, col), shape=(*,*))
Atilde = sparse.csr_matrix((CSRdata, Atilde.indices, Atilde.indptr),
shape=(Atilde.shape[0] / numPDEs,
Atilde.shape[1] / numPDEs))
# Standardized strength values require small values be weak and large
# values be strong. So, we invert the algebraic distances computed here
Atilde.data = 1.0/Atilde.data
# Scale C by the largest magnitude entry in each row
Atilde = scale_rows_by_largest_entry(Atilde)
return Atilde
def setup_FC_block_jacobi(lvl,
F_iterations=DEFAULT_NITER,
C_iterations=DEFAULT_NITER,
iterations=DEFAULT_NITER,
omega=1.0,
Dinv=None,
blocksize=None,
withrho=False):
# Determine Blocksize
if blocksize is None and Dinv is None:
if sp.sparse.isspmatrix_csr(lvl.A):
blocksize = 1
elif sp.sparse.isspmatrix_bsr(lvl.A):
blocksize = lvl.A.blocksize[0]
elif blocksize is None:
if sp.sparse.isspmatrix_bsr(Dinv):
blocksize = Dinv.blocksize[1]
else:
blocksize = 1
# Check for compatible dimensions
if (lvl.A.shape[0] % blocksize) != 0:
raise ValueError("Blocksize does not divide size of matrix.")
elif (len(lvl.splitting) * blocksize != lvl.A.shape[0]):
raise ValueError(
"Blocksize not compatible with CF-splitting and matrix size.")
if blocksize == 1:
# Block Jacobi is equivalent to normal Jacobi
return setup_FC_jacobi(lvl,
iterations=iterations,
omega=omega,
withrho=withrho)
else:
# Get C-points and F-points from splitting
try:
Fpts = np.array(np.where(lvl.splitting == 0)[0], dtype='int32')
Cpts = np.array(np.where(lvl.splitting == 1)[0], dtype='int32')
lvl.nf = len(Fpts)
lvl.nc = len(Cpts)
except:
raise ValueError(
"CF-splitting array needs to be stored in hierarchy.")
# Use Block Jacobi
if Dinv is None:
Dinv = get_block_diag(lvl.A, blocksize=blocksize, inv_flag=True)
if withrho:
omega = omega / rho_block_D_inv_A(lvl.A, Dinv)
def smoother(A, x, b):
relaxation.FC_block_jacobi(A,
x,
b,
Cpts=Cpts,
Fpts=Fpts,
iterations=iterations,
F_iterations=F_iterations,
C_iterations=C_iterations,
omega=omega,
Dinv=Dinv,
blocksize=blocksize)
return smoother
def gmres_prolongation_smoothing(A, T, B, BtBinv, Sparsity_Pattern, maxiter, tol, weighting='local', Cpt_params=None):
'''
Helper function for energy_prolongation_smoother(...).
Use GMRES to smooth T by solving A T = 0, subject to nullspace
and sparsity constraints.
Parameters
----------
A : {csr_matrix, bsr_matrix}
SPD sparse NxN matrix
Should be at least nonsymmetric or indefinite
T : {bsr_matrix}
Tentative prolongator, a NxM sparse matrix (M < N).
This is initial guess for the equation A T = 0.
Assumed that T B_c = B_f
B : {array}
Near-nullspace modes for coarse grid, i.e., B_c.
Has shape (M,k) where k is the number of coarse candidate vectors.
BtBinv : {array}
3 dimensional array such that,
BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted
to the neighborhood (in the matrix graph) of dof of i.
Sparsity_Pattern : {csr_matrix, bsr_matrix}
Sparse NxM matrix
This is the sparsity pattern constraint to enforce on the
eventual prolongator
maxiter : int
maximum number of iterations
tol : float
residual tolerance for A T = 0
weighting : {string}
'block', 'diagonal' or 'local' construction of the diagonal preconditioning
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
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.
Returns
-------
T : {bsr_matrix}
Smoothed prolongator using GMRES to solve A T = 0,
subject to the constraints, T B_c = B_f, and T has no nonzero
outside of the sparsity pattern in Sparsity_Pattern.
See Also
--------
The principal calling routine,
pyamg.aggregation.smooth.energy_prolongation_smoother
'''
#For non-SPD system, apply GMRES with Diagonal Preconditioning
# Preallocate space for new search directions
AV = bsr_matrix((numpy.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype), Sparsity_Pattern.indices, Sparsity_Pattern.indptr), shape=(Sparsity_Pattern.shape) )
# Preallocate for Givens Rotations, Hessenberg matrix and Krylov Space
xtype = scipy.sparse.sputils.upcast(A.dtype, T.dtype, B.dtype)
Q = [] # Givens Rotations
V = [] # Krylov Space
vs = [] # vs store the pointers to each column of V.
# This saves a considerable amount of time.
H = numpy.zeros( (maxiter+1, maxiter+1), dtype=xtype) # Upper Hessenberg matrix, which is then
# converted to upper tri with Givens Rots
# GMRES will be run with diagonal preconditioning
if weighting == 'diagonal':
Dinv = get_diagonal(A, norm_eq=False, inv=True)
elif weighting == 'block':
Dinv = get_block_diag(A, blocksize=A.blocksize[0], inv_flag=True)
Dinv = bsr_matrix( (Dinv, numpy.arange(Dinv.shape[0]), numpy.arange(Dinv.shape[0]+1)), shape = A.shape)
elif weighting == 'local':
# Based on Gershgorin estimate
D = numpy.abs(A)*numpy.ones((A.shape[0],1), dtype=A.dtype)
Dinv = numpy.zeros_like(D)
Dinv[D != 0] = 1.0 / numpy.abs(D[D != 0])
else:
raise ValueError('weighting value is invalid')
# Calculate initial residual
# Equivalent to R = -A*T; R = R.multiply(Sparsity_Pattern)
# with the added constraint that R has an explicit 0 wherever
# R is 0 and Sparsity_Pattern is not
R = bsr_matrix((numpy.zeros(Sparsity_Pattern.data.shape, dtype=T.dtype),
Sparsity_Pattern.indices, Sparsity_Pattern.indptr),
shape=(Sparsity_Pattern.shape) )
pyamg.amg_core.incomplete_mat_mult_bsr(A.indptr, A.indices, numpy.ravel(A.data),
T.indptr, T.indices, numpy.ravel(T.data),
R.indptr, R.indices, numpy.ravel(R.data),
T.shape[0]/T.blocksize[0], T.shape[1]/T.blocksize[1],
A.blocksize[0], A.blocksize[1], T.blocksize[1])
R.data *= -1.0
#Apply diagonal preconditioner
if weighting == 'local' or weighting == 'diagonal':
R = scale_rows(R, Dinv)
else:
R = Dinv*R
# Enforce R*B = 0
Satisfy_Constraints(R, B, BtBinv)
if R.nnz == 0:
print "Error in sa_energy_min(..). Initial R no nonzeros on a level. Returning tentative prolongator\n"
return T
# This is the RHS vector for the problem in the Krylov Space
normr = numpy.sqrt((R.data.conjugate()*R.data).sum())
g = numpy.zeros((maxiter+1,), dtype=xtype)
g[0] = normr
# First Krylov vector
# V[0] = r/normr
if normr > 0.0:
V.append((1.0/normr)*R)
#print "Energy Minimization of Prolongator --- Iteration 0 --- r = " + str(normr)
i = -1
#vect = numpy.ravel((A*T).data)
#print "Iteration " + str(i+1) + " Energy = %1.3e"%numpy.sqrt( (vect.conjugate()*vect).sum() )
#print "Iteration " + str(i+1) + " Normr %1.3e"%normr
while i < maxiter-1 and normr > tol:
i = i+1
# Calculate new search direction
# Equivalent to: AV = A*V; AV = AV.multiply(Sparsity_Pattern)
# with the added constraint that explicit zeros are in AP wherever
# AP = 0 and Sparsity_Pattern does not
AV.data[:] = 0.0
pyamg.amg_core.incomplete_mat_mult_bsr(A.indptr, A.indices, numpy.ravel(A.data),
V[i].indptr, V[i].indices, numpy.ravel(V[i].data),
AV.indptr, AV.indices, numpy.ravel(AV.data),
T.shape[0]/T.blocksize[0], T.shape[1]/T.blocksize[1],
A.blocksize[0], A.blocksize[1], T.blocksize[1])
if weighting == 'local' or weighting == 'diagonal':
AV = scale_rows(AV, Dinv)
else:
AV = Dinv*AV
# Enforce AV*B = 0
Satisfy_Constraints(AV, B, BtBinv)
V.append(AV.copy())
# Modified Gram-Schmidt
for j in xrange(i+1):
# Frobenius inner-product
H[j,i] = (V[j].conjugate().multiply(V[i+1])).sum()
V[i+1] = V[i+1] - H[j,i]*V[j]
# Frobenius Norm
H[i+1,i] = numpy.sqrt( (V[i+1].data.conjugate()*V[i+1].data).sum() )
# Check for breakdown
if H[i+1,i] != 0.0:
V[i+1] = (1.0/H[i+1,i])*V[i+1]
# Apply previous Givens rotations to H
if i > 0:
apply_givens(Q, H[:,i], i)
# Calculate and apply next complex-valued Givens Rotation
if H[i+1, i] != 0:
h1 = H[i, i];
h2 = H[i+1, i];
h1_mag = numpy.abs(h1)
h2_mag = numpy.abs(h2)
if h1_mag < h2_mag:
mu = h1/h2
tau = numpy.conjugate(mu)/numpy.abs(mu)
else:
mu = h2/h1
tau = mu/numpy.abs(mu)
denom = numpy.sqrt( h1_mag**2 + h2_mag**2 )
c = h1_mag/denom
s = h2_mag*tau/denom;
Qblock = numpy.array([[c, numpy.conjugate(s)], [-s, c]], dtype=xtype)
Q.append(Qblock)
# Apply Givens Rotation to g,
# the RHS for the linear system in the Krylov Subspace.
g[i:i+2] = scipy.dot(Qblock, g[i:i+2])
# Apply effect of Givens Rotation to H
H[i, i] = scipy.dot(Qblock[0,:], H[i:i+2, i])
H[i+1, i] = 0.0
normr = numpy.abs(g[i+1])
#print "Iteration " + str(i+1) + " Normr %1.3e"%normr
# End while loop
# Find best update to x in Krylov Space, V. Solve (i x i) system.
if i != -1:
y = scipy.linalg.solve(H[0:i+1,0:i+1], g[0:i+1])
for j in range(i+1):
T = T + y[j]*V[j]
#vect = numpy.ravel((A*T).data)
#print "Final Iteration " + str(i) + " Energy = %1.3e"%numpy.sqrt( (vect.conjugate()*vect).sum() )
# Ensure identity at C-pts
if Cpt_params[0]:
T = Cpt_params[1]['I_F']*T + Cpt_params[1]['P_I']
return T
def block_jacobi(A, x, b, Dinv=None, blocksize=1, iterations=1, omega=1.0):
"""Perform block Jacobi iteration on the linear system Ax=b
Parameters
----------
A : csr_matrix or bsr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
iterations : int
Number of iterations to perform
omega : scalar
Damping parameter
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use block Jacobi as a Stand-Alone Solver
>>> from pyamg.relaxation import *
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((10,10), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> block_jacobi(A, x0, b, blocksize=4, iterations=10, omega=1.0)
>>> print norm(b-A*x0)
4.66474230129
>>> #
>>> ## Use block Jacobi as the Multigrid Smoother
>>> from pyamg import smoothed_aggregation_solver
>>> sa = smoothed_aggregation_solver(A, B=numpy.ones((A.shape[0],1)),
... coarse_solver='pinv2', max_coarse=50,
... presmoother=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4}),
... postsmoother=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4}))
>>> x0=numpy.zeros((A.shape[0],1))
>>> residuals=[]
>>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals)
"""
A, x, b = make_system(A, x, b, formats=['csr', 'bsr'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv == None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != A.shape[0] / blocksize:
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
sweep = slice(None)
(row_start, row_stop, row_step) = sweep.indices(A.shape[0] / blocksize)
if (row_stop - row_start) * row_step <= 0: #no work to do
return
temp = numpy.empty_like(x)
# Create uniform type, and convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
for iter in xrange(iterations):
amg_core.block_jacobi(A.indptr, A.indices, numpy.ravel(A.data), x, b,
numpy.ravel(Dinv), temp, row_start, row_stop,
row_step, omega, blocksize)
def block_jacobi(A, x, b, Dinv=None, blocksize=1, iterations=1, omega=1.0):
"""Perform block Jacobi iteration on the linear system Ax=b
Parameters
----------
A : csr_matrix or bsr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
iterations : int
Number of iterations to perform
omega : scalar
Damping parameter
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use block Jacobi as a Stand-Alone Solver
>>> from pyamg.relaxation import *
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((10,10), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> block_jacobi(A, x0, b, blocksize=4, iterations=10, omega=1.0)
>>> print norm(b-A*x0)
4.66474230129
>>> #
>>> ## Use block Jacobi as the Multigrid Smoother
>>> from pyamg import smoothed_aggregation_solver
>>> sa = smoothed_aggregation_solver(A, B=numpy.ones((A.shape[0],1)),
... coarse_solver='pinv2', max_coarse=50,
... presmoother=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4}),
... postsmoother=('block_jacobi', {'omega': 4.0/3.0, 'iterations' : 2, 'blocksize' : 4}))
>>> x0=numpy.zeros((A.shape[0],1))
>>> residuals=[]
>>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals)
"""
A,x,b = make_system(A, x, b, formats=['csr', 'bsr'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv == None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != A.shape[0]/blocksize:
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
sweep = slice(None)
(row_start,row_stop,row_step) = sweep.indices(A.shape[0]/blocksize)
if (row_stop - row_start) * row_step <= 0: #no work to do
return
temp = numpy.empty_like(x)
# Create uniform type, and convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
for iter in xrange(iterations):
amg_core.block_jacobi(A.indptr, A.indices, numpy.ravel(A.data),
x, b, numpy.ravel(Dinv), temp,
row_start, row_stop, row_step,
omega, blocksize)
def block_gauss_seidel(A,
x,
b,
iterations=1,
sweep='forward',
blocksize=1,
Dinv=None):
"""Perform block Gauss-Seidel iteration on the linear system Ax=b
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
iterations : int
Number of iterations to perform
sweep : {'forward','backward','symmetric'}
Direction of sweep
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
Returns
-------
Nothing, x will be modified in place.
Examples
--------
>>> ## Use Gauss-Seidel as a Stand-Alone Solver
>>> from pyamg.relaxation import *
>>> from pyamg.gallery import poisson
>>> from pyamg.util.linalg import norm
>>> import numpy
>>> A = poisson((10,10), format='csr')
>>> x0 = numpy.zeros((A.shape[0],1))
>>> b = numpy.ones((A.shape[0],1))
>>> block_gauss_seidel(A, x0, b, iterations=10, blocksize=4, sweep='symmetric')
>>> print norm(b-A*x0)
0.958333817624
>>> #
>>> ## Use Gauss-Seidel as the Multigrid Smoother
>>> from pyamg import smoothed_aggregation_solver
>>> sa = smoothed_aggregation_solver(A, B=numpy.ones((A.shape[0],1)),
... coarse_solver='pinv2', max_coarse=50,
... presmoother=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}),
... postsmoother=('block_gauss_seidel', {'sweep':'symmetric', 'blocksize' : 4}))
>>> x0=numpy.zeros((A.shape[0],1))
>>> residuals=[]
>>> x = sa.solve(b, x0=x0, tol=1e-8, residuals=residuals)
"""
A, x, b = make_system(A, x, b, formats=['csr', 'bsr'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv == None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != A.shape[0] / blocksize:
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
if sweep == 'forward':
row_start, row_stop, row_step = 0, len(x) / blocksize, 1
elif sweep == 'backward':
row_start, row_stop, row_step = len(x) / blocksize - 1, -1, -1
elif sweep == 'symmetric':
for iter in xrange(iterations):
block_gauss_seidel(A,
x,
b,
iterations=1,
sweep='forward',
blocksize=blocksize,
Dinv=Dinv)
block_gauss_seidel(A,
x,
b,
iterations=1,
sweep='backward',
blocksize=blocksize,
Dinv=Dinv)
return
else:
raise ValueError(
"valid sweep directions are 'forward', 'backward', and 'symmetric'"
)
for iter in xrange(iterations):
amg_core.block_gauss_seidel(A.indptr,
A.indices, numpy.ravel(A.data), x, b,
numpy.ravel(Dinv), row_start, row_stop,
row_step, blocksize)
def evolution_strength_of_connection(A,
B=None,
epsilon=4.0,
k=2,
proj_type="l2",
weighting='diagonal',
symmetrize_measure=True,
cost=[0]):
"""
Construct strength of connection matrix using an Evolution-based measure
Parameters
----------
A : {csr_matrix, bsr_matrix}
Sparse NxN matrix
B : {string, array}
If B=None, then the near nullspace vector used is all ones. If B is
an (NxK) array, then B is taken to be the near nullspace vectors.
epsilon : scalar
Drop tolerance
k : integer
ODE num time steps, step size is assumed to be 1/rho(DinvA)
proj_type : {'l2','D_A'}
Define norm for constrained min prob, i.e. define projection
weighting : {string}
'block', 'diagonal' or 'local' construction of the D-inverse
used to precondition A before "evolving" delta-functions. The
local option is the cheapest.
Returns
-------
Atilde : {csr_matrix}
Sparse matrix of strength values
References
----------
.. [1] Olson, L. N., Schroder, J., Tuminaro, R. S.,
"A New Perspective on Strength Measures in Algebraic Multigrid",
submitted, June, 2008.
Examples
--------
>>> import numpy as np
>>> from pyamg.gallery import stencil_grid
>>> from pyamg.strength import evolution_strength_of_connection
>>> n=3
>>> stencil = np.array([[-1.0,-1.0,-1.0],
... [-1.0, 8.0,-1.0],
... [-1.0,-1.0,-1.0]])
>>> A = stencil_grid(stencil, (n,n), format='csr')
>>> S = evolution_strength_of_connection(A, np.ones((A.shape[0],1)))
"""
# local imports for evolution_strength_of_connection
from pyamg.util.utils import scale_rows, get_block_diag, scale_columns
from pyamg.util.linalg import approximate_spectral_radius
# ====================================================================
# Check inputs
if epsilon < 1.0:
raise ValueError("expected epsilon > 1.0")
if k <= 0:
raise ValueError("number of time steps must be > 0")
if proj_type not in ['l2', 'D_A']:
raise ValueError("proj_type must be 'l2' or 'D_A'")
if (not sparse.isspmatrix_csr(A)) and (not sparse.isspmatrix_bsr(A)):
raise TypeError("expected csr_matrix or bsr_matrix")
# ====================================================================
# Format A and B correctly.
# B must be in mat format, this isn't a deep copy
if B is None:
Bmat = np.mat(np.ones((A.shape[0], 1), dtype=A.dtype))
else:
Bmat = np.mat(B)
# Is matrix A CSR?
if (not sparse.isspmatrix_csr(A)):
numPDEs = A.blocksize[0]
csrflag = False
else:
numPDEs = 1
csrflag = True
# Pre-process A. We need A in CSR, to be devoid of explicit 0's, have
# sorted indices and be scaled by D-inverse
if weighting == 'block':
Dinv = get_block_diag(A, blocksize=numPDEs, inv_flag=True)
Dinv = sparse.bsr_matrix(
(Dinv, np.arange(Dinv.shape[0]), np.arange(Dinv.shape[0] + 1)),
shape=A.shape)
Dinv_A = (Dinv * A).tocsr()
cost[0] += 1
elif weighting == 'diagonal':
D = A.diagonal()
Dinv = get_diagonal(A, norm_eq=False, inv=True)
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
cost[0] += 1
elif weighting == 'local':
D = np.abs(A) * np.ones((A.shape[0], 1), dtype=A.dtype)
Dinv = np.zeros_like(D)
Dinv[D != 0] = 1.0 / np.abs(D[D != 0])
Dinv[D == 0] = 1.0
Dinv_A = scale_rows(A, Dinv, copy=True)
cost[0] += 1
else:
raise ValueError('Unrecognized weighting for Evolution measure')
A = A.tocsr()
A.eliminate_zeros()
A.sort_indices()
# Handle preliminaries for the algorithm
dimen = A.shape[1]
NullDim = Bmat.shape[1]
if weighting == 'diagonal' or weighting == 'block':
# Get spectral radius of Dinv*A, scales the time step size for the ODE
rho_DinvA = approximate_spectral_radius(Dinv_A)
cost[0] += 15 # 15 lanczos iterations to approximate spectral radius
else:
# Using local weighting, no need for spectral radius
rho_DinvA = 1.0
# Calculate D_A for later use in the minimization problem
if proj_type == "D_A":
D = A.diagonal()
D_A = sparse.spdiags([D], [0], dimen, dimen, format='csr')
else:
D_A = sparse.eye(dimen, dimen, format="csr", dtype=A.dtype)
# Calculate (I - delta_t Dinv A)^k
# We transpose the product, so that we can efficiently access
# the columns in CSR format. We want the columns (not rows) because
# strength is based on the columns of (I - delta_t Dinv A)^k, i.e.,
# relaxed delta functions
# Calculate the number of time steps that can be done by squaring, and
# the number of time steps that must be done incrementally
nsquare = int(np.log2(k))
ninc = k - 2**nsquare
# Calculate one time step
I = sparse.eye(dimen, dimen, format="csr", dtype=A.dtype)
Atilde = (I - (1.0 / rho_DinvA) * Dinv_A)
Atilde = Atilde.T.tocsr()
cost[0] += 1
# Construct a sparsity mask for Atilde that will restrict Atilde^T to the
# nonzero pattern of A, with the added constraint that row i of Atilde^T
# retains only the nonzeros that are also in the same PDE as i.
mask = A.copy()
# Restrict to same PDE
if numPDEs > 1:
row_length = np.diff(mask.indptr)
my_pde = np.mod(np.arange(dimen), numPDEs)
my_pde = np.repeat(my_pde, row_length)
mask.data[np.mod(mask.indices, numPDEs) != my_pde] = 0.0
del row_length, my_pde
mask.eliminate_zeros()
# If the total number of time steps is a power of two, then there is
# a very efficient computational short-cut. Otherwise, we support
# other numbers of time steps, through an inefficient algorithm.
if ninc > 0:
warn("The most efficient time stepping for the Evolution Strength\
Method is done in powers of two.\nYou have chosen " + str(k) +
" time steps.")
JacobiStep = csr_matrix(Atilde, copy=True)
# Calculate (Atilde^nsquare)^T = (Atilde^T)^nsquare
for i in range(nsquare):
cost[0] += mat_mat_complexity(Atilde, Atilde)
Atilde = Atilde * Atilde
for i in range(ninc):
cost[0] += mat_mat_complexity(Atilde, JacobiStep)
Atilde = Atilde * JacobiStep
del JacobiStep
# Apply mask to Atilde, zeros in mask have already been eliminated at
# start of routine.
mask.data[:] = 1.0
Atilde = Atilde.multiply(mask)
Atilde.eliminate_zeros()
Atilde.sort_indices()
cost[0] += Atilde.nnz / float(A.nnz)
elif nsquare == 0:
if numPDEs > 1:
# Apply mask to Atilde, zeros in mask have already been eliminated
# at start of routine.
mask.data[:] = 1.0
Atilde = Atilde.multiply(mask)
Atilde.eliminate_zeros()
Atilde.sort_indices()
else:
# Use computational short-cut for case (ninc == 0) and (nsquare > 0)
# Calculate Atilde^k only at the sparsity pattern of mask.
for i in range(nsquare - 1):
cost[0] += mat_mat_complexity(Atilde, Atilde)
Atilde = Atilde * Atilde
# Call incomplete mat-mat mult
AtildeCSC = Atilde.tocsc()
AtildeCSC.sort_indices()
mask.sort_indices()
Atilde.sort_indices()
amg_core.incomplete_mat_mult_csr(Atilde.indptr, Atilde.indices,
Atilde.data, AtildeCSC.indptr,
AtildeCSC.indices, AtildeCSC.data,
mask.indptr, mask.indices, mask.data,
dimen)
cost[0] += mat_mat_complexity(Atilde, mask, incomplete=True) / float(
A.nnz)
del AtildeCSC, Atilde
Atilde = mask
Atilde.eliminate_zeros()
Atilde.sort_indices()
del Dinv, Dinv_A, mask
# Calculate strength based on constrained min problem of
# min( z - B*x ), such that
# (B*x)|_i = z|_i, i.e. they are equal at point i
# z = (I - (t/k) Dinv A)^k delta_i
#
# Strength is defined as the relative point-wise approx. error between
# B*x and z. We don't use the full z in this problem, only that part of
# z that is in the sparsity pattern of A.
#
# Can use either the D-norm, and inner product, or l2-norm and inner-prod
# to solve the constrained min problem. Using D gives scale invariance.
#
# This is a quadratic minimization problem with a linear constraint, so
# we can build a linear system and solve it to find the critical point,
# i.e. minimum.
#
# We exploit a known shortcut for the case of NullDim = 1. The shortcut is
# mathematically equivalent to the longer constrained min. problem
if NullDim == 1:
# Use shortcut to solve constrained min problem if B is only a vector
# Strength(i,j) = | 1 - (z(i)/b(j))/(z(j)/b(i)) |
# These ratios can be calculated by diagonal row and column scalings
# Create necessary vectors for scaling Atilde
# Its not clear what to do where B == 0. This is an
# an easy programming solution, that may make sense.
Bmat_forscaling = np.ravel(Bmat)
Bmat_forscaling[Bmat_forscaling == 0] = 1.0
DAtilde = Atilde.diagonal()
DAtildeDivB = np.ravel(DAtilde) / Bmat_forscaling
cost[0] += Atilde.shape[0] / float(A.nnz)
# Calculate best approximation, z_tilde, in span(B)
# Importantly, scale_rows and scale_columns leave zero entries
# in the matrix. For previous implementations this was useful
# because we assume data and Atilde.data are the same length below
data = Atilde.data.copy()
Atilde.data[:] = 1.0
Atilde = scale_rows(Atilde, DAtildeDivB)
Atilde = scale_columns(Atilde, np.ravel(Bmat_forscaling))
cost[0] += 2.0 * Atilde.nnz / float(A.nnz)
# If angle in the complex plane between z and z_tilde is
# greater than 90 degrees, then weak. We can just look at the
# dot product to determine if angle is greater than 90 degrees.
angle = np.real(Atilde.data) * np.real(data) +\
np.imag(Atilde.data) * np.imag(data)
angle = angle < 0.0
angle = np.array(angle, dtype=bool)
cost[0] += Atilde.nnz / float(A.nnz)
if Atilde.dtype is 'complex':
cost[0] += Atilde.nnz / float(A.nnz)
# Calculate Approximation ratio
Atilde.data = Atilde.data / data
cost[0] += Atilde.nnz / float(A.nnz)
# If approximation ratio is less than tol, then weak connection
weak_ratio = (np.abs(Atilde.data) < 1e-4)
# Calculate Approximation error
Atilde.data = abs(1.0 - Atilde.data)
cost[0] += Atilde.nnz / float(A.nnz)
# Set small ratios and large angles to weak
Atilde.data[weak_ratio] = 0.0
Atilde.data[angle] = 0.0
# Set near perfect connections to 1e-4
Atilde.eliminate_zeros()
Atilde.data[Atilde.data < np.sqrt(np.finfo(float).eps)] = 1e-4
del data, weak_ratio, angle
else:
# For use in computing local B_i^H*B, precompute the element-wise
# multiply of each column of B with each other column. We also scale
# by 2.0 to account for BDB's eventual use in a constrained
# minimization problem
BDBCols = int(np.sum(np.arange(NullDim + 1)))
BDB = np.zeros((dimen, BDBCols), dtype=A.dtype)
counter = 0
for i in range(NullDim):
for j in range(i, NullDim):
BDB[:, counter] = 2.0 *\
(np.conjugate(np.ravel(np.asarray(B[:, i]))) *
np.ravel(np.asarray(D_A * B[:, j])))
counter = counter + 1
cost[0] += B.shape[0] / float(A.nnz)
# Choose tolerance for dropping "numerically zero" values later
t = Atilde.dtype.char
eps = np.finfo(np.float).eps
feps = np.finfo(np.single).eps
geps = np.finfo(np.longfloat).eps
_array_precision = {'f': 0, 'd': 1, 'g': 2, 'F': 0, 'D': 1, 'G': 2}
tol = {0: feps * 1e3, 1: eps * 1e6, 2: geps * 1e6}[_array_precision[t]]
# Use constrained min problem to define strength.
# This function is doing similar to NullDim=1 with more bad guys.
# Complexity accounts for computing the block inverse, and
# hat{z_i} = B_i*x, hat{z_i} .* hat{z_i},
# hat{z_i} = hat{z_i} / z_i, and abs(1.0 - hat{z_i}).
cost[0] += (Atilde.nnz * (3 + NullDim) +
(NullDim**3) * dimen) / float(A.nnz)
amg_core.evolution_strength_helper(
Atilde.data, Atilde.indptr, Atilde.indices, Atilde.shape[0],
np.ravel(np.asarray(B)),
np.ravel(np.asarray((D_A * np.conjugate(B)).T)),
np.ravel(np.asarray(BDB)), BDBCols, NullDim, tol)
Atilde.eliminate_zeros()
# All of the strength values are real by this point, so ditch the complex
# part
Atilde.data = np.array(np.real(Atilde.data), dtype=float)
# Apply drop tolerance
if epsilon != np.inf:
cost[0] += Atilde.nnz / float(A.nnz)
amg_core.apply_distance_filter(dimen, epsilon, Atilde.indptr,
Atilde.indices, Atilde.data)
Atilde.eliminate_zeros()
# Set diagonal to 1.0, as each point is strongly connected to itself.
I = sparse.eye(dimen, dimen, format="csr")
I.data -= Atilde.diagonal()
Atilde = Atilde + I
cost[0] += Atilde.shape[0] / float(A.nnz)
# If converted BSR to CSR, convert back and return amalgamated matrix,
# i.e. the sparsity structure of the blocks of Atilde
if not csrflag:
Atilde = Atilde.tobsr(blocksize=(numPDEs, numPDEs))
n_blocks = Atilde.indices.shape[0]
blocksize = Atilde.blocksize[0] * Atilde.blocksize[1]
CSRdata = np.zeros((n_blocks, ))
amg_core.min_blocks(n_blocks, blocksize,
np.ravel(np.asarray(Atilde.data)), CSRdata)
# Atilde = sparse.csr_matrix((data, row, col), shape=(*,*))
Atilde = sparse.csr_matrix((CSRdata, Atilde.indices, Atilde.indptr),
shape=(int(Atilde.shape[0] / numPDEs),
int(Atilde.shape[1] / numPDEs)))
# Standardized strength values require small values be weak and large
# values be strong. So, we invert the algebraic distances computed here
Atilde.data = 1.0 / Atilde.data
cost[0] += Atilde.nnz / float(A.nnz)
# Scale C by the largest magnitude entry in each row
Atilde = scale_rows_by_largest_entry(Atilde)
cost[0] += Atilde.nnz / float(A.nnz)
# Symmetrize
if symmetrize_measure:
Atilde = 0.5 * (Atilde + Atilde.T)
cost[0] += Atilde.nnz / float(A.nnz)
return Atilde
def FC_block_jacobi(A, x, b, Cpts, Fpts, Dinv=None, blocksize=1, iterations=1,
F_iterations=1, C_iterations=1, omega=1.0):
"""Perform FC block Jacobi iteration on the linear system Ax=b, that is
x_f = (1-omega)x_f + omega*Dff^{-1}(b_f - Aff*xf - Afc*xc)
x_c = (1-omega)x_c + omega*Dff^{-1}(b_c - Acf*xf - Acc*xc)
where xf is x restricted to F-blocks, and Dff^{-1} the block inverse
of the block diagonal Dff, and likewise for c subscripts.
Parameters
----------
A : csr_matrix or bsr_matrix
Sparse NxN matrix
x : ndarray
Approximate solution (length N)
b : ndarray
Right-hand side (length N)
Cpts : array ints
List of C-blocks in A
Fpts : array ints
List of F-blocks in A
Dinv : array
Array holding block diagonal inverses of A
size (N/blocksize, blocksize, blocksize)
blocksize : int
Desired dimension of blocks
iterations : int
Number of iterations to perform of total FC-cycle
F_iterations : int
Number of sweeps of F-relaxation to perform
C_iterations : int
Number of sweeps of C-relaxation to perform
omega : scalar
Damping parameter
Returns
-------
Nothing, x will be modified in place.
"""
A, x, b = make_system(A, x, b, formats=['csr', 'bsr'])
A = A.tobsr(blocksize=(blocksize, blocksize))
if Dinv is None:
Dinv = get_block_diag(A, blocksize=blocksize, inv_flag=True)
elif Dinv.shape[0] != int(A.shape[0]/blocksize):
raise ValueError('Dinv and A have incompatible dimensions')
elif (Dinv.shape[1] != blocksize) or (Dinv.shape[2] != blocksize):
raise ValueError('Dinv and blocksize are incompatible')
# Create uniform type, convert possibly complex scalars to length 1 arrays
[omega] = type_prep(A.dtype, [omega])
# Perform block C-relaxation then block F-relaxation
for iter in range(iterations):
for Fiter in range(F_iterations):
amg_core.block_jacobi_indexed(A.indptr, A.indices, np.ravel(A.data),
x, b, np.ravel(Dinv), Fpts, omega,
blocksize)
for Citer in range(C_iterations):
amg_core.block_jacobi_indexed(A.indptr, A.indices, np.ravel(A.data),
x, b, np.ravel(Dinv), Cpts, omega,
blocksize)
