def relax(A, x):
fn, kwargs = unpack_arg(prepostsmoother, cost=False)
if fn == 'gauss_seidel':
gauss_seidel(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(A, x, np.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A, x, np.zeros_like(x), iterations=1,
coefficients=[1.0/approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
def relax(A, x):
fn, kwargs = unpack_arg(prepostsmoother, cost=False)
if fn == 'gauss_seidel':
gauss_seidel(A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'jacobi':
jacobi(A,
x,
np.zeros_like(x),
iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A,
x,
np.zeros_like(x),
iterations=1,
coefficients=[1.0 / approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
def initial_setup_stage(A,
symmetry,
pdef,
candidate_iters,
epsilon,
max_levels,
max_coarse,
aggregate,
prepostsmoother,
smooth,
strength,
work,
B=None):
"""
Computes a complete aggregation and the first near-nullspace candidate
following Algorithm 3 in Brezina et al.
Parameters
----------
candidate_iters
number of test relaxation iterations
epsilon
minimum acceptable relaxation convergence factor
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
# Define relaxation routine
def relax(A, x):
fn, kwargs = unpack_arg(prepostsmoother, cost=False)
if fn == 'gauss_seidel':
gauss_seidel(A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'jacobi':
jacobi(A,
x,
np.zeros_like(x),
iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A,
x,
np.zeros_like(x),
iterations=1,
coefficients=[1.0 / approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
# flag for skipping steps f-i in step 4
skip_f_to_i = True
# step 1
A_l = A
if B is None:
x = sp.rand(A_l.shape[0], 1).astype(A_l.dtype)
# The following type check matches the usual 'complex' type,
# but also numpy data types such as 'complex64', 'complex128'
# and 'complex256'.
if A_l.dtype.name.startswith('complex'):
x = x + 1.0j * sp.rand(A_l.shape[0], 1)
else:
x = np.array(B, dtype=A_l.dtype)
# step 2
relax(A_l, x)
work[:] += A_l.nnz * candidate_iters * 2
# step 3
# not advised to stop the iteration here: often the first relaxation pass
# _is_ good, but the remaining passes are poor
# if x_A_x/x_A_x_old < epsilon:
# # relaxation alone is sufficient
# print 'relaxation alone works: %g'%(x_A_x/x_A_x_old)
# return x, []
# step 4
As = [A]
xs = [x]
Ps = []
AggOps = []
StrengthOps = []
while A.shape[0] > max_coarse and max_levels > 1:
# The real check to break from the while loop is below
# Begin constructing next level
fn, kwargs = unpack_arg(strength[len(As) - 1]) # step 4b
if fn == 'symmetric':
C_l = symmetric_strength_of_connection(A_l, **kwargs)
# Diagonal must be nonzero
C_l = C_l + eye(C_l.shape[0], C_l.shape[1], format='csr')
elif fn == 'classical':
C_l = classical_strength_of_connection(A_l, **kwargs)
# Diagonal must be nonzero
C_l = C_l + eye(C_l.shape[0], C_l.shape[1], format='csr')
if isspmatrix_bsr(A_l):
C_l = amalgamate(C_l, A_l.blocksize[0])
elif (fn == 'ode') or (fn == 'evolution'):
C_l = evolution_strength_of_connection(
A_l, np.ones((A_l.shape[0], 1), dtype=A.dtype), **kwargs)
elif fn == 'predefined':
C_l = kwargs['C'].tocsr()
elif fn is None:
C_l = A_l.tocsr()
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
# In SA, strength represents "distance", so we take magnitude of
# complex values
if C_l.dtype.name.startswith('complex'):
C_l.data = np.abs(C_l.data)
# Create a unified strength framework so that large values represent
# strong connections and small values represent weak connections
if (fn == 'ode') or (fn == 'evolution') or (fn == 'energy_based'):
C_l.data = 1.0 / C_l.data
# aggregation
fn, kwargs = unpack_arg(aggregate[len(As) - 1])
if fn == 'standard':
AggOp = standard_aggregation(C_l, **kwargs)[0]
elif fn == 'lloyd':
AggOp = lloyd_aggregation(C_l, **kwargs)[0]
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
else:
raise ValueError('unrecognized aggregation method %s' % str(fn))
T_l, x = fit_candidates(AggOp, x) # step 4c
fn, kwargs = unpack_arg(smooth[len(As) - 1]) # step 4d
if fn == 'jacobi':
P_l = jacobi_prolongation_smoother(A_l, T_l, C_l, x, **kwargs)
elif fn == 'richardson':
P_l = richardson_prolongation_smoother(A_l, T_l, **kwargs)
elif fn == 'energy':
P_l = energy_prolongation_smoother(A_l, T_l, C_l, x, None,
(False, {}), **kwargs)
elif fn is None:
P_l = T_l
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# R should reflect A's structure # step 4e
if symmetry == 'symmetric':
A_l = P_l.T.asformat(P_l.format) * A_l * P_l
elif symmetry == 'hermitian':
A_l = P_l.H.asformat(P_l.format) * A_l * P_l
StrengthOps.append(C_l)
AggOps.append(AggOp)
Ps.append(P_l)
As.append(A_l)
# skip to step 5 as in step 4e
if (A_l.shape[0] <= max_coarse) or (len(AggOps) + 1 >= max_levels):
break
if not skip_f_to_i:
x_hat = x.copy() # step 4g
relax(A_l, x) # step 4h
work[:] += A_l.nnz * candidate_iters * 2
if pdef is True:
x_A_x = np.dot(np.conjugate(x).T, A_l * x)
xhat_A_xhat = np.dot(np.conjugate(x_hat).T, A_l * x_hat)
err_ratio = (x_A_x / xhat_A_xhat)**(1.0 / candidate_iters)
else:
# use A.H A inner-product
Ax = A_l * x
# Axhat = A_l * x_hat
x_A_x = np.dot(np.conjugate(Ax).T, Ax)
xhat_A_xhat = np.dot(np.conjugate(x_hat).T, A_l * x_hat)
err_ratio = (x_A_x / xhat_A_xhat)**(1.0 / candidate_iters)
if err_ratio < epsilon: # step 4i
# print "sufficient convergence, skipping"
skip_f_to_i = True
if x_A_x == 0:
x = x_hat # need to restore x
else:
# just carry out relaxation, don't check for convergence
relax(A_l, x) # step 4h
work[:] += 2 * A_l.nnz * candidate_iters
# store xs for diagnostic use and for use in step 5
xs.append(x)
# step 5
# Extend coarse-level candidate to the finest level
# --> note that we start with the x from the second coarsest level
x = xs[-1]
# make sure that xs[-1] has been relaxed by step 4h, i.e. relax(As[-2], x)
for lev in range(len(Ps) - 2, -1, -1): # lev = coarsest ... finest-1
P = Ps[lev] # I: lev --> lev+1
A = As[lev] # A on lev+1
x = P * x
relax(A, x)
work[:] += A.nnz * candidate_iters * 2
# Set predefined strength of connection and aggregation
if len(AggOps) > 1:
aggregate = [('predefined', {
'AggOp': AggOps[i]
}) for i in range(len(AggOps))]
strength = [('predefined', {
'C': StrengthOps[i]
}) for i in range(len(StrengthOps))]
return x, aggregate, strength # first candidate
def adaptive_sa_solver(A,
B=None,
symmetry='hermitian',
pdef=True,
num_candidates=1,
candidate_iters=5,
improvement_iters=0,
epsilon=0.1,
max_levels=10,
max_coarse=10,
aggregate='standard',
prepostsmoother=('gauss_seidel', {
'sweep': 'symmetric'
}),
smooth=('jacobi', {}),
strength='symmetric',
coarse_solver='pinv2',
eliminate_local=(False, {
'Ca': 1.0
}),
keep=False,
**kwargs):
"""
Create a multilevel solver using Adaptive Smoothed Aggregation (aSA)
Parameters
----------
A : {csr_matrix, bsr_matrix}
Square matrix in CSR or BSR format
B : {None, n x m dense matrix}
If a matrix, then this forms the basis for the first m candidates.
Also in this case, the initial setup stage is skipped, because this
provides the first candidate(s). If None, then a random initial guess
and relaxation are used to inform the initial candidate.
symmetry : {string}
'symmetric' refers to both real and complex symmetric
'hermitian' refers to both complex Hermitian and real Hermitian
Note that for the strictly real case, these two options are the same
Note that this flag does not denote definiteness of the operator
pdef : {bool}
True or False, whether A is known to be positive definite.
num_candidates : {integer} : default 1
Number of near-nullspace candidates to generate
candidate_iters : {integer} : default 5
Number of smoothing passes/multigrid cycles used at each level of
the adaptive setup phase
improvement_iters : {integer} : default 0
Number of times each candidate is improved
epsilon : {float} : default 0.1
Target convergence factor
max_levels : {integer} : default 10
Maximum number of levels to be used in the multilevel solver.
max_coarse : {integer} : default 500
Maximum number of variables permitted on the coarse grid.
prepostsmoother : {string or dict}
Pre- and post-smoother used in the adaptive method
strength : ['symmetric', 'classical', 'evolution',
('predefined', {'C': csr_matrix}), None]
Method used to determine the strength of connection between unknowns of
the linear system. See smoothed_aggregation_solver(...) documentation.
aggregate : ['standard', 'lloyd', 'naive',
('predefined', {'AggOp': csr_matrix})]
Method used to aggregate nodes. See smoothed_aggregation_solver(...)
documentation.
smooth : ['jacobi', 'richardson', 'energy', None]
Method used used to smooth the tentative prolongator. See
smoothed_aggregation_solver(...) documentation
coarse_solver : ['splu', 'lu', 'cholesky, 'pinv', 'gauss_seidel', ... ]
Solver used at the coarsest level of the MG hierarchy.
Optionally, may be a tuple (fn, args), where fn is a string such as
['splu', 'lu', ...] or a callable function, and args is a dictionary of
arguments to be passed to fn.
eliminate_local : {tuple}
Length 2 tuple. If the first entry is True, then eliminate candidates
where they aren't needed locally, using the second entry of the tuple
to contain arguments to local elimination routine. Given the rigid
sparse data structures, this doesn't help much, if at all, with
complexity. Its more of a diagnostic utility.
keep: {bool} : default False
Flag to indicate keeping extra operators in the hierarchy for
diagnostics. For example, if True, then strength of connection (C),
tentative prolongation (T), and aggregation (AggOp) are kept.
Returns
-------
multilevel_solver : multilevel_solver
Smoothed aggregation solver with adaptively generated candidates
Notes
-----
- Floating point value representing the "work" required to generate
the solver. This value is the total cost of just relaxation, relative
to the fine grid. The relaxation method used is assumed to symmetric
Gauss-Seidel.
- Unlike the standard Smoothed Aggregation (SA) method, adaptive SA does
not require knowledge of near-nullspace candidate vectors. Instead, an
adaptive procedure computes one or more candidates 'from scratch'. This
approach is useful when no candidates are known or the candidates have
been invalidated due to changes to matrix A.
Examples
--------
>>> from pyamg.gallery import stencil_grid
>>> from pyamg.aggregation import adaptive_sa_solver
>>> import numpy as np
>>> A=stencil_grid([[-1,-1,-1],[-1,8.0,-1],[-1,-1,-1]],\
(31,31),format='csr')
>>> [asa,work] = adaptive_sa_solver(A,num_candidates=1)
>>> residuals=[]
>>> x=asa.solve(b=np.ones((A.shape[0],)), x0=np.ones((A.shape[0],)),\
residuals=residuals)
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
if not (isspmatrix_csr(A) or isspmatrix_bsr(A)):
try:
A = csr_matrix(A)
warn("Implicit conversion of A to CSR", SparseEfficiencyWarning)
except:
raise TypeError('Argument A must have type csr_matrix or\
bsr_matrix, or be convertible to csr_matrix')
A = A.asfptype()
if A.shape[0] != A.shape[1]:
raise ValueError('expected square matrix')
# Track work in terms of relaxation
work = np.zeros((1, ))
# Levelize the user parameters, so that they become lists describing the
# desired user option on each level.
max_levels, max_coarse, strength =\
levelize_strength_or_aggregation(strength, max_levels, max_coarse)
max_levels, max_coarse, aggregate =\
levelize_strength_or_aggregation(aggregate, max_levels, max_coarse)
smooth = levelize_smooth_or_improve_candidates(smooth, max_levels)
# Develop initial candidate(s). Note that any predefined aggregation is
# preserved.
if B is None:
B, aggregate, strength =\
initial_setup_stage(A, symmetry, pdef, candidate_iters, epsilon,
max_levels, max_coarse, aggregate,
prepostsmoother, smooth, strength, work)
# Normalize B
B = (1.0 / norm(B, 'inf')) * B
num_candidates -= 1
else:
# Otherwise, use predefined candidates
num_candidates -= B.shape[1]
# Generate Aggregation and Strength Operators (the brute force way)
sa = smoothed_aggregation_solver(A,
B=B,
symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=True,
**kwargs)
if len(sa.levels) > 1:
# Set strength-of-connection and aggregation
aggregate = [('predefined', {
'AggOp': sa.levels[i].AggOp.tocsr()
}) for i in range(len(sa.levels) - 1)]
strength = [('predefined', {
'C': sa.levels[i].C.tocsr()
}) for i in range(len(sa.levels) - 1)]
# Develop additional candidates
for i in range(num_candidates):
x = general_setup_stage(
smoothed_aggregation_solver(A,
B=B,
symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=True,
**kwargs), symmetry, candidate_iters,
prepostsmoother, smooth, eliminate_local, coarse_solver, work)
# Normalize x and add to candidate list
x = x / norm(x, 'inf')
if np.isinf(x[0]) or np.isnan(x[0]):
raise ValueError('Adaptive candidate is all 0.')
B = np.hstack((B, x.reshape(-1, 1)))
# Improve candidates
if B.shape[1] > 1 and improvement_iters > 0:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
for i in range(improvement_iters):
for j in range(B.shape[1]):
# Run a V-cycle built on everything except candidate j, while
# using candidate j as the initial guess
x0 = B[:, 0]
B = B[:, 1:]
sa_temp =\
smoothed_aggregation_solver(A, B=B, symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=True, **kwargs)
x = sa_temp.solve(b,
x0=x0,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters,
cycle='V')
work[:] += 2 * sa_temp.operator_complexity() *\
sa_temp.levels[0].A.nnz * candidate_iters
# Apply local elimination
elim, elim_kwargs = unpack_arg(eliminate_local)
if elim is True:
x = x / norm(x, 'inf')
eliminate_local_candidates(x, sa_temp.levels[0].AggOp, A,
sa_temp.levels[0].T,
**elim_kwargs)
# Normalize x and add to candidate list
x = x / norm(x, 'inf')
if np.isinf(x[0]) or np.isnan(x[0]):
raise ValueError('Adaptive candidate is all 0.')
B = np.hstack((B, x.reshape(-1, 1)))
elif improvement_iters > 0:
# Special case for improving a single candidate
max_levels = len(aggregate) + 1
max_coarse = 0
for i in range(improvement_iters):
B, aggregate, strength =\
initial_setup_stage(A, symmetry, pdef, candidate_iters,
epsilon, max_levels, max_coarse,
aggregate, prepostsmoother, smooth,
strength, work, B=B)
# Normalize B
B = (1.0 / norm(B, 'inf')) * B
# Return smoother. Note, solver parameters are passed in via params
# argument to be stored in final solver hierarchy and used to estimate
# complexity.
return [
smoothed_aggregation_solver(A,
B=B,
symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=keep,
**kwargs), work[0] / A.nnz
]
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates,
diagonal_dominance=False, keep=True):
"""Service routine to implement the strength of connection, aggregation,
tentative prolongation construction, and prolongation smoothing. Called by
smoothed_aggregation_solver.
"""
A = levels[-1].A
B = levels[-1].B
if A.symmetry == "nonsymmetric":
AH = A.H.asformat(A.format)
BH = levels[-1].BH
# Compute the strength-of-connection matrix C, where larger
# C[i, j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength[len(levels)-1])
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == 'ode') or (fn == 'evolution'):
if 'B' in kwargs:
C = evolution_strength_of_connection(A, **kwargs)
else:
C = evolution_strength_of_connection(A, B, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'predefined':
C = kwargs['C'].tocsr()
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A.tocsr()
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] = kwargs['cost'][0]
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
levels[-1].complexity['diag_dom'] = kwargs['cost'][0]
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate[len(levels)-1])
if fn == 'standard':
AggOp, Cnodes = standard_aggregation(C, **kwargs)
elif fn == 'naive':
AggOp, Cnodes = naive_aggregation(C, **kwargs)
elif fn == 'lloyd':
AggOp, Cnodes = lloyd_aggregation(C, **kwargs)
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
Cnodes = kwargs['Cnodes']
else:
raise ValueError('unrecognized aggregation method %s' % str(fn))
levels[-1].complexity['aggregation'] = kwargs['cost'][0] * (float(C.nnz)/A.nnz)
# Improve near nullspace candidates by relaxing on A B = 0
temp_cost = [0.0]
fn, kwargs = unpack_arg(improve_candidates[len(levels)-1],cost=False)
if fn is not None:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
B = relaxation_as_linear_operator((fn, kwargs), A, b, temp_cost) * B
levels[-1].B = B
if A.symmetry == "nonsymmetric":
BH = relaxation_as_linear_operator((fn, kwargs), AH, b, temp_cost) * BH
levels[-1].BH = BH
levels[-1].complexity['candidates'] = temp_cost[0] * B.shape[1]
# Compute the tentative prolongator, T, which is a tentative interpolation
# matrix from the coarse-grid to the fine-grid. T exactly interpolates
# B_fine[:, 0:blocksize(A)] = T B_coarse[:, 0:blocksize(A)].
# Orthogonalization complexity ~ 2nk^2, k = blocksize(A).
temp_cost=[0.0]
T, dummy = fit_candidates(AggOp, B[:, 0:blocksize(A)], cost=temp_cost)
del dummy
if A.symmetry == "nonsymmetric":
TH, dummyH = fit_candidates(AggOp, BH[:, 0:blocksize(A)], cost=temp_cost)
del dummyH
levels[-1].complexity['tentative'] = temp_cost[0]/A.nnz
# Create necessary root node matrices
Cpt_params = (True, get_Cpt_params(A, Cnodes, AggOp, T))
T = scale_T(T, Cpt_params[1]['P_I'], Cpt_params[1]['I_F'])
levels[-1].complexity['tentative'] += T.nnz / float(A.nnz)
if A.symmetry == "nonsymmetric":
TH = scale_T(TH, Cpt_params[1]['P_I'], Cpt_params[1]['I_F'])
levels[-1].complexity['tentative'] += TH.nnz / float(A.nnz)
# Set coarse grid near nullspace modes as injected fine grid near
# null-space modes
B = Cpt_params[1]['P_I'].T*levels[-1].B
if A.symmetry == "nonsymmetric":
BH = Cpt_params[1]['P_I'].T*levels[-1].BH
# Smooth the tentative prolongator, so that it's accuracy is greatly
# improved for algebraically smooth error.
fn, kwargs = unpack_arg(smooth[len(levels)-1])
if fn == 'energy':
P = energy_prolongation_smoother(A, T, C, B, levels[-1].B,
Cpt_params=Cpt_params, **kwargs)
elif fn is None:
P = T
else:
raise ValueError('unrecognized prolongation smoother \
method %s' % str(fn))
levels[-1].complexity['smooth_P'] = kwargs['cost'][0]
# Compute the restriction matrix R, which interpolates from the fine-grid
# to the coarse-grid. If A is nonsymmetric, then R must be constructed
# based on A.H. Otherwise R = P.H or P.T.
symmetry = A.symmetry
if symmetry == 'hermitian':
R = P.H
elif symmetry == 'symmetric':
R = P.T
elif symmetry == 'nonsymmetric':
fn, kwargs = unpack_arg(smooth[len(levels)-1])
if fn == 'energy':
R = energy_prolongation_smoother(AH, TH, C, BH, levels[-1].BH,
Cpt_params=Cpt_params, **kwargs)
R = R.H
levels[-1].complexity['smooth_R'] = kwargs['cost'][0]
elif fn is None:
R = T.H
else:
raise ValueError('unrecognized prolongation smoother \
method %s' % str(fn))
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].AggOp = AggOp # aggregation operator
levels[-1].T = T # tentative prolongator
levels[-1].Fpts = Cpt_params[1]['Fpts'] # Fpts
levels[-1].P_I = Cpt_params[1]['P_I'] # Injection operator
levels[-1].I_F = Cpt_params[1]['I_F'] # Identity on F-pts
levels[-1].I_C = Cpt_params[1]['I_C'] # Identity on C-pts
levels[-1].P = P # smoothed prolongator
levels[-1].R = R # restriction operator
levels[-1].Cpts = Cpt_params[1]['Cpts'] # Cpts (i.e., rootnodes)
# Form coarse grid operator, get complexity
levels[-1].complexity['RAP'] = mat_mat_complexity(R,A) / float(A.nnz)
RA = R * A
levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P) / float(A.nnz)
A = RA * P # Galerkin operator, Ac = RAP
A.symmetry = symmetry
levels.append(multilevel_solver.level())
levels[-1].A = A
levels[-1].B = B # right near nullspace candidates
if A.symmetry == "nonsymmetric":
levels[-1].BH = BH # left near nullspace candidates
def extend_hierarchy(levels, strength, CF, interp, restrict, filter_operator,
coarse_grid_P, coarse_grid_R, keep):
""" helper function for local methods """
# Filter operator. Need to keep original matrix on fineest level for
# computing residuals
if (filter_operator is not None) and (filter_operator[1] != 0):
if len(levels) == 1:
A = deepcopy(levels[-1].A)
else:
A = levels[-1].A
filter_matrix_rows(A, filter_operator[1], diagonal=True, lump=filter_operator[0])
else:
A = levels[-1].A
# Check if matrix was filtered to be diagonal --> coarsest grid
if A.nnz == A.shape[0]:
return 1
# Zero initial complexities for strength, splitting and interpolation
levels[-1].complexity['CF'] = 0.0
levels[-1].complexity['strength'] = 0.0
levels[-1].complexity['interpolate'] = 0.0
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == 'ode') or (fn == 'evolution'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] += kwargs['cost'][0] * A.nnz / float(A.nnz)
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = RS(C, **kwargs)
elif fn == 'PMIS':
splitting = PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
elif fn == 'weighted_matching':
splitting, soc = weighted_matching(C, **kwargs)
if soc is not None:
C = soc
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] += kwargs['cost'][0] * C.nnz / float(A.nnz)
temp = np.sum(splitting)
if (temp == len(splitting)) or (temp == 0):
return 1
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
r_flag = False
fn, kwargs = unpack_arg(interp)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'scaledAfc':
P = scaled_Afc_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = bsr_matrix(P.T)
else:
temp_A = csr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = csr_matrix(P.T)
elif fn == 'restrict':
r_flag = True
else:
raise ValueError('unknown interpolation method (%s)' % interp)
levels[-1].complexity['interpolate'] += kwargs['cost'][0] * A.nnz / float(A.nnz)
# Build restriction operator
fn, kwargs = unpack_arg(restrict)
if fn is None:
R = P.T
elif fn == 'air':
R = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown restriction method (%s)' % restrict)
# If set P = R^T
if r_flag:
P = R.T
# Optional different interpolation for RAP
fn, kwargs = unpack_arg(coarse_grid_P)
if fn == 'standard':
P_temp = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P_temp = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P_temp = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P_temp = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P_temp = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P_temp = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = bsr_matrix(P_temp.T)
else:
temp_A = csr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = csr_matrix(P_temp.T)
else:
P_temp = P
# Optional different restriction for RAP
fn, kwargs = unpack_arg(coarse_grid_R)
if fn == 'air':
R_temp = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R_temp = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R_temp = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R_temp = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
else:
R_temp = R
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# Form coarse grid operator, get complexity
#levels[-1].complexity['RAP'] = mat_mat_complexity(R_temp,A) / float(A.nnz)
#RA = R_temp * A
#levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P_temp) / float(A.nnz)
#A = RA * P_temp
# RL: RAP = R*(A*P)
levels[-1].complexity['RAP'] = mat_mat_complexity(A, P_temp) / float(A.nnz)
AP = A * P_temp
levels[-1].complexity['RAP'] += mat_mat_complexity(R_temp, AP) / float(A.nnz)
A = R_temp * AP
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
A.eliminate_zeros()
levels.append(multilevel_solver.level())
levels[-1].A = A
return 0
def change_smoothers(ml, presmoother, postsmoother):
'''
Initialize pre- and post- smoothers throughout a multilevel_solver, with
the option of having different smoothers at different levels
For each level of the multilevel_solver 'ml' (except the coarsest level),
initialize the .presmoother() and .postsmoother() methods used in the
multigrid cycle.
Parameters
----------
ml : {pyamg multilevel hierarchy}
Data structure that stores the multigrid hierarchy.
presmoother : {None, string, tuple, list}
presmoother can be (1) the name of a supported smoother, e.g.
"gauss_seidel", (2) a tuple of the form ('method','opts') where
'method' is the name of a supported smoother and 'opts' a dict of
keyword arguments to the smoother, or (3) a list of instances of
options 1 or 2. See the Examples section for illustrations of the
format.
If presmoother is a list, presmoother[i] determines the smoothing
strategy for level i. Else, presmoother defines the same strategy
for all levels.
If len(presmoother) < len(ml.levels), then
presmoother[-1] is used for all remaining levels
If len(presmoother) > len(ml.levels), then
the remaining smoothing strategies are ignored
postsmoother : {string, tuple, list}
Defines postsmoother in identical fashion to presmoother
Returns
-------
ml changed in place
ml.level[i].smoothers['presmoother'] <=== presmoother[i]
ml.level[i].smoothers['postsmoother'] <=== postsmoother[i]
ml.symmetric_smoothing is marked True/False depending on whether
the smoothing scheme is symmetric.
Notes
-----
- Parameter 'omega' of the Jacobi, Richardson, and jacobi_ne
methods is scaled by the spectral radius of the matrix on
each level. Therefore 'omega' should be in the interval (0,2).
- Parameter 'withrho' (default: True) controls whether the omega is
rescaled by the spectral radius in jacobi, block_jacobi, and jacobi_ne
- By initializing the smoothers after the hierarchy has been setup, allows
for "algebraically" directed relaxation, such as strength_based_schwarz,
which uses only the strong connections of a degree-of-freedom to define
overlapping regions
- Available smoother methods::
gauss_seidel
block_gauss_seidel
jacobi
block_jacobi
richardson
sor
chebyshev
gauss_seidel_nr
gauss_seidel_ne
jacobi_ne
cg
gmres
cgne
cgnr
schwarz
strength_based_schwarz
None
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.aggregation import smoothed_aggregation_solver
>>> from pyamg.relaxation.smoothing import change_smoothers
>>> from pyamg.util.linalg import norm
>>> from scipy import rand, array, mean
>>> A = poisson((10,10), format='csr')
>>> b = rand(A.shape[0],)
>>> ml = smoothed_aggregation_solver(A, max_coarse=10)
>>> #
>>> # Set all levels to use gauss_seidel's defaults
>>> smoothers = 'gauss_seidel'
>>> change_smoothers(ml, presmoother=smoothers, postsmoother=smoothers)
>>> residuals=[]
>>> x = ml.solve(b, tol=1e-8, residuals=residuals)
>>> #
>>> # Set all levels to use three iterations of gauss_seidel's defaults
>>> smoothers = ('gauss_seidel', {'iterations' : 3})
>>> change_smoothers(ml, presmoother=smoothers, postsmoother=None)
>>> residuals=[]
>>> x = ml.solve(b, tol=1e-8, residuals=residuals)
>>> #
>>> # Set level 0 to use gauss_seidel's defaults, and all
>>> # subsequent levels to use 5 iterations of cgnr
>>> smoothers = ['gauss_seidel', ('cgnr', {'maxiter' : 5})]
>>> change_smoothers(ml, presmoother=smoothers, postsmoother=smoothers)
>>> residuals=[]
>>> x = ml.solve(b, tol=1e-8, residuals=residuals)
'''
ml.symmetric_smoothing = True
# interpret arguments into list
if isinstance(presmoother, str) or isinstance(presmoother, tuple) or\
(presmoother is None):
presmoother = [presmoother]
elif not isinstance(presmoother, list):
raise ValueError('Unrecognized presmoother')
if isinstance(postsmoother, str) or isinstance(postsmoother, tuple) or\
(postsmoother is None):
postsmoother = [postsmoother]
elif not isinstance(postsmoother, list):
raise ValueError('Unrecognized postsmoother')
# set ml.levels[i].presmoother = presmoother[i],
# ml.levels[i].postsmoother = postsmoother[i]
fn1 = None # Predefine to keep scope beyond first loop
fn2 = None
kwargs1 = {}
kwargs2 = {}
min_len = min(len(presmoother), len(postsmoother), len(ml.levels[:-1]))
same = (len(presmoother) == len(postsmoother))
for i in range(0, min_len):
# unpack presmoother[i]
fn1, kwargs1 = unpack_arg(presmoother[i], cost=False)
# get function handle
try:
setup_presmoother = eval('setup_' + str(fn1))
except NameError:
raise NameError("invalid presmoother method: ", fn1)
ml.levels[i].presmoother = setup_presmoother(ml.levels[i], **kwargs1)
# unpack postsmoother[i]
fn2, kwargs2 = unpack_arg(postsmoother[i], cost=False)
# get function handle
try:
setup_postsmoother = eval('setup_' + str(fn2))
except NameError:
raise NameError("invalid postsmoother method: ", fn2)
ml.levels[i].postsmoother = setup_postsmoother(ml.levels[i], **kwargs2)
# Save tuples in ml to check cycle complexity
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
# Check if symmetric smoothing scheme
try:
it1 = kwargs1['iterations']
except:
it1 = DEFAULT_NITER
try:
it2 = kwargs2['iterations']
except:
it2 = DEFAULT_NITER
if (it1 != it2):
ml.symmetric_smoothing = False
elif (fn1 != fn2):
if ((fn1 == 'CF_jacobi' and fn2 == 'FC_jacobi') or \
(fn1 == 'CF_block_jacobi' and fn2 == 'FC_block_jacobi')):
try:
fit1 = kwargs1['F_iterations']
except:
fit1 = DEFAULT_NITER
try:
fit2 = kwargs2['F_iterations']
except:
fit2 = DEFAULT_NITER
try:
cit1 = kwargs1['C_iterations']
except:
cit1 = DEFAULT_NITER
try:
cit2 = kwargs2['C_iterations']
except:
cit2 = DEFAULT_NITER
if ((fit1 == fit2) and (cit1 == cit2)):
pass
else:
ml.symmetric_smoothing = False
else:
ml.symmetric_smoothing = False
elif fn1 in KRYLOV_RELAXATION or fn2 in KRYLOV_RELAXATION:
ml.symmetric_smoothing = False
elif fn1 not in SYMMETRIC_RELAXATION:
if ('CF' in fn1) or ('FC' in fn1):
ml.symmetric_smoothing = False
else:
try:
sweep1 = kwargs1['sweep']
except:
sweep1 = DEFAULT_SWEEP
try:
sweep2 = kwargs2['sweep']
except:
sweep2 = DEFAULT_SWEEP
if (sweep1 == 'forward' and sweep2 == 'backward') or \
(sweep1 == 'backward' and sweep2 == 'forward') or \
(sweep1 == 'symmetric' and sweep2 == 'symmetric'):
pass
else:
ml.symmetric_smoothing = False
if len(presmoother) < len(postsmoother):
mid_len = min(len(postsmoother), len(ml.levels[:-1]))
for i in range(min_len, mid_len):
# Set up presmoother
ml.levels[i].presmoother = setup_presmoother(
ml.levels[i], **kwargs1)
# unpack postsmoother[i]
fn2, kwargs2 = unpack_arg(postsmoother[i], cost=False)
# get function handle
try:
setup_postsmoother = eval('setup_' + str(fn2))
except NameError:
raise NameError("invalid postsmoother method: ", fn2)
ml.levels[i].postsmoother = setup_postsmoother(
ml.levels[i], **kwargs2)
# Save tuples in ml to check cycle complexity
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
# Check if symmetric smoothing scheme
try:
it1 = kwargs1['iterations']
except:
it1 = DEFAULT_NITER
try:
it2 = kwargs2['iterations']
except:
it2 = DEFAULT_NITER
if (it1 != it2):
ml.symmetric_smoothing = False
elif (fn1 != fn2):
if ((fn1 == 'CF_jacobi' and fn2 == 'FC_jacobi') or \
(fn1 == 'CF_block_jacobi' and fn2 == 'FC_block_jacobi')):
try:
fit1 = kwargs1['F_iterations']
except:
fit1 = DEFAULT_NITER
try:
fit2 = kwargs2['F_iterations']
except:
fit2 = DEFAULT_NITER
try:
cit1 = kwargs1['C_iterations']
except:
cit1 = DEFAULT_NITER
try:
cit2 = kwargs2['C_iterations']
except:
cit2 = DEFAULT_NITER
if ((fit1 == fit2) and (cit1 == cit2)):
pass
else:
ml.symmetric_smoothing = False
else:
ml.symmetric_smoothing = False
elif fn1 in KRYLOV_RELAXATION or fn2 in KRYLOV_RELAXATION:
ml.symmetric_smoothing = False
elif fn1 not in SYMMETRIC_RELAXATION:
if ('CF' in fn1) or ('FC' in fn1):
ml.symmetric_smoothing = False
else:
try:
sweep1 = kwargs1['sweep']
except:
sweep1 = DEFAULT_SWEEP
try:
sweep2 = kwargs2['sweep']
except:
sweep2 = DEFAULT_SWEEP
if (sweep1 == 'forward' and sweep2 == 'backward') or \
(sweep1 == 'backward' and sweep2 == 'forward') or \
(sweep1 == 'symmetric' and sweep2 == 'symmetric'):
pass
else:
ml.symmetric_smoothing = False
elif len(presmoother) > len(postsmoother):
mid_len = min(len(presmoother), len(ml.levels[:-1]))
for i in range(min_len, mid_len):
# unpack presmoother[i]
fn1, kwargs1 = unpack_arg(presmoother[i], cost=False)
# get function handle
try:
setup_presmoother = eval('setup_' + str(fn1))
except NameError:
raise NameError("invalid presmoother method: ", fn1)
ml.levels[i].presmoother = setup_presmoother(
ml.levels[i], **kwargs1)
# Set up postsmoother
ml.levels[i].postsmoother = setup_postsmoother(
ml.levels[i], **kwargs2)
# Save tuples in ml to check cycle complexity
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
# Check if symmetric smoothing scheme
try:
it1 = kwargs1['iterations']
except:
it1 = DEFAULT_NITER
try:
it2 = kwargs2['iterations']
except:
it2 = DEFAULT_NITER
if (it1 != it2):
ml.symmetric_smoothing = False
elif (fn1 != fn2):
if ((fn1 == 'CF_jacobi' and fn2 == 'FC_jacobi') or \
(fn1 == 'CF_block_jacobi' and fn2 == 'FC_block_jacobi')):
try:
fit1 = kwargs1['F_iterations']
except:
fit1 = DEFAULT_NITER
try:
fit2 = kwargs2['F_iterations']
except:
fit2 = DEFAULT_NITER
try:
cit1 = kwargs1['C_iterations']
except:
cit1 = DEFAULT_NITER
try:
cit2 = kwargs2['C_iterations']
except:
cit2 = DEFAULT_NITER
if ((fit1 == fit2) and (cit1 == cit2)):
pass
else:
ml.symmetric_smoothing = False
else:
ml.symmetric_smoothing = False
elif fn1 in KRYLOV_RELAXATION or fn2 in KRYLOV_RELAXATION:
ml.symmetric_smoothing = False
elif fn1 not in SYMMETRIC_RELAXATION:
if ('CF' in fn1) or ('FC' in fn1):
ml.symmetric_smoothing = False
else:
try:
sweep1 = kwargs1['sweep']
except:
sweep1 = DEFAULT_SWEEP
try:
sweep2 = kwargs2['sweep']
except:
sweep2 = DEFAULT_SWEEP
if (sweep1 == 'forward' and sweep2 == 'backward') or \
(sweep1 == 'backward' and sweep2 == 'forward') or \
(sweep1 == 'symmetric' and sweep2 == 'symmetric'):
pass
else:
ml.symmetric_smoothing = False
else:
mid_len = min_len
# Fill in remaining levels
for i in range(mid_len, len(ml.levels[:-1])):
ml.levels[i].presmoother = setup_presmoother(ml.levels[i], **kwargs1)
ml.levels[i].postsmoother = setup_postsmoother(ml.levels[i], **kwargs2)
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
def initial_setup_stage(A, symmetry, pdef, candidate_iters, epsilon,
max_levels, max_coarse, aggregate, prepostsmoother,
smooth, strength, work, B=None):
"""
Computes a complete aggregation and the first near-nullspace candidate
following Algorithm 3 in Brezina et al.
Parameters
----------
candidate_iters
number of test relaxation iterations
epsilon
minimum acceptable relaxation convergence factor
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
# Define relaxation routine
def relax(A, x):
fn, kwargs = unpack_arg(prepostsmoother, cost=False)
if fn == 'gauss_seidel':
gauss_seidel(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(A, x, np.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A, x, np.zeros_like(x), iterations=1,
coefficients=[1.0/approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
# flag for skipping steps f-i in step 4
skip_f_to_i = True
# step 1
A_l = A
if B is None:
x = sp.rand(A_l.shape[0], 1).astype(A_l.dtype)
# The following type check matches the usual 'complex' type,
# but also numpy data types such as 'complex64', 'complex128'
# and 'complex256'.
if A_l.dtype.name.startswith('complex'):
x = x + 1.0j*sp.rand(A_l.shape[0], 1)
else:
x = np.array(B, dtype=A_l.dtype)
# step 2
relax(A_l, x)
work[:] += A_l.nnz * candidate_iters*2
# step 3
# not advised to stop the iteration here: often the first relaxation pass
# _is_ good, but the remaining passes are poor
# if x_A_x/x_A_x_old < epsilon:
# # relaxation alone is sufficient
# print 'relaxation alone works: %g'%(x_A_x/x_A_x_old)
# return x, []
# step 4
As = [A]
xs = [x]
Ps = []
AggOps = []
StrengthOps = []
while A.shape[0] > max_coarse and max_levels > 1:
# The real check to break from the while loop is below
# Begin constructing next level
fn, kwargs = unpack_arg(strength[len(As)-1]) # step 4b
if fn == 'symmetric':
C_l = symmetric_strength_of_connection(A_l, **kwargs)
# Diagonal must be nonzero
C_l = C_l + eye(C_l.shape[0], C_l.shape[1], format='csr')
elif fn == 'classical':
C_l = classical_strength_of_connection(A_l, **kwargs)
# Diagonal must be nonzero
C_l = C_l + eye(C_l.shape[0], C_l.shape[1], format='csr')
if isspmatrix_bsr(A_l):
C_l = amalgamate(C_l, A_l.blocksize[0])
elif (fn == 'ode') or (fn == 'evolution'):
C_l = evolution_strength_of_connection(A_l,
np.ones(
(A_l.shape[0], 1),
dtype=A.dtype),
**kwargs)
elif fn == 'predefined':
C_l = kwargs['C'].tocsr()
elif fn is None:
C_l = A_l.tocsr()
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
# In SA, strength represents "distance", so we take magnitude of
# complex values
if C_l.dtype.name.startswith('complex'):
C_l.data = np.abs(C_l.data)
# Create a unified strength framework so that large values represent
# strong connections and small values represent weak connections
if (fn == 'ode') or (fn == 'evolution') or (fn == 'energy_based'):
C_l.data = 1.0 / C_l.data
# aggregation
fn, kwargs = unpack_arg(aggregate[len(As) - 1])
if fn == 'standard':
AggOp = standard_aggregation(C_l, **kwargs)[0]
elif fn == 'lloyd':
AggOp = lloyd_aggregation(C_l, **kwargs)[0]
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
else:
raise ValueError('unrecognized aggregation method %s' % str(fn))
T_l, x = fit_candidates(AggOp, x) # step 4c
fn, kwargs = unpack_arg(smooth[len(As)-1]) # step 4d
if fn == 'jacobi':
P_l = jacobi_prolongation_smoother(A_l, T_l, C_l, x, **kwargs)
elif fn == 'richardson':
P_l = richardson_prolongation_smoother(A_l, T_l, **kwargs)
elif fn == 'energy':
P_l = energy_prolongation_smoother(A_l, T_l, C_l, x, None,
(False, {}), **kwargs)
elif fn is None:
P_l = T_l
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# R should reflect A's structure # step 4e
if symmetry == 'symmetric':
A_l = P_l.T.asformat(P_l.format) * A_l * P_l
elif symmetry == 'hermitian':
A_l = P_l.H.asformat(P_l.format) * A_l * P_l
StrengthOps.append(C_l)
AggOps.append(AggOp)
Ps.append(P_l)
As.append(A_l)
# skip to step 5 as in step 4e
if (A_l.shape[0] <= max_coarse) or (len(AggOps) + 1 >= max_levels):
break
if not skip_f_to_i:
x_hat = x.copy() # step 4g
relax(A_l, x) # step 4h
work[:] += A_l.nnz*candidate_iters*2
if pdef is True:
x_A_x = np.dot(np.conjugate(x).T, A_l*x)
xhat_A_xhat = np.dot(np.conjugate(x_hat).T, A_l*x_hat)
err_ratio = (x_A_x/xhat_A_xhat)**(1.0/candidate_iters)
else:
# use A.H A inner-product
Ax = A_l * x
# Axhat = A_l * x_hat
x_A_x = np.dot(np.conjugate(Ax).T, Ax)
xhat_A_xhat = np.dot(np.conjugate(x_hat).T, A_l*x_hat)
err_ratio = (x_A_x/xhat_A_xhat)**(1.0/candidate_iters)
if err_ratio < epsilon: # step 4i
# print "sufficient convergence, skipping"
skip_f_to_i = True
if x_A_x == 0:
x = x_hat # need to restore x
else:
# just carry out relaxation, don't check for convergence
relax(A_l, x) # step 4h
work[:] += 2 * A_l.nnz * candidate_iters
# store xs for diagnostic use and for use in step 5
xs.append(x)
# step 5
# Extend coarse-level candidate to the finest level
# --> note that we start with the x from the second coarsest level
x = xs[-1]
# make sure that xs[-1] has been relaxed by step 4h, i.e. relax(As[-2], x)
for lev in range(len(Ps)-2, -1, -1): # lev = coarsest ... finest-1
P = Ps[lev] # I: lev --> lev+1
A = As[lev] # A on lev+1
x = P * x
relax(A, x)
work[:] += A.nnz*candidate_iters*2
# Set predefined strength of connection and aggregation
if len(AggOps) > 1:
aggregate = [('predefined', {'AggOp': AggOps[i]})
for i in range(len(AggOps))]
strength = [('predefined', {'C': StrengthOps[i]})
for i in range(len(StrengthOps))]
return x, aggregate, strength # first candidate
def coarse_grid_solver(solver):
"""Return a coarse grid solver suitable for multilevel_solver
Parameters
----------
solver : {string, callable, tuple}
The solver method is either (1) a string such as 'splu' or 'pinv' of a
callable object which receives only parameters (A, b) and returns an
(approximate or exact) solution to the linear system Ax = b, or (2) a
callable object that takes parameters (A,b) and returns an (approximate
or exact) solution to Ax = b, or (3) a tuple of the form
(string|callable, args), where args is a dictionary of arguments to
be passed to the function denoted by string or callable.
The set of valid string arguments is:
- Sparse direct methods:
+ splu : sparse LU solver
- Sparse iterative methods:
+ the name of any method in scipy.sparse.linalg.isolve or
pyamg.krylov (e.g. 'cg').
Methods in pyamg.krylov take precedence.
+ relaxation method, such as 'gauss_seidel' or 'jacobi',
present in pyamg.relaxation
- Dense methods:
+ pinv : pseudoinverse (QR)
+ pinv2 : pseudoinverse (SVD)
+ lu : LU factorization
+ cholesky : Cholesky factorization
Returns
-------
ptr : generic_solver
A class for use as a standalone or coarse grids solver
Examples
--------
>>> import numpy as np
>>> from scipy.sparse import spdiags
>>> from pyamg.gallery import poisson
>>> from pyamg import coarse_grid_solver
>>> A = poisson((10, 10), format='csr')
>>> b = A * np.ones(A.shape[0])
>>> cgs = coarse_grid_solver('lu')
>>> x = cgs(A, b)
"""
solver, kwargs = unpack_arg(solver, cost=False)
if solver in ['pinv', 'pinv2']:
def solve(self, A, b):
if not hasattr(self, 'P'):
self.P = getattr(sp.linalg, solver)(A.todense(), **kwargs)
return np.dot(self.P, b)
elif solver == 'lu':
def solve(self, A, b):
if not hasattr(self, 'LU'):
self.LU = sp.linalg.lu_factor(A.todense(), **kwargs)
return sp.linalg.lu_solve(self.LU, b)
elif solver == 'cholesky':
def solve(self, A, b):
if not hasattr(self, 'L'):
self.L = sp.linalg.cho_factor(A.todense(), **kwargs)
return sp.linalg.cho_solve(self.L, b)
elif solver == 'splu':
def solve(self, A, b):
if not hasattr(self, 'LU'):
# for multiple candidates in B, A will often have a couple zero
# rows/columns that must be removed
Acsc = A.tocsc()
Acsc.eliminate_zeros()
diffptr = Acsc.indptr[:-1] - Acsc.indptr[1:]
nonzero_cols = (diffptr != 0).nonzero()[0]
Map = sp.sparse.eye(Acsc.shape[0], Acsc.shape[1], format='csc')
Map = Map[:, nonzero_cols]
Acsc = Map.T.tocsc() * Acsc * Map
self.LU = sp.sparse.linalg.splu(Acsc, **kwargs)
self.LU_Map = Map
return self.LU_Map * self.LU.solve(np.ravel(self.LU_Map.T * b))
elif solver in ['bicg', 'bicgstab', 'cg', 'cgs', 'gmres', 'qmr', 'minres']:
from pyamg import krylov
if hasattr(krylov, solver):
fn = getattr(krylov, solver)
else:
fn = getattr(sp.sparse.linalg.isolve, solver)
def solve(self, A, b):
if 'tol' not in kwargs:
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}
kwargs['tol'] = {0: feps * 1e3, 1: eps * 1e6,
2: geps * 1e6}[_array_precision[A.dtype.char]]
return fn(A, b, **kwargs)[0]
elif solver in ['gauss_seidel', 'jacobi', 'block_gauss_seidel', 'schwarz',
'block_jacobi', 'richardson', 'sor', 'chebyshev',
'jacobi_ne', 'gauss_seidel_ne', 'gauss_seidel_nr']:
if 'iterations' not in kwargs:
kwargs['iterations'] = 10
def solve(self, A, b):
from pyamg.relaxation import smoothing
from pyamg import multilevel_solver
lvl = multilevel_solver.level()
lvl.A = A
fn = getattr(smoothing, 'setup_' + str(solver))
relax = fn(lvl, **kwargs)
x = np.zeros_like(b)
relax(A, x, b)
return x
elif solver is None:
# No coarse grid solve
def solve(self, A, b):
return 0 * b # should this return b instead?
elif callable(solver):
def solve(self, A, b):
return solver(A, b, **kwargs)
else:
raise ValueError('unknown solver: %s' % solver)
class generic_solver:
def __call__(self, A, b):
# make sure x is same dimensions and type as b
b = np.asanyarray(b)
if A.nnz == 0:
# if A.nnz = 0, then we expect no correction
x = np.zeros(b.shape)
else:
x = solve(self, A, b)
if isinstance(b, np.ndarray):
x = np.asarray(x)
elif isinstance(b, np.matrix):
x = np.asmatrix(x)
else:
raise ValueError('unrecognized type')
return x.reshape(b.shape)
def __repr__(self):
return 'coarse_grid_solver(' + repr(solver) + ')'
def name(self):
return repr(solver)
return generic_solver()
def extend_hierarchy(levels, strength, CF, interpolation, restriction, keep):
""" helper function for local methods """
A = levels[-1].A
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == 'ode') or (fn == 'evolution'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] = kwargs['cost'][0]
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = split.RS(C, **kwargs)
elif fn == 'PMIS':
splitting = split.PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = split.PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = split.CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = split.CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] = kwargs['cost'][0]
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
fn, kwargs = unpack_arg(interpolation)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'injection':
P = injection_interpolation(A, splitting, **kwargs)
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
levels[-1].complexity['interpolate'] = kwargs['cost'][0]
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid. Must make sure transpose matrices remain in CSR or BSR
fn, kwargs = unpack_arg(restriction)
if isspmatrix_csr(A):
if restriction == 'galerkin':
R = P.T.tocsr()
elif fn == 'standard':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
else:
if restriction == 'galerkin':
R = P.T.tobsr()
elif fn == 'standard':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'distance_two':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'direct':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tobsr()
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
levels[-1].complexity['restriction'] = kwargs['cost'][0]
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# Form coarse grid operator, get complexity
levels[-1].complexity['RAP'] = mat_mat_complexity(R,A) / float(A.nnz)
RA = R * A
levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P) / float(A.nnz)
A = RA * P # Galerkin operator, Ac = RAP
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
# Form next level through Galerkin product
levels.append(multilevel_solver.level())
levels[-1].A = A
def change_smoothers(ml, presmoother, postsmoother):
'''
Initialize pre- and post- smoothers throughout a multilevel_solver, with
the option of having different smoothers at different levels
For each level of the multilevel_solver 'ml' (except the coarsest level),
initialize the .presmoother() and .postsmoother() methods used in the
multigrid cycle.
Parameters
----------
ml : {pyamg multilevel hierarchy}
Data structure that stores the multigrid hierarchy.
presmoother : {None, string, tuple, list}
presmoother can be (1) the name of a supported smoother, e.g.
"gauss_seidel", (2) a tuple of the form ('method','opts') where
'method' is the name of a supported smoother and 'opts' a dict of
keyword arguments to the smoother, or (3) a list of instances of
options 1 or 2. See the Examples section for illustrations of the
format.
If presmoother is a list, presmoother[i] determines the smoothing
strategy for level i. Else, presmoother defines the same strategy
for all levels.
If len(presmoother) < len(ml.levels), then
presmoother[-1] is used for all remaining levels
If len(presmoother) > len(ml.levels), then
the remaining smoothing strategies are ignored
postsmoother : {string, tuple, list}
Defines postsmoother in identical fashion to presmoother
Returns
-------
ml changed in place
ml.level[i].smoothers['presmoother'] <=== presmoother[i]
ml.level[i].smoothers['postsmoother'] <=== postsmoother[i]
ml.symmetric_smoothing is marked True/False depending on whether
the smoothing scheme is symmetric.
Notes
-----
- Parameter 'omega' of the Jacobi, Richardson, and jacobi_ne
methods is scaled by the spectral radius of the matrix on
each level. Therefore 'omega' should be in the interval (0,2).
- Parameter 'withrho' (default: True) controls whether the omega is
rescaled by the spectral radius in jacobi, block_jacobi, and jacobi_ne
- By initializing the smoothers after the hierarchy has been setup, allows
for "algebraically" directed relaxation, such as strength_based_schwarz,
which uses only the strong connections of a degree-of-freedom to define
overlapping regions
- Available smoother methods::
gauss_seidel
block_gauss_seidel
jacobi
block_jacobi
richardson
sor
chebyshev
gauss_seidel_nr
gauss_seidel_ne
jacobi_ne
cg
gmres
cgne
cgnr
schwarz
strength_based_schwarz
None
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.aggregation import smoothed_aggregation_solver
>>> from pyamg.relaxation.smoothing import change_smoothers
>>> from pyamg.util.linalg import norm
>>> from scipy import rand, array, mean
>>> A = poisson((10,10), format='csr')
>>> b = rand(A.shape[0],)
>>> ml = smoothed_aggregation_solver(A, max_coarse=10)
>>> #
>>> # Set all levels to use gauss_seidel's defaults
>>> smoothers = 'gauss_seidel'
>>> change_smoothers(ml, presmoother=smoothers, postsmoother=smoothers)
>>> residuals=[]
>>> x = ml.solve(b, tol=1e-8, residuals=residuals)
>>> #
>>> # Set all levels to use three iterations of gauss_seidel's defaults
>>> smoothers = ('gauss_seidel', {'iterations' : 3})
>>> change_smoothers(ml, presmoother=smoothers, postsmoother=None)
>>> residuals=[]
>>> x = ml.solve(b, tol=1e-8, residuals=residuals)
>>> #
>>> # Set level 0 to use gauss_seidel's defaults, and all
>>> # subsequent levels to use 5 iterations of cgnr
>>> smoothers = ['gauss_seidel', ('cgnr', {'maxiter' : 5})]
>>> change_smoothers(ml, presmoother=smoothers, postsmoother=smoothers)
>>> residuals=[]
>>> x = ml.solve(b, tol=1e-8, residuals=residuals)
'''
ml.symmetric_smoothing = True
# interpret arguments into list
if isinstance(presmoother, str) or isinstance(presmoother, tuple) or\
(presmoother is None):
presmoother = [presmoother]
elif not isinstance(presmoother, list):
raise ValueError('Unrecognized presmoother')
if isinstance(postsmoother, str) or isinstance(postsmoother, tuple) or\
(postsmoother is None):
postsmoother = [postsmoother]
elif not isinstance(postsmoother, list):
raise ValueError('Unrecognized postsmoother')
# set ml.levels[i].presmoother = presmoother[i],
# ml.levels[i].postsmoother = postsmoother[i]
fn1 = None # Predefine to keep scope beyond first loop
fn2 = None
kwargs1 = {}
kwargs2 = {}
min_len = min(len(presmoother), len(postsmoother), len(ml.levels[:-1]))
same = (len(presmoother) == len(postsmoother))
for i in range(0, min_len):
# unpack presmoother[i]
fn1, kwargs1 = unpack_arg(presmoother[i], cost=False)
# get function handle
try:
setup_presmoother = eval('setup_' + str(fn1))
except NameError:
raise NameError("invalid presmoother method: ", fn1)
ml.levels[i].presmoother = setup_presmoother(ml.levels[i], **kwargs1)
# unpack postsmoother[i]
fn2, kwargs2 = unpack_arg(postsmoother[i], cost=False)
# get function handle
try:
setup_postsmoother = eval('setup_' + str(fn2))
except NameError:
raise NameError("invalid postsmoother method: ", fn2)
ml.levels[i].postsmoother = setup_postsmoother(ml.levels[i], **kwargs2)
# Save tuples in ml to check cycle complexity
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
# Check if symmetric smoothing scheme
try:
it1 = kwargs1['iterations']
except:
it1 = DEFAULT_NITER
try:
it2 = kwargs2['iterations']
except:
it2 = DEFAULT_NITER
if (it1 != it2):
ml.symmetric_smoothing = False
elif (fn1 != fn2):
if ((fn1 == 'CF_jacobi' and fn2 == 'FC_jacobi') or \
(fn1 == 'CF_block_jacobi' and fn2 == 'FC_block_jacobi')):
try:
fit1 = kwargs1['F_iterations']
except:
fit1 = DEFAULT_NITER
try:
fit2 = kwargs2['F_iterations']
except:
fit2 = DEFAULT_NITER
try:
cit1 = kwargs1['C_iterations']
except:
cit1 = DEFAULT_NITER
try:
cit2 = kwargs2['C_iterations']
except:
cit2 = DEFAULT_NITER
if ((fit1 == fit2) and (cit1 == cit2)):
pass
else:
ml.symmetric_smoothing = False
else:
ml.symmetric_smoothing = False
elif fn1 in KRYLOV_RELAXATION or fn2 in KRYLOV_RELAXATION:
ml.symmetric_smoothing = False
elif fn1 not in SYMMETRIC_RELAXATION:
if ('CF' in fn1) or ('FC' in fn1):
ml.symmetric_smoothing = False
else:
try:
sweep1 = kwargs1['sweep']
except:
sweep1 = DEFAULT_SWEEP
try:
sweep2 = kwargs2['sweep']
except:
sweep2 = DEFAULT_SWEEP
if (sweep1 == 'forward' and sweep2 == 'backward') or \
(sweep1 == 'backward' and sweep2 == 'forward') or \
(sweep1 == 'symmetric' and sweep2 == 'symmetric'):
pass
else:
ml.symmetric_smoothing = False
if len(presmoother) < len(postsmoother):
mid_len = min(len(postsmoother), len(ml.levels[:-1]))
for i in range(min_len, mid_len):
# Set up presmoother
ml.levels[i].presmoother = setup_presmoother(ml.levels[i], **kwargs1)
# unpack postsmoother[i]
fn2, kwargs2 = unpack_arg(postsmoother[i], cost=False)
# get function handle
try:
setup_postsmoother = eval('setup_' + str(fn2))
except NameError:
raise NameError("invalid postsmoother method: ", fn2)
ml.levels[i].postsmoother = setup_postsmoother(ml.levels[i], **kwargs2)
# Save tuples in ml to check cycle complexity
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
# Check if symmetric smoothing scheme
try:
it1 = kwargs1['iterations']
except:
it1 = DEFAULT_NITER
try:
it2 = kwargs2['iterations']
except:
it2 = DEFAULT_NITER
if (it1 != it2):
ml.symmetric_smoothing = False
elif (fn1 != fn2):
if ((fn1 == 'CF_jacobi' and fn2 == 'FC_jacobi') or \
(fn1 == 'CF_block_jacobi' and fn2 == 'FC_block_jacobi')):
try:
fit1 = kwargs1['F_iterations']
except:
fit1 = DEFAULT_NITER
try:
fit2 = kwargs2['F_iterations']
except:
fit2 = DEFAULT_NITER
try:
cit1 = kwargs1['C_iterations']
except:
cit1 = DEFAULT_NITER
try:
cit2 = kwargs2['C_iterations']
except:
cit2 = DEFAULT_NITER
if ((fit1 == fit2) and (cit1 == cit2)):
pass
else:
ml.symmetric_smoothing = False
else:
ml.symmetric_smoothing = False
elif fn1 in KRYLOV_RELAXATION or fn2 in KRYLOV_RELAXATION:
ml.symmetric_smoothing = False
elif fn1 not in SYMMETRIC_RELAXATION:
if ('CF' in fn1) or ('FC' in fn1):
ml.symmetric_smoothing = False
else:
try:
sweep1 = kwargs1['sweep']
except:
sweep1 = DEFAULT_SWEEP
try:
sweep2 = kwargs2['sweep']
except:
sweep2 = DEFAULT_SWEEP
if (sweep1 == 'forward' and sweep2 == 'backward') or \
(sweep1 == 'backward' and sweep2 == 'forward') or \
(sweep1 == 'symmetric' and sweep2 == 'symmetric'):
pass
else:
ml.symmetric_smoothing = False
elif len(presmoother) > len(postsmoother):
mid_len = min(len(presmoother), len(ml.levels[:-1]))
for i in range(min_len, mid_len):
# unpack presmoother[i]
fn1, kwargs1 = unpack_arg(presmoother[i], cost=False)
# get function handle
try:
setup_presmoother = eval('setup_' + str(fn1))
except NameError:
raise NameError("invalid presmoother method: ", fn1)
ml.levels[i].presmoother = setup_presmoother(ml.levels[i], **kwargs1)
# Set up postsmoother
ml.levels[i].postsmoother = setup_postsmoother(ml.levels[i], **kwargs2)
# Save tuples in ml to check cycle complexity
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
# Check if symmetric smoothing scheme
try:
it1 = kwargs1['iterations']
except:
it1 = DEFAULT_NITER
try:
it2 = kwargs2['iterations']
except:
it2 = DEFAULT_NITER
if (it1 != it2):
ml.symmetric_smoothing = False
elif (fn1 != fn2):
if ((fn1 == 'CF_jacobi' and fn2 == 'FC_jacobi') or \
(fn1 == 'CF_block_jacobi' and fn2 == 'FC_block_jacobi')):
try:
fit1 = kwargs1['F_iterations']
except:
fit1 = DEFAULT_NITER
try:
fit2 = kwargs2['F_iterations']
except:
fit2 = DEFAULT_NITER
try:
cit1 = kwargs1['C_iterations']
except:
cit1 = DEFAULT_NITER
try:
cit2 = kwargs2['C_iterations']
except:
cit2 = DEFAULT_NITER
if ((fit1 == fit2) and (cit1 == cit2)):
pass
else:
ml.symmetric_smoothing = False
else:
ml.symmetric_smoothing = False
elif fn1 in KRYLOV_RELAXATION or fn2 in KRYLOV_RELAXATION:
ml.symmetric_smoothing = False
elif fn1 not in SYMMETRIC_RELAXATION:
if ('CF' in fn1) or ('FC' in fn1):
ml.symmetric_smoothing = False
else:
try:
sweep1 = kwargs1['sweep']
except:
sweep1 = DEFAULT_SWEEP
try:
sweep2 = kwargs2['sweep']
except:
sweep2 = DEFAULT_SWEEP
if (sweep1 == 'forward' and sweep2 == 'backward') or \
(sweep1 == 'backward' and sweep2 == 'forward') or \
(sweep1 == 'symmetric' and sweep2 == 'symmetric'):
pass
else:
ml.symmetric_smoothing = False
else:
mid_len = min_len
# Fill in remaining levels
for i in range(mid_len, len(ml.levels[:-1])):
ml.levels[i].presmoother = setup_presmoother(ml.levels[i], **kwargs1)
ml.levels[i].postsmoother = setup_postsmoother(ml.levels[i], **kwargs2)
ml.levels[i].smoothers['presmoother'] = [fn1, kwargs1]
ml.levels[i].smoothers['postsmoother'] = [fn2, kwargs2]
def extend_hierarchy(levels, strength, CF, interpolation, restriction, keep):
""" helper function for local methods """
A = levels[-1].A
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == 'ode') or (fn == 'evolution'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] = kwargs['cost'][0]
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = split.RS(C, **kwargs)
elif fn == 'PMIS':
splitting = split.PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = split.PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = split.CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = split.CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] = kwargs['cost'][0]
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
fn, kwargs = unpack_arg(interpolation)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'injection':
P = injection_interpolation(A, splitting, **kwargs)
else:
raise ValueError('unknown interpolation method (%s)' % interpolation)
levels[-1].complexity['interpolate'] = kwargs['cost'][0]
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid. Must make sure transpose matrices remain in CSR or BSR
fn, kwargs = unpack_arg(restriction)
if isspmatrix_csr(A):
if restriction == 'galerkin':
R = P.T.tocsr()
elif fn == 'standard':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown interpolation method (%s)' %
interpolation)
else:
if restriction == 'galerkin':
R = P.T.tobsr()
elif fn == 'standard':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'distance_two':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'direct':
temp_A = A.T.tobsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
elif fn == 'injection': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
R = R.T.tobsr()
else:
raise ValueError('unknown interpolation method (%s)' %
interpolation)
levels[-1].complexity['restriction'] = kwargs['cost'][0]
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# Form coarse grid operator, get complexity
levels[-1].complexity['RAP'] = mat_mat_complexity(R, A) / float(A.nnz)
RA = R * A
levels[-1].complexity['RAP'] += mat_mat_complexity(RA, P) / float(A.nnz)
A = RA * P # Galerkin operator, Ac = RAP
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
# Form next level through Galerkin product
levels.append(multilevel_solver.level())
levels[-1].A = A
def general_setup_stage(ml, symmetry, candidate_iters, prepostsmoother,
smooth, eliminate_local, coarse_solver, work):
"""
Computes additional candidates and improvements
following Algorithm 4 in Brezina et al.
Parameters
----------
candidate_iters
number of test relaxation iterations
epsilon
minimum acceptable relaxation convergence factor
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation (alphaSA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
def make_bridge(T):
M, N = T.shape
K = T.blocksize[0]
bnnz = T.indptr[-1]
# the K+1 represents the new dof introduced by the new candidate. the
# bridge 'T' ignores this new dof and just maps zeros there
data = np.zeros((bnnz, K+1, K), dtype=T.dtype)
data[:, :-1, :] = T.data
return bsr_matrix((data, T.indices, T.indptr),
shape=((K + 1) * int(M / K), N))
def expand_candidates(B_old, nodesize):
# insert a new dof that is always zero, to create NullDim+1 dofs per
# node in B
NullDim = B_old.shape[1]
nnodes = int(B_old.shape[0] / nodesize)
Bnew = np.zeros((nnodes, nodesize+1, NullDim), dtype=B_old.dtype)
Bnew[:, :-1, :] = B_old.reshape(nnodes, nodesize, NullDim)
return Bnew.reshape(-1, NullDim)
levels = ml.levels
x = sp.rand(levels[0].A.shape[0], 1)
if levels[0].A.dtype.name.startswith('complex'):
x = x + 1.0j*sp.rand(levels[0].A.shape[0], 1)
b = np.zeros_like(x)
x = ml.solve(b, x0=x, tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += ml.operator_complexity()*ml.levels[0].A.nnz*candidate_iters*2
T0 = levels[0].T.copy()
# TEST FOR CONVERGENCE HERE
for i in range(len(ml.levels) - 2):
# alpha-SA paper does local elimination here, but after talking
# to Marian, its not clear that this helps things
# fn, kwargs = unpack_arg(eliminate_local)
# if fn == True:
# eliminate_local_candidates(x,levels[i].AggOp,levels[i].A,
# levels[i].T, **kwargs)
# add candidate to B
B = np.hstack((levels[i].B, x.reshape(-1, 1)))
# construct Ptent
T, R = fit_candidates(levels[i].AggOp, B)
levels[i].T = T
x = R[:, -1].reshape(-1, 1)
# smooth P
fn, kwargs = unpack_arg(smooth[i])
if fn == 'jacobi':
levels[i].P = jacobi_prolongation_smoother(levels[i].A, T,
levels[i].C, R,
**kwargs)
elif fn == 'richardson':
levels[i].P = richardson_prolongation_smoother(levels[i].A, T,
**kwargs)
elif fn == 'energy':
levels[i].P = energy_prolongation_smoother(levels[i].A, T,
levels[i].C, R, None,
(False, {}), **kwargs)
x = R[:, -1].reshape(-1, 1)
elif fn is None:
levels[i].P = T
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i].R = levels[i].P.T.asformat(levels[i].P.format)
elif symmetry == 'hermitian':
levels[i].R = levels[i].P.H.asformat(levels[i].P.format)
# construct coarse A
levels[i+1].A = levels[i].R * levels[i].A * levels[i].P
# construct bridging P
T_bridge = make_bridge(levels[i+1].T)
R_bridge = levels[i+2].B
# smooth bridging P
fn, kwargs = unpack_arg(smooth[i+1])
if fn == 'jacobi':
levels[i+1].P = jacobi_prolongation_smoother(levels[i+1].A,
T_bridge,
levels[i+1].C,
R_bridge, **kwargs)
elif fn == 'richardson':
levels[i+1].P = richardson_prolongation_smoother(levels[i+1].A,
T_bridge,
**kwargs)
elif fn == 'energy':
levels[i+1].P = energy_prolongation_smoother(levels[i+1].A,
T_bridge,
levels[i+1].C,
R_bridge, None,
(False, {}), **kwargs)
elif fn is None:
levels[i+1].P = T_bridge
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct the "bridging" R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i+1].R = levels[i+1].P.T.asformat(levels[i+1].P.format)
elif symmetry == 'hermitian':
levels[i+1].R = levels[i+1].P.H.asformat(levels[i+1].P.format)
# run solver on candidate
solver = multilevel_solver(levels[i+1:], coarse_solver=coarse_solver)
change_smoothers(solver, presmoother=prepostsmoother,
postsmoother=prepostsmoother)
x = solver.solve(np.zeros_like(x), x0=x,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += 2 * solver.operator_complexity() * solver.levels[0].A.nnz *\
candidate_iters*2
# update values on next level
levels[i+1].B = R[:, :-1].copy()
levels[i+1].T = T_bridge
# note that we only use the x from the second coarsest level
fn, kwargs = unpack_arg(prepostsmoother, cost=False)
for lvl in reversed(levels[:-2]):
x = lvl.P * x
work[:] += lvl.A.nnz*candidate_iters*2
if fn == 'gauss_seidel':
# only relax at nonzeros, so as not to mess up any locally dropped
# candidates
indices = np.ravel(x).nonzero()[0]
gauss_seidel_indexed(lvl.A, x, np.zeros_like(x), indices,
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(lvl.A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(lvl.A, x, np.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(lvl.A, x, np.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(lvl.A))
elif fn == 'richardson':
polynomial(lvl.A, x, np.zeros_like(x), iterations=1,
coefficients=[1.0/approximate_spectral_radius(lvl.A)])
elif fn == 'gmres':
x[:] = (gmres(lvl.A, np.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
# x will be dense again, so we have to drop locally again
elim, elim_kwargs = unpack_arg(eliminate_local)
if elim is True:
x = x/norm(x, 'inf')
eliminate_local_candidates(x, levels[0].AggOp, levels[0].A, T0,
**elim_kwargs)
return x.reshape(-1, 1)
def general_setup_stage(ml, symmetry, candidate_iters, prepostsmoother, smooth,
eliminate_local, coarse_solver, work):
"""
Computes additional candidates and improvements
following Algorithm 4 in Brezina et al.
Parameters
----------
candidate_iters
number of test relaxation iterations
epsilon
minimum acceptable relaxation convergence factor
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation (alphaSA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
def make_bridge(T):
M, N = T.shape
K = T.blocksize[0]
bnnz = T.indptr[-1]
# the K+1 represents the new dof introduced by the new candidate. the
# bridge 'T' ignores this new dof and just maps zeros there
data = np.zeros((bnnz, K + 1, K), dtype=T.dtype)
data[:, :-1, :] = T.data
return bsr_matrix((data, T.indices, T.indptr),
shape=((K + 1) * int(M / K), N))
def expand_candidates(B_old, nodesize):
# insert a new dof that is always zero, to create NullDim+1 dofs per
# node in B
NullDim = B_old.shape[1]
nnodes = int(B_old.shape[0] / nodesize)
Bnew = np.zeros((nnodes, nodesize + 1, NullDim), dtype=B_old.dtype)
Bnew[:, :-1, :] = B_old.reshape(nnodes, nodesize, NullDim)
return Bnew.reshape(-1, NullDim)
levels = ml.levels
x = sp.rand(levels[0].A.shape[0], 1)
if levels[0].A.dtype.name.startswith('complex'):
x = x + 1.0j * sp.rand(levels[0].A.shape[0], 1)
b = np.zeros_like(x)
x = ml.solve(b,
x0=x,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += ml.operator_complexity(
) * ml.levels[0].A.nnz * candidate_iters * 2
T0 = levels[0].T.copy()
# TEST FOR CONVERGENCE HERE
for i in range(len(ml.levels) - 2):
# alpha-SA paper does local elimination here, but after talking
# to Marian, its not clear that this helps things
# fn, kwargs = unpack_arg(eliminate_local)
# if fn == True:
# eliminate_local_candidates(x,levels[i].AggOp,levels[i].A,
# levels[i].T, **kwargs)
# add candidate to B
B = np.hstack((levels[i].B, x.reshape(-1, 1)))
# construct Ptent
T, R = fit_candidates(levels[i].AggOp, B)
levels[i].T = T
x = R[:, -1].reshape(-1, 1)
# smooth P
fn, kwargs = unpack_arg(smooth[i])
if fn == 'jacobi':
levels[i].P = jacobi_prolongation_smoother(levels[i].A, T,
levels[i].C, R,
**kwargs)
elif fn == 'richardson':
levels[i].P = richardson_prolongation_smoother(
levels[i].A, T, **kwargs)
elif fn == 'energy':
levels[i].P = energy_prolongation_smoother(levels[i].A, T,
levels[i].C, R, None,
(False, {}), **kwargs)
x = R[:, -1].reshape(-1, 1)
elif fn is None:
levels[i].P = T
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i].R = levels[i].P.T.asformat(levels[i].P.format)
elif symmetry == 'hermitian':
levels[i].R = levels[i].P.H.asformat(levels[i].P.format)
# construct coarse A
levels[i + 1].A = levels[i].R * levels[i].A * levels[i].P
# construct bridging P
T_bridge = make_bridge(levels[i + 1].T)
R_bridge = levels[i + 2].B
# smooth bridging P
fn, kwargs = unpack_arg(smooth[i + 1])
if fn == 'jacobi':
levels[i + 1].P = jacobi_prolongation_smoother(
levels[i + 1].A, T_bridge, levels[i + 1].C, R_bridge, **kwargs)
elif fn == 'richardson':
levels[i + 1].P = richardson_prolongation_smoother(
levels[i + 1].A, T_bridge, **kwargs)
elif fn == 'energy':
levels[i + 1].P = energy_prolongation_smoother(
levels[i + 1].A, T_bridge, levels[i + 1].C, R_bridge, None,
(False, {}), **kwargs)
elif fn is None:
levels[i + 1].P = T_bridge
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
# construct the "bridging" R
if symmetry == 'symmetric': # R should reflect A's structure
levels[i + 1].R = levels[i + 1].P.T.asformat(levels[i +
1].P.format)
elif symmetry == 'hermitian':
levels[i + 1].R = levels[i + 1].P.H.asformat(levels[i +
1].P.format)
# run solver on candidate
solver = multilevel_solver(levels[i + 1:], coarse_solver=coarse_solver)
change_smoothers(solver,
presmoother=prepostsmoother,
postsmoother=prepostsmoother)
x = solver.solve(np.zeros_like(x),
x0=x,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters)
work[:] += 2 * solver.operator_complexity() * solver.levels[0].A.nnz *\
candidate_iters*2
# update values on next level
levels[i + 1].B = R[:, :-1].copy()
levels[i + 1].T = T_bridge
# note that we only use the x from the second coarsest level
fn, kwargs = unpack_arg(prepostsmoother, cost=False)
for lvl in reversed(levels[:-2]):
x = lvl.P * x
work[:] += lvl.A.nnz * candidate_iters * 2
if fn == 'gauss_seidel':
# only relax at nonzeros, so as not to mess up any locally dropped
# candidates
indices = np.ravel(x).nonzero()[0]
gauss_seidel_indexed(lvl.A,
x,
np.zeros_like(x),
indices,
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(lvl.A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(lvl.A,
x,
np.zeros_like(x),
iterations=candidate_iters,
sweep='symmetric')
elif fn == 'jacobi':
jacobi(lvl.A,
x,
np.zeros_like(x),
iterations=1,
omega=1.0 / rho_D_inv_A(lvl.A))
elif fn == 'richardson':
polynomial(lvl.A,
x,
np.zeros_like(x),
iterations=1,
coefficients=[1.0 / approximate_spectral_radius(lvl.A)])
elif fn == 'gmres':
x[:] = (gmres(lvl.A,
np.zeros_like(x),
x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
# x will be dense again, so we have to drop locally again
elim, elim_kwargs = unpack_arg(eliminate_local)
if elim is True:
x = x / norm(x, 'inf')
eliminate_local_candidates(x, levels[0].AggOp, levels[0].A, T0,
**elim_kwargs)
return x.reshape(-1, 1)
def extend_hierarchy(levels, strength, CF, interp, restrict, filter_operator,
coarse_grid_P, coarse_grid_R, keep):
""" helper function for local methods """
# Filter operator. Need to keep original matrix on fineest level for
# computing residuals
if (filter_operator is not None) and (filter_operator[1] != 0):
if len(levels) == 1:
A = deepcopy(levels[-1].A)
else:
A = levels[-1].A
filter_matrix_rows(A,
filter_operator[1],
diagonal=True,
lump=filter_operator[0])
else:
A = levels[-1].A
# Check if matrix was filtered to be diagonal --> coarsest grid
if A.nnz == A.shape[0]:
return 1
# Zero initial complexities for strength, splitting and interpolation
levels[-1].complexity['CF'] = 0.0
levels[-1].complexity['strength'] = 0.0
levels[-1].complexity['interpolate'] = 0.0
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength)
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == 'ode') or (fn == 'evolution'):
C = evolution_strength_of_connection(A, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] += kwargs['cost'][0] * A.nnz / float(
A.nnz)
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = RS(C, **kwargs)
elif fn == 'PMIS':
splitting = PMIS(C, **kwargs)
elif fn == 'PMISc':
splitting = PMISc(C, **kwargs)
elif fn == 'CLJP':
splitting = CLJP(C, **kwargs)
elif fn == 'CLJPc':
splitting = CLJPc(C, **kwargs)
elif fn == 'CR':
splitting = CR(C, **kwargs)
elif fn == 'weighted_matching':
splitting, soc = weighted_matching(C, **kwargs)
if soc is not None:
C = soc
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
levels[-1].complexity['CF'] += kwargs['cost'][0] * C.nnz / float(A.nnz)
temp = np.sum(splitting)
if (temp == len(splitting)) or (temp == 0):
return 1
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
r_flag = False
fn, kwargs = unpack_arg(interp)
if fn == 'standard':
P = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'scaledAfc':
P = scaled_Afc_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = bsr_matrix(P.T)
else:
temp_A = csr_matrix(A.T)
P = local_AIR(temp_A, splitting, **kwargs)
P = csr_matrix(P.T)
elif fn == 'restrict':
r_flag = True
else:
raise ValueError('unknown interpolation method (%s)' % interp)
levels[-1].complexity['interpolate'] += kwargs['cost'][0] * A.nnz / float(
A.nnz)
# Build restriction operator
fn, kwargs = unpack_arg(restrict)
if fn is None:
R = P.T
elif fn == 'air':
R = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R = R.T.tobsr()
else:
R = R.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = standard_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = distance_two_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R = R.T.tocsr()
else:
raise ValueError('unknown restriction method (%s)' % restrict)
# If set P = R^T
if r_flag:
P = R.T
# Optional different interpolation for RAP
fn, kwargs = unpack_arg(coarse_grid_P)
if fn == 'standard':
P_temp = standard_interpolation(A, C, splitting, **kwargs)
elif fn == 'distance_two':
P_temp = distance_two_interpolation(A, C, splitting, **kwargs)
elif fn == 'direct':
P_temp = direct_interpolation(A, C, splitting, **kwargs)
elif fn == 'one_point':
P_temp = one_point_interpolation(A, C, splitting, **kwargs)
elif fn == 'inject':
P_temp = injection_interpolation(A, splitting, **kwargs)
elif fn == 'neumann':
P_temp = neumann_ideal_interpolation(A, splitting, **kwargs)
elif fn == 'air':
if isspmatrix_bsr(A):
temp_A = bsr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = bsr_matrix(P_temp.T)
else:
temp_A = csr_matrix(A.T)
P_temp = local_AIR(temp_A, splitting, **kwargs)
P_temp = csr_matrix(P_temp.T)
else:
P_temp = P
# Optional different restriction for RAP
fn, kwargs = unpack_arg(coarse_grid_R)
if fn == 'air':
R_temp = local_AIR(A, splitting, **kwargs)
elif fn == 'neumann':
R_temp = neumann_AIR(A, splitting, **kwargs)
elif fn == 'one_point': # Don't need A^T here
temp_C = C.T.tocsr()
R_temp = one_point_interpolation(A, temp_C, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'inject': # Don't need A^T or C^T here
R_temp = injection_interpolation(A, splitting, **kwargs)
if isspmatrix_bsr(A):
R_temp = R_temp.T.tobsr()
else:
R_temp = R_temp.T.tocsr()
elif fn == 'standard':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = standard_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'distance_two':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = distance_two_interpolation(temp_A, temp_C, splitting,
**kwargs)
R_temp = R_temp.T.tocsr()
elif fn == 'direct':
if isspmatrix_bsr(A):
temp_A = A.T.tobsr()
temp_C = C.T.tobsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tobsr()
else:
temp_A = A.T.tocsr()
temp_C = C.T.tocsr()
R_temp = direct_interpolation(temp_A, temp_C, splitting, **kwargs)
R_temp = R_temp.T.tocsr()
else:
R_temp = R
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels[-1].splitting = splitting # C/F splitting
# Form coarse grid operator, get complexity
#levels[-1].complexity['RAP'] = mat_mat_complexity(R_temp,A) / float(A.nnz)
#RA = R_temp * A
#levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P_temp) / float(A.nnz)
#A = RA * P_temp
# RL: RAP = R*(A*P)
levels[-1].complexity['RAP'] = mat_mat_complexity(A, P_temp) / float(A.nnz)
AP = A * P_temp
levels[-1].complexity['RAP'] += mat_mat_complexity(R_temp, AP) / float(
A.nnz)
A = R_temp * AP
# Make sure coarse-grid operator is in correct sparse format
if (isspmatrix_csr(P) and (not isspmatrix_csr(A))):
A = A.tocsr()
elif (isspmatrix_bsr(P) and (not isspmatrix_bsr(A))):
A = A.tobsr()
A.eliminate_zeros()
levels.append(multilevel_solver.level())
levels[-1].A = A
return 0
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates,
diagonal_dominance=False, keep=True):
"""Service routine to implement the strength of connection, aggregation,
tentative prolongation construction, and prolongation smoothing. Called by
smoothed_aggregation_solver.
"""
A = levels[-1].A
B = levels[-1].B
if A.symmetry == "nonsymmetric":
AH = A.H.asformat(A.format)
BH = levels[-1].BH
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength[len(levels)-1])
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == 'ode') or (fn == 'evolution'):
if 'B' in kwargs:
C = evolution_strength_of_connection(A, **kwargs)
else:
C = evolution_strength_of_connection(A, B, **kwargs)
elif fn == 'energy_based':
C = energy_based_strength_of_connection(A, **kwargs)
elif fn == 'predefined':
C = kwargs['C'].tocsr()
elif fn == 'algebraic_distance':
C = algebraic_distance(A, **kwargs)
elif fn == 'affinity':
C = affinity_distance(A, **kwargs)
elif fn is None:
C = A.tocsr()
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
levels[-1].complexity['strength'] = kwargs['cost'][0]
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
levels[-1].complexity['diag_dom'] = kwargs['cost'][0]
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate[len(levels)-1])
if fn == 'standard':
AggOp = standard_aggregation(C, **kwargs)[0]
elif fn == 'naive':
AggOp = naive_aggregation(C, **kwargs)[0]
elif fn == 'lloyd':
AggOp = lloyd_aggregation(C, **kwargs)[0]
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
else:
raise ValueError('unrecognized aggregation method %s' % str(fn))
levels[-1].complexity['aggregation'] = kwargs['cost'][0] * (float(C.nnz)/A.nnz)
# Improve near nullspace candidates by relaxing on A B = 0
temp_cost = [0.0]
fn, kwargs = unpack_arg(improve_candidates[len(levels)-1], cost=False)
if fn is not None:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
B = relaxation_as_linear_operator((fn, kwargs), A, b, temp_cost) * B
levels[-1].B = B
if A.symmetry == "nonsymmetric":
BH = relaxation_as_linear_operator((fn, kwargs), AH, b, temp_cost) * BH
levels[-1].BH = BH
levels[-1].complexity['candidates'] = temp_cost[0] * B.shape[1]
# Compute the tentative prolongator, T, which is a tentative interpolation
# matrix from the coarse-grid to the fine-grid. T exactly interpolates
# B_fine = T B_coarse. Orthogonalization complexity ~ 2nk^2, k=B.shape[1].
temp_cost=[0.0]
T, B = fit_candidates(AggOp, B, cost=temp_cost)
if A.symmetry == "nonsymmetric":
TH, BH = fit_candidates(AggOp, BH, cost=temp_cost)
levels[-1].complexity['tentative'] = temp_cost[0]/A.nnz
# Smooth the tentative prolongator, so that it's accuracy is greatly
# improved for algebraically smooth error.
fn, kwargs = unpack_arg(smooth[len(levels)-1])
if fn == 'jacobi':
P = jacobi_prolongation_smoother(A, T, C, B, **kwargs)
elif fn == 'richardson':
P = richardson_prolongation_smoother(A, T, **kwargs)
elif fn == 'energy':
P = energy_prolongation_smoother(A, T, C, B, None, (False, {}),
**kwargs)
elif fn is None:
P = T
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
levels[-1].complexity['smooth_P'] = kwargs['cost'][0]
# Compute the restriction matrix, R, which interpolates from the fine-grid
# to the coarse-grid. If A is nonsymmetric, then R must be constructed
# based on A.H. Otherwise R = P.H or P.T.
symmetry = A.symmetry
if symmetry == 'hermitian':
R = P.H
elif symmetry == 'symmetric':
R = P.T
elif symmetry == 'nonsymmetric':
fn, kwargs = unpack_arg(smooth[len(levels)-1])
if fn == 'jacobi':
R = jacobi_prolongation_smoother(AH, TH, C, BH, **kwargs).H
elif fn == 'richardson':
R = richardson_prolongation_smoother(AH, TH, **kwargs).H
elif fn == 'energy':
R = energy_prolongation_smoother(AH, TH, C, BH, None, (False, {}),
**kwargs)
R = R.H
elif fn is None:
R = T.H
else:
raise ValueError('unrecognized prolongation smoother method %s' %
str(fn))
levels[-1].complexity['smooth_R'] = kwargs['cost'][0]
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].AggOp = AggOp # aggregation operator
levels[-1].T = T # tentative prolongator
levels[-1].P = P # smoothed prolongator
levels[-1].R = R # restriction operator
# Form coarse grid operator, get complexity
levels[-1].complexity['RAP'] = mat_mat_complexity(R,A) / float(A.nnz)
RA = R * A
levels[-1].complexity['RAP'] += mat_mat_complexity(RA,P) / float(A.nnz)
A = RA * P # Galerkin operator, Ac = RAP
A.symmetry = symmetry
levels.append(multilevel_solver.level())
levels[-1].A = A
levels[-1].B = B # right near nullspace candidates
if A.symmetry == "nonsymmetric":
levels[-1].BH = BH # left near nullspace candidates
def adaptive_sa_solver(A, B=None, symmetry='hermitian',
pdef=True, num_candidates=1, candidate_iters=5,
improvement_iters=0, epsilon=0.1,
max_levels=10, max_coarse=10, aggregate='standard',
prepostsmoother=('gauss_seidel',
{'sweep': 'symmetric'}),
smooth=('jacobi', {}), strength='symmetric',
coarse_solver='pinv2',
eliminate_local=(False, {'Ca': 1.0}), keep=False,
**kwargs):
"""
Create a multilevel solver using Adaptive Smoothed Aggregation (aSA)
Parameters
----------
A : {csr_matrix, bsr_matrix}
Square matrix in CSR or BSR format
B : {None, n x m dense matrix}
If a matrix, then this forms the basis for the first m candidates.
Also in this case, the initial setup stage is skipped, because this
provides the first candidate(s). If None, then a random initial guess
and relaxation are used to inform the initial candidate.
symmetry : {string}
'symmetric' refers to both real and complex symmetric
'hermitian' refers to both complex Hermitian and real Hermitian
Note that for the strictly real case, these two options are the same
Note that this flag does not denote definiteness of the operator
pdef : {bool}
True or False, whether A is known to be positive definite.
num_candidates : {integer} : default 1
Number of near-nullspace candidates to generate
candidate_iters : {integer} : default 5
Number of smoothing passes/multigrid cycles used at each level of
the adaptive setup phase
improvement_iters : {integer} : default 0
Number of times each candidate is improved
epsilon : {float} : default 0.1
Target convergence factor
max_levels : {integer} : default 10
Maximum number of levels to be used in the multilevel solver.
max_coarse : {integer} : default 500
Maximum number of variables permitted on the coarse grid.
prepostsmoother : {string or dict}
Pre- and post-smoother used in the adaptive method
strength : ['symmetric', 'classical', 'evolution',
('predefined', {'C': csr_matrix}), None]
Method used to determine the strength of connection between unknowns of
the linear system. See smoothed_aggregation_solver(...) documentation.
aggregate : ['standard', 'lloyd', 'naive',
('predefined', {'AggOp': csr_matrix})]
Method used to aggregate nodes. See smoothed_aggregation_solver(...)
documentation.
smooth : ['jacobi', 'richardson', 'energy', None]
Method used used to smooth the tentative prolongator. See
smoothed_aggregation_solver(...) documentation
coarse_solver : ['splu', 'lu', 'cholesky, 'pinv', 'gauss_seidel', ... ]
Solver used at the coarsest level of the MG hierarchy.
Optionally, may be a tuple (fn, args), where fn is a string such as
['splu', 'lu', ...] or a callable function, and args is a dictionary of
arguments to be passed to fn.
eliminate_local : {tuple}
Length 2 tuple. If the first entry is True, then eliminate candidates
where they aren't needed locally, using the second entry of the tuple
to contain arguments to local elimination routine. Given the rigid
sparse data structures, this doesn't help much, if at all, with
complexity. Its more of a diagnostic utility.
keep: {bool} : default False
Flag to indicate keeping extra operators in the hierarchy for
diagnostics. For example, if True, then strength of connection (C),
tentative prolongation (T), and aggregation (AggOp) are kept.
Returns
-------
multilevel_solver : multilevel_solver
Smoothed aggregation solver with adaptively generated candidates
Notes
-----
- Floating point value representing the "work" required to generate
the solver. This value is the total cost of just relaxation, relative
to the fine grid. The relaxation method used is assumed to symmetric
Gauss-Seidel.
- Unlike the standard Smoothed Aggregation (SA) method, adaptive SA does
not require knowledge of near-nullspace candidate vectors. Instead, an
adaptive procedure computes one or more candidates 'from scratch'. This
approach is useful when no candidates are known or the candidates have
been invalidated due to changes to matrix A.
Examples
--------
>>> from pyamg.gallery import stencil_grid
>>> from pyamg.aggregation import adaptive_sa_solver
>>> import numpy as np
>>> A=stencil_grid([[-1,-1,-1],[-1,8.0,-1],[-1,-1,-1]],\
(31,31),format='csr')
>>> [asa,work] = adaptive_sa_solver(A,num_candidates=1)
>>> residuals=[]
>>> x=asa.solve(b=np.ones((A.shape[0],)), x0=np.ones((A.shape[0],)),\
residuals=residuals)
References
----------
.. [1] Brezina, Falgout, MacLachlan, Manteuffel, McCormick, and Ruge
"Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid"
SIAM Review Volume 47, Issue 2 (2005)
http://www.cs.umn.edu/~maclach/research/aSA2.pdf
"""
if not (isspmatrix_csr(A) or isspmatrix_bsr(A)):
try:
A = csr_matrix(A)
warn("Implicit conversion of A to CSR", SparseEfficiencyWarning)
except:
raise TypeError('Argument A must have type csr_matrix or\
bsr_matrix, or be convertible to csr_matrix')
A = A.asfptype()
if A.shape[0] != A.shape[1]:
raise ValueError('expected square matrix')
# Track work in terms of relaxation
work = np.zeros((1,))
# Levelize the user parameters, so that they become lists describing the
# desired user option on each level.
max_levels, max_coarse, strength =\
levelize_strength_or_aggregation(strength, max_levels, max_coarse)
max_levels, max_coarse, aggregate =\
levelize_strength_or_aggregation(aggregate, max_levels, max_coarse)
smooth = levelize_smooth_or_improve_candidates(smooth, max_levels)
# Develop initial candidate(s). Note that any predefined aggregation is
# preserved.
if B is None:
B, aggregate, strength =\
initial_setup_stage(A, symmetry, pdef, candidate_iters, epsilon,
max_levels, max_coarse, aggregate,
prepostsmoother, smooth, strength, work)
# Normalize B
B = (1.0/norm(B, 'inf')) * B
num_candidates -= 1
else:
# Otherwise, use predefined candidates
num_candidates -= B.shape[1]
# Generate Aggregation and Strength Operators (the brute force way)
sa = smoothed_aggregation_solver(A, B=B, symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=True, **kwargs)
if len(sa.levels) > 1:
# Set strength-of-connection and aggregation
aggregate = [('predefined', {'AggOp': sa.levels[i].AggOp.tocsr()})
for i in range(len(sa.levels) - 1)]
strength = [('predefined', {'C': sa.levels[i].C.tocsr()})
for i in range(len(sa.levels) - 1)]
# Develop additional candidates
for i in range(num_candidates):
x = general_setup_stage(
smoothed_aggregation_solver(A, B=B, symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=True, **kwargs),
symmetry, candidate_iters, prepostsmoother, smooth,
eliminate_local, coarse_solver, work)
# Normalize x and add to candidate list
x = x/norm(x, 'inf')
if np.isinf(x[0]) or np.isnan(x[0]):
raise ValueError('Adaptive candidate is all 0.')
B = np.hstack((B, x.reshape(-1, 1)))
# Improve candidates
if B.shape[1] > 1 and improvement_iters > 0:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
for i in range(improvement_iters):
for j in range(B.shape[1]):
# Run a V-cycle built on everything except candidate j, while
# using candidate j as the initial guess
x0 = B[:, 0]
B = B[:, 1:]
sa_temp =\
smoothed_aggregation_solver(A, B=B, symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=True, **kwargs)
x = sa_temp.solve(b, x0=x0,
tol=float(np.finfo(np.float).tiny),
maxiter=candidate_iters, cycle='V')
work[:] += 2 * sa_temp.operator_complexity() *\
sa_temp.levels[0].A.nnz * candidate_iters
# Apply local elimination
elim, elim_kwargs = unpack_arg(eliminate_local)
if elim is True:
x = x/norm(x, 'inf')
eliminate_local_candidates(x, sa_temp.levels[0].AggOp, A,
sa_temp.levels[0].T,
**elim_kwargs)
# Normalize x and add to candidate list
x = x/norm(x, 'inf')
if np.isinf(x[0]) or np.isnan(x[0]):
raise ValueError('Adaptive candidate is all 0.')
B = np.hstack((B, x.reshape(-1, 1)))
elif improvement_iters > 0:
# Special case for improving a single candidate
max_levels = len(aggregate) + 1
max_coarse = 0
for i in range(improvement_iters):
B, aggregate, strength =\
initial_setup_stage(A, symmetry, pdef, candidate_iters,
epsilon, max_levels, max_coarse,
aggregate, prepostsmoother, smooth,
strength, work, B=B)
# Normalize B
B = (1.0/norm(B, 'inf'))*B
# Return smoother. Note, solver parameters are passed in via params
# argument to be stored in final solver hierarchy and used to estimate
# complexity.
return [smoothed_aggregation_solver(A, B=B, symmetry=symmetry,
presmoother=prepostsmoother,
postsmoother=prepostsmoother,
smooth=smooth,
strength=strength,
max_levels=max_levels,
max_coarse=max_coarse,
aggregate=aggregate,
coarse_solver=coarse_solver,
improve_candidates=None,
keep=keep,
**kwargs),
work[0]/A.nnz]