def get_aggregate(A, strength, aggregate, diagonal_dominance, B, **kwargs):
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
# 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'):
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))
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate)
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 == 'pairwise':
AggOp = pairwise_aggregation(A, B, **kwargs)[0]
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
else:
raise ValueError('unrecognized aggregation method %s' % str(fn))
return AggOp
def test_direct_interpolation(self):
for A in self.cases:
S = classical_strength_of_connection(A, 0.0)
splitting = split.RS(S)
result = direct_interpolation(A, S, splitting)
expected = reference_direct_interpolation(A, S, splitting)
assert_almost_equal(result.todense(), expected.todense())
def test_direct_interpolation(self):
for A in self.cases:
S = classical_strength_of_connection(A, 0.0)
splitting = split.RS( S )
result = direct_interpolation(A,S,splitting)
expected = reference_direct_interpolation( A, S, splitting )
assert_almost_equal( result.todense(), expected.todense() )
def test_cljpc_splitting(self):
for A in self.cases:
S = classical_strength_of_connection(A, 0.0)
splitting = split.CLJPc(S)
assert (splitting.min() >= 0) # could be all 1s
assert_equal(splitting.max(), 1)
S.data[:] = 1
# check that all F-nodes are strongly connected to a C-node
assert ((splitting + S * splitting).min() > 0)
def test_cljpc_splitting(self):
for A in self.cases:
S = classical_strength_of_connection(A, 0.0)
splitting = split.CLJPc( S )
assert( splitting.min() >= 0 ) #could be all 1s
assert_equal( splitting.max(), 1 )
S.data[:] = 1
# check that all F-nodes are strongly connected to a C-node
assert( (splitting + S*splitting).min() > 0 )
def test_standard_interpolation(self):
for A in self.cases:
# the reference code is very slow, so just take a small block of A
mini = min(100, A.shape[0])
A = ((A.tocsr()[0:mini, :])[:,0:mini]).tocsr()
S = classical_strength_of_connection(A, 0.0)
splitting = split.RS(S)
result = standard_interpolation(A, S, splitting)
expected = reference_standard_interpolation(A, S, splitting)
# elasticity produces large entries, so normalize
Diff = result - expected
Diff.data = abs(Diff.data)
expected.data = 1./abs(expected.data)
Diff = Diff.multiply(expected)
Diff.data[ Diff.data < 1e-7] = 0.0
Diff.eliminate_zeros()
assert( Diff.nnz == 0)
def test_classical(self):
for A in self.cases:
# theta = 0, no entries dropped
theta = 0.0
cost = [0]
classical_strength_of_connection(A, theta, cost)
assert_almost_equal(cost[0], 3.0)
for theta in [0.1, 0.25, 0.5, 0.75, 0.9]:
cost = [0]
classical_strength_of_connection(A, theta, cost)
assert (cost[0] <= 3.0)
assert (cost[0] > 0.0)
# theta = 1, only largest entries in each row remain
theta = 1.0
cost = [0]
classical_strength_of_connection(A, theta, cost)
est = 2.0 + float(A.shape[0]) / A.nnz
assert_almost_equal(cost[0], est)
def test_classical(self):
for A in self.cases:
# theta = 0, no entries dropped
theta = 0.0
cost = [0]
classical_strength_of_connection(A, theta, cost)
assert_almost_equal(cost[0], 3.0)
for theta in [0.1,0.25,0.5,0.75,0.9]:
cost = [0]
classical_strength_of_connection(A, theta, cost)
assert(cost[0] <= 3.0)
assert(cost[0] > 0.0)
# theta = 1, only largest entries in each row remain
theta = 1.0
cost = [0]
classical_strength_of_connection(A, theta, cost)
est = 2.0 + float(A.shape[0]) / A.nnz
assert_almost_equal(cost[0],est)
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.
"""
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
A = levels[-1].A
B = levels[-1].B
if A.symmetry == "nonsymmetric":
AH = A.H.asformat(A.format)
BH = levels[-1].BH
# Compute the strength-of-connection matrix C, where larger
# C[i, j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength[len(levels) - 1])
if fn == "symmetric":
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == "classical":
C = classical_strength_of_connection(A, **kwargs)
elif fn == "distance":
C = distance_strength_of_connection(A, **kwargs)
elif (fn == "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 is None:
C = A.tocsr()
else:
raise ValueError("unrecognized strength of connection method: %s" % str(fn))
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate[len(levels) - 1])
if fn == "standard":
AggOp, Cnodes = standard_aggregation(C, **kwargs)
elif fn == "naive":
AggOp, Cnodes = naive_aggregation(C, **kwargs)
elif fn == "lloyd":
AggOp, Cnodes = lloyd_aggregation(C, **kwargs)
elif fn == "predefined":
AggOp = kwargs["AggOp"].tocsr()
Cnodes = kwargs["Cnodes"]
else:
raise ValueError("unrecognized aggregation method %s" % str(fn))
# Improve near nullspace candidates by relaxing on A B = 0
fn, kwargs = unpack_arg(improve_candidates[len(levels) - 1])
if fn is not None:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
B = relaxation_as_linear_operator((fn, kwargs), A, b) * B
levels[-1].B = B
if A.symmetry == "nonsymmetric":
BH = relaxation_as_linear_operator((fn, kwargs), AH, b) * BH
levels[-1].BH = BH
# Compute the tentative prolongator, T, which is a tentative interpolation
# matrix from the coarse-grid to the fine-grid. T exactly interpolates
# B_fine[:, 0:blocksize(A)] = T B_coarse[:, 0:blocksize(A)].
T, dummy = fit_candidates(AggOp, B[:, 0 : blocksize(A)])
del dummy
if A.symmetry == "nonsymmetric":
TH, dummyH = fit_candidates(AggOp, BH[:, 0 : blocksize(A)])
del dummyH
# Create necessary root node matrices
Cpt_params = (True, get_Cpt_params(A, Cnodes, AggOp, T))
T = scale_T(T, Cpt_params[1]["P_I"], Cpt_params[1]["I_F"])
if A.symmetry == "nonsymmetric":
TH = scale_T(TH, Cpt_params[1]["P_I"], Cpt_params[1]["I_F"])
# Set coarse grid near nullspace modes as injected fine grid near
# null-space modes
B = Cpt_params[1]["P_I"].T * levels[-1].B
if A.symmetry == "nonsymmetric":
BH = Cpt_params[1]["P_I"].T * levels[-1].BH
# Smooth the tentative prolongator, so that it's accuracy is greatly
# improved for algebraically smooth error.
fn, kwargs = unpack_arg(smooth[len(levels) - 1])
if fn == "energy":
P = energy_prolongation_smoother(A, T, C, B, levels[-1].B, Cpt_params=Cpt_params, **kwargs)
elif fn is None:
P = T
else:
raise ValueError(
"unrecognized prolongation smoother \
method %s"
% str(fn)
)
# Compute the restriction matrix R, which interpolates from the fine-grid
# to the coarse-grid. If A is nonsymmetric, then R must be constructed
# based on A.H. Otherwise R = P.H or P.T.
symmetry = A.symmetry
if symmetry == "hermitian":
R = P.H
elif symmetry == "symmetric":
R = P.T
elif symmetry == "nonsymmetric":
fn, kwargs = unpack_arg(smooth[len(levels) - 1])
if fn == "energy":
R = energy_prolongation_smoother(AH, TH, C, BH, levels[-1].BH, Cpt_params=Cpt_params, **kwargs)
R = R.H
elif fn is None:
R = T.H
else:
raise ValueError(
"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)
levels.append(multilevel_solver.level())
A = R * A * P # Galerkin operator
A.symmetry = symmetry
levels[-1].A = A
levels[-1].B = B # right near nullspace candidates
if A.symmetry == "nonsymmetric":
levels[-1].BH = BH # left near nullspace candidates
def initial_setup_stage(A,
symmetry,
pdef,
candidate_iters,
epsilon,
max_levels,
max_coarse,
aggregate,
prepostsmoother,
smooth,
strength,
work,
initial_candidate=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)
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 initial_candidate 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(initial_candidate, 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 extend_hierarchy(levels,
strength,
aggregate,
smooth,
improve_candidates,
diagonal_dominance=False,
keep=True):
"""Extend the multigrid hierarchy.
Service routine to implement the strength of connection, aggregation,
tentative prolongation construction, and prolongation smoothing. Called by
smoothed_aggregation_solver.
"""
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
A = levels[-1].A
B = levels[-1].B
if A.symmetry == "nonsymmetric":
AH = A.H.asformat(A.format)
BH = levels[-1].BH
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
fn, kwargs = unpack_arg(strength[len(levels) - 1])
if fn == 'symmetric':
C = symmetric_strength_of_connection(A, **kwargs)
elif fn == 'classical':
C = classical_strength_of_connection(A, **kwargs)
elif fn == 'distance':
C = distance_strength_of_connection(A, **kwargs)
elif (fn == '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))
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate[len(levels) - 1])
if fn == 'standard':
AggOp = 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))
# Improve near nullspace candidates by relaxing on A B = 0
fn, kwargs = unpack_arg(improve_candidates[len(levels) - 1])
if fn is not None:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
B = relaxation_as_linear_operator((fn, kwargs), A, b) * B
levels[-1].B = B
if A.symmetry == "nonsymmetric":
BH = relaxation_as_linear_operator((fn, kwargs), AH, b) * BH
levels[-1].BH = BH
# Compute the tentative prolongator, T, which is a tentative interpolation
# matrix from the coarse-grid to the fine-grid. T exactly interpolates
# B_fine = T B_coarse.
T, B = fit_candidates(AggOp, B)
if A.symmetry == "nonsymmetric":
TH, BH = fit_candidates(AggOp, 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 == '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))
# 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))
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
levels.append(multilevel_solver.level())
A = R * A * P # Galerkin operator
A.symmetry = symmetry
levels[-1].A = A
levels[-1].B = B # right near nullspace candidates
if A.symmetry == "nonsymmetric":
levels[-1].BH = BH # left near nullspace candidates
def extend_hierarchy(levels, strength, CF, keep):
""" helper function for local methods """
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
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 is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = split.RS(C)
elif fn == 'PMIS':
splitting = split.PMIS(C)
elif fn == 'PMISc':
splitting = split.PMISc(C)
elif fn == 'CLJP':
splitting = split.CLJP(C)
elif fn == 'CLJPc':
splitting = split.CLJPc(C)
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
P = direct_interpolation(A, C, splitting)
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid
R = P.T.tocsr()
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].splitting = splitting # C/F splitting
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels.append(multilevel_solver.level())
# Form next level through Galerkin product
A = R * A * P
levels[-1].A = A
def distance_two_interpolation(A,
C,
splitting,
theta=None,
norm='min',
plus_i=True,
cost=[0]):
"""Create prolongator using distance-two AMG interpolation (extended+i interpolaton).
Parameters
----------
A : {csr_matrix}
NxN matrix in CSR format
C : {csr_matrix}
Strength-of-Connection matrix
Must have zero diagonal
splitting : array
C/F splitting stored in an array of length N
theta : float in [0,1), default None
theta value defining strong connections in a classical AMG sense. Provide if
different SOC used for P than for CF-splitting; otherwise, theta = None.
norm : string, default 'abs'
Norm used in redefining classical SOC. Options are 'min' and 'abs' for CSR matrices,
and 'min', 'abs', and 'fro' for BSR matrices. See strength.py for more information.
plus_i : bool, default True
Use "Extended+i" interpolation from [0] as opposed to "Extended" interpolation. Typically
gives better interpolation with minimal added expense.
Returns
-------
P : {csr_matrix}
Prolongator using standard interpolation
References
----------
[0] "Distance-Two Interpolation for Parallel Algebraic Multigrid,"
H. De Sterck, R. Falgout, J. Nolting, U. M. Yang, (2007).
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.classical import standard_interpolation
>>> import numpy as np
>>> A = poisson((5,),format='csr')
>>> splitting = np.array([1,0,1,0,1], dtype='intc')
>>> P = standard_interpolation(A, A, splitting)
>>> print P.todense()
[[ 1. 0. 0. ]
[ 0.5 0.5 0. ]
[ 0. 1. 0. ]
[ 0. 0.5 0.5]
[ 0. 0. 1. ]]
"""
if not isspmatrix_csr(C):
raise TypeError('Expected csr_matrix SOC matrix, C.')
nc = np.sum(splitting)
n = A.shape[0]
# Block BSR format. Transfer A to CSR and the splitting and SOC matrix to have
# DOFs corresponding to CSR A
if isspmatrix_bsr(A):
temp_A = A.tocsr()
splitting0 = splitting * np.ones((A.blocksize[0], 1), dtype='intc')
splitting0 = np.reshape(splitting0, (np.prod(splitting0.shape), ),
order='F')
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
C0 = UnAmal(C0, A.blocksize[0], A.blocksize[1])
else:
C0 = UnAmal(C, A.blocksize[0], A.blocksize[1])
C0 = C0.tocsr()
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(temp_A)
P_indptr = np.empty_like(temp_A.indptr)
amg_core.distance_two_amg_interpolation_pass1(temp_A.shape[0],
C0.indptr, C0.indices,
splitting0, P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=temp_A.dtype)
if plus_i:
amg_core.extended_plusi_interpolation_pass2(
temp_A.shape[0], temp_A.indptr, temp_A.indices, temp_A.data,
C0.indptr, C0.indices, C0.data, splitting0, P_indptr,
P_colinds, P_data)
else:
amg_core.extended_interpolation_pass2(temp_A.shape[0],
temp_A.indptr,
temp_A.indices, temp_A.data,
C0.indptr, C0.indices,
C0.data, splitting0,
P_indptr, P_colinds, P_data)
nc = np.sum(splitting0)
n = A.shape[0]
P = csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
return P.tobsr(blocksize=A.blocksize)
# CSR format
else:
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
else:
C0 = C.copy()
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(A)
P_indptr = np.empty_like(A.indptr)
amg_core.distance_two_amg_interpolation_pass1(A.shape[0], C0.indptr,
C0.indices, splitting,
P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=A.dtype)
if plus_i:
amg_core.extended_plusi_interpolation_pass2(
A.shape[0], A.indptr, A.indices, A.data, C0.indptr, C0.indices,
C0.data, splitting, P_indptr, P_colinds, P_data)
else:
amg_core.extended_interpolation_pass2(A.shape[0], A.indptr,
A.indices, A.data, C0.indptr,
C0.indices, C0.data,
splitting, P_indptr,
P_colinds, P_data)
nc = np.sum(splitting)
n = A.shape[0]
return csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
def weighted_matching(A,
B=None,
theta=0.5,
use_weights=True,
get_SOC=False,
cost=[0.0],
**kwargs):
""" Pairwise aggregation of nodes using Drake approximate
1/2-matching algorithm.
Parameters
----------
A : csr_matrix or bsr_matrix
matrix for linear system.
B : array_like : default None
Right near-nullspace candidates stored in the columns of an NxK array.
If no target vector provided, constant vector is used. In the case of
multiple targets, k>1, only the first is used to construct coarse grid
matrices for pairwise aggregations.
use_weights : {bool} : default True
Optional function handle to compute weights used in the matching,
e.g. a strength of connection routine. Additional arguments for
this routine should be provided in **kwargs.
get_SOC : {bool} : default False
TODO
theta : float
connections deemed "strong" if |a_ij| > theta*|a_ii|
NOTES
-----
- Not implemented for block systems or complex.
+ Need to define how a matching is done nodally.
+ Also must consider what targets are used to form coarse grid
in nodal approach...
+ Drake should be accessible in complex, but not Notay due to the
hard minimum. Is there a < operator overloaded for complex?
Could I overload it perhaps? Probably would do magnitude or something
though, which is not what we want...
REFERENCES
----------
[1] D'Ambra, Pasqua, and Panayot S. Vassilevski. "Adaptive AMG with
coarsening based on compatible weighted matching." Computing and
Visualization in Science 16.2 (2013): 59-76.
[2] Drake, Doratha E., and Stefan Hougardy. "A simple approximation
algorithm for the weighted matching problem." Information Processing
Letters 85.4 (2003): 211-213.
"""
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
if A.dtype == 'complex':
raise TypeError("Not currently implemented for complex.")
if (A.getformat() != 'csr'):
try:
A = A.tocsr()
except:
raise TypeError("Must pass in CSR matrix, or sparse matrix "
"which can be converted to CSR.")
n = A.shape[0]
# If target vectors provided, take first.
if B is not None:
if len(B.shape) == 2:
target = B[:, 0]
else:
target = B[:, ]
else:
target = None
# Compute weights if function provided, otherwise let W = A
if use_weights:
weights = np.empty((A.nnz, ), dtype=A.dtype)
temp_cost = np.ones((1, ), dtype=A.dtype)
if target is None:
amg_core.compute_weights(A.indptr, A.indices, A.data, weights,
temp_cost)
else:
amg_core.compute_weights(A.indptr, A.indices, A.data, weights,
target, temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
else:
weights = A.data
# Get CF splitting
temp_cost = np.ones((1, ), dtype=A.dtype)
splitting = np.empty(n, dtype='int32')
amg_core.drake_CF_matching(A.indptr, A.indices, weights, splitting, theta,
temp_cost)
cost[0] += temp_cost[0] / float(A.nnz)
if get_SOC:
temp = csr_matrix((weights, A.indices, A.indptr), copy=True)
C = classical_strength_of_connection(temp, theta, cost=cost)
return splitting, C
else:
return splitting, None
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 extend_hierarchy(levels, strength, CF, interp, keep):
""" helper function for local methods """
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
A = levels[-1].A
block_starts = levels[-1].block_starts
verts = levels[-1].verts
# If this is a system, apply the unknown approach by coarsening and generating interpolation based on each diagonal block of A
if (block_starts):
A_diag = extract_diagonal_blocks(A, block_starts)
else:
A_diag = [A]
# Compute the strength-of-connection matrix C, where larger
# C[i,j] denote stronger couplings between i and j.
C_diag = []
P_diag = []
splitting = []
next_lvl_block_starts = [0]
block_cnt = 0
for mat in A_diag:
fn, kwargs = unpack_arg(strength)
if fn == 'symmetric':
C_diag.append( symmetric_strength_of_connection(mat, **kwargs) )
elif fn == 'classical':
C_diag.append( classical_strength_of_connection(mat, **kwargs) )
elif fn == 'distance':
C_diag.append( distance_strength_of_connection(mat, **kwargs) )
elif (fn == 'ode') or (fn == 'evolution'):
C_diag.append( evolution_strength_of_connection(mat, **kwargs) )
elif fn == 'energy_based':
C_diag.append( energy_based_strength_of_connection(mat, **kwargs) )
elif fn == 'algebraic_distance':
C_diag.append( algebraic_distance(mat, **kwargs) )
elif fn == 'affinity':
C_diag.append( affinity_distance(mat, **kwargs) )
elif fn is None:
C_diag.append( mat )
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting.append( split.RS(C_diag[-1]) )
elif fn == 'PMIS':
splitting.append( split.PMIS(C_diag[-1]) )
elif fn == 'PMISc':
splitting.append( split.PMISc(C_diag[-1]) )
elif fn == 'CLJP':
splitting.append( split.CLJP(C_diag[-1]) )
elif fn == 'CLJPc':
splitting.append( split.CLJPc(C_diag[-1]) )
elif fn == 'Shifted2DCoarsening':
splitting.append( split.Shifted2DCoarsening(C_diag[-1]) )
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
fn, kwargs = unpack_arg(interp)
if fn == 'standard':
P_diag.append( standard_interpolation(mat, C_diag[-1], splitting[-1]) )
elif fn == 'direct':
P_diag.append( direct_interpolation(mat, C_diag[-1], splitting[-1]) )
else:
raise ValueError('unknown interpolation method (%s)' % interp)
next_lvl_block_starts.append( next_lvl_block_starts[-1] + P_diag[-1].shape[1])
block_cnt = block_cnt + 1
P = block_diag(P_diag)
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid
R = P.T.tocsr()
# Store relevant information for this level
splitting = numpy.concatenate(splitting)
if keep:
C = block_diag(C_diag)
levels[-1].C = C # strength of connection matrix
levels[-1].splitting = splitting # C/F splitting
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels.append(multilevel_solver.level())
# Form next level through Galerkin product
# !!! For systems, how do I propogate the block structure information down to the next grid? Especially if the blocks are different sizes? !!!
A = R * A * P
levels[-1].A = A
if (block_starts):
levels[-1].block_starts = next_lvl_block_starts
else:
levels[-1].block_starts = None
# If called for, output a visualization of the C/F splitting
if (verts.any()):
new_verts = numpy.empty([P.shape[1], 2])
cnt = 0
for i in range(len(splitting)):
if (splitting[i]):
new_verts[cnt] = verts[i]
cnt = cnt + 1
levels[-1].verts = new_verts
else:
levels[-1].verts = numpy.zeros(1)
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 standard_interpolation(A,
C,
splitting,
theta=None,
norm='min',
modified=True,
cost=[0]):
"""Create prolongator using standard interpolation
Parameters
----------
A : {csr_matrix}
NxN matrix in CSR format
C : {csr_matrix}
Strength-of-Connection matrix
Must have zero diagonal
splitting : array
C/F splitting stored in an array of length N
theta : float in [0,1), default None
theta value defining strong connections in a classical AMG sense. Provide if
different SOC used for P than for CF-splitting; otherwise, theta = None.
norm : string, default 'abs'
Norm used in redefining classical SOC. Options are 'min' and 'abs' for CSR matrices,
and 'min', 'abs', and 'fro' for BSR matrices. See strength.py for more information.
modified : bool, default True
Use modified classical interpolation. More robust if RS coarsening with second
pass is not used for CF splitting. Ignores interpolating from strong F-connections
without a common C-neighbor.
Returns
-------
P : {csr_matrix}
Prolongator using standard interpolation
Examples
--------
>>> from pyamg.gallery import poisson
>>> from pyamg.classical import standard_interpolation
>>> import numpy as np
>>> A = poisson((5,),format='csr')
>>> splitting = np.array([1,0,1,0,1], dtype='intc')
>>> P = standard_interpolation(A, A, splitting)
>>> print P.todense()
[[ 1. 0. 0. ]
[ 0.5 0.5 0. ]
[ 0. 1. 0. ]
[ 0. 0.5 0.5]
[ 0. 0. 1. ]]
"""
if not isspmatrix_csr(C):
raise TypeError('Expected csr_matrix SOC matrix, C.')
nc = np.sum(splitting)
n = A.shape[0]
# Block BSR format. Transfer A to CSR and the splitting and SOC matrix to have
# DOFs corresponding to CSR A
if isspmatrix_bsr(A):
temp_A = A.tocsr()
splitting0 = splitting * np.ones((A.blocksize[0], 1), dtype='intc')
splitting0 = np.reshape(splitting0, (np.prod(splitting0.shape), ),
order='F')
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
C0 = UnAmal(C0, A.blocksize[0], A.blocksize[1])
else:
C0 = UnAmal(C, A.blocksize[0], A.blocksize[1])
C0 = C0.tocsr()
# Use modified standard interpolation by ignoring strong F-connections that do
# not have a common C-point.
if modified:
amg_core.remove_strong_FF_connections(temp_A.shape[0], C0.indptr,
C0.indices, C0.data,
splitting)
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(temp_A)
P_indptr = np.empty_like(temp_A.indptr)
amg_core.rs_standard_interpolation_pass1(temp_A.shape[0], C0.indptr,
C0.indices, splitting0,
P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=temp_A.dtype)
if modified:
amg_core.mod_standard_interpolation_pass2(
temp_A.shape[0], temp_A.indptr, temp_A.indices, temp_A.data,
C0.indptr, C0.indices, C0.data, splitting0, P_indptr,
P_colinds, P_data)
else:
amg_core.rs_standard_interpolation_pass2(
temp_A.shape[0], temp_A.indptr, temp_A.indices, temp_A.data,
C0.indptr, C0.indices, C0.data, splitting0, P_indptr,
P_colinds, P_data)
nc = np.sum(splitting0)
n = A.shape[0]
P = csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
return P.tobsr(blocksize=A.blocksize)
# CSR format
else:
if theta is not None:
C0 = classical_strength_of_connection(A,
theta=theta,
norm=norm,
cost=cost)
else:
C0 = C.copy()
# Use modified standard interpolation by ignoring strong F-connections that do
# not have a common C-point.
if modified:
amg_core.remove_strong_FF_connections(A.shape[0], C0.indptr,
C0.indices, C0.data,
splitting)
C0.eliminate_zeros()
# Interpolation weights are computed based on entries in A, but subject to
# the sparsity pattern of C. So, copy the entries of A into the
# sparsity pattern of C.
C0.data[:] = 1.0
C0 = C0.multiply(A)
P_indptr = np.empty_like(A.indptr)
amg_core.rs_standard_interpolation_pass1(A.shape[0], C0.indptr,
C0.indices, splitting,
P_indptr)
nnz = P_indptr[-1]
P_colinds = np.empty(nnz, dtype=P_indptr.dtype)
P_data = np.empty(nnz, dtype=A.dtype)
if modified:
amg_core.mod_standard_interpolation_pass2(
A.shape[0], A.indptr, A.indices, A.data, C0.indptr, C0.indices,
C0.data, splitting, P_indptr, P_colinds, P_data)
else:
amg_core.rs_standard_interpolation_pass2(
A.shape[0], A.indptr, A.indices, A.data, C0.indptr, C0.indices,
C0.data, splitting, P_indptr, P_colinds, P_data)
nc = np.sum(splitting)
n = A.shape[0]
return csr_matrix((P_data, P_colinds, P_indptr), shape=[n, nc])
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 local_AIR(A,
splitting,
theta=0.1,
norm='abs',
degree=1,
use_gmres=False,
maxiter=10,
precondition=True,
cost=[0]):
""" Compute approximate ideal restriction by setting RA = 0, within the
sparsity pattern of R. Sparsity pattern of R for the ith row (i.e. ith
C-point) is the set of all strongly connected F-points, or the max_row
*most* strongly connected F-points.
Parameters
----------
A : {csr_matrix}
NxN matrix in CSR or BSR format
splitting : array
C/F splitting stored in an array of length N
theta : float, default 0.1
Solve local system for each row of R for all values
|A_ij| >= 0.1 * max_{i!=k} |A_ik|
degree : int, default 1
Expand sparsity pattern for R by considering strongly connected
neighbors within 'degree' of a given node. Only supports degree 1 and 2.
use_gmres : bool
Solve local linear system for each row of R using GMRES
maxiter : int
Maximum number of GMRES iterations
precondition : bool
Diagonally precondition GMRES
Returns
-------
Approximate ideal restriction, R, in same sparse format as A.
Notes
-----
- This was the original idea for approximating ideal restriction. In practice,
however, a Neumann approximation is typically used.
- Supports block bsr matrices as well.
"""
# Get SOC matrix containing neighborhood to be included in local solve
if isspmatrix_bsr(A):
C = classical_strength_of_connection(A=A,
theta=theta,
block='amalgamate',
norm=norm)
blocksize = A.blocksize[0]
elif isspmatrix_csr(A):
blocksize = 1
C = classical_strength_of_connection(A=A,
theta=theta,
block=None,
norm=norm)
else:
try:
A = A.tocsr()
warn("Implicit conversion of A to csr", SparseEfficiencyWarning)
C = classical_strength_of_connection(A=A,
theta=theta,
block=None,
norm=norm)
blocksize = 1
except:
raise TypeError("Invalid matrix type, must be CSR or BSR.")
Cpts = np.array(np.where(splitting == 1)[0], dtype='int32')
nc = Cpts.shape[0]
n = C.shape[0]
R_rowptr = np.empty(nc + 1, dtype='int32')
amg_core.approx_ideal_restriction_pass1(R_rowptr, C.indptr, C.indices,
Cpts, splitting, degree)
# Build restriction operator
nnz = R_rowptr[-1]
R_colinds = np.zeros(nnz, dtype='int32')
# Block matrix
if isspmatrix_bsr(A):
R_data = np.zeros(nnz * blocksize * blocksize, dtype=A.dtype)
amg_core.block_approx_ideal_restriction_pass2(
R_rowptr, R_colinds, R_data, A.indptr, A.indices, A.data.ravel(),
C.indptr, C.indices, C.data, Cpts, splitting, blocksize, degree,
use_gmres, maxiter, precondition)
R = bsr_matrix(
(R_data.reshape(nnz, blocksize, blocksize), R_colinds, R_rowptr),
blocksize=[blocksize, blocksize],
shape=[nc * blocksize, A.shape[0]])
# Not block matrix
else:
R_data = np.zeros(nnz, dtype=A.dtype)
amg_core.approx_ideal_restriction_pass2(R_rowptr, R_colinds, R_data,
A.indptr, A.indices, A.data,
C.indptr, C.indices, C.data,
Cpts, splitting, degree,
use_gmres, maxiter,
precondition)
R = csr_matrix((R_data, R_colinds, R_rowptr), shape=[nc, A.shape[0]])
R.eliminate_zeros()
return R
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
import scipy
from pyamg.gallery import stencil_grid
from pyamg.gallery.diffusion import diffusion_stencil_2d
from pyamg.strength import classical_strength_of_connection
from pyamg.classical.classical import ruge_stuben_solver
from convergence_tools import print_cycle_history
if __name__ == '__main__':
n = 100
nx = n
ny = n
# Rotated Anisotropic Diffusion
stencil = diffusion_stencil_2d(type='FE',epsilon=0.001,theta=scipy.pi/3)
A = stencil_grid(stencil, (nx,ny), format='csr')
S = classical_strength_of_connection(A, 0.0)
numpy.random.seed(625)
x = scipy.rand(A.shape[0])
b = A*scipy.rand(A.shape[0])
ml = ruge_stuben_solver(A, max_coarse=10)
resvec = []
x = ml.solve(b, x0=x, maxiter=20, tol=1e-14, residuals=resvec)
print_cycle_history(resvec, ml, verbose=True, plotting=True)
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, 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 initial_setup_stage(A, symmetry, pdef, candidate_iters, epsilon,
max_levels, max_coarse, aggregate, prepostsmoother,
smooth, strength, work, initial_candidate=None):
"""Compute 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 (aSA) 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)
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 initial_candidate 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(initial_candidate, 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 extend_hierarchy(levels, strength, CF, keep):
""" helper function for local methods """
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
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 is None:
C = A
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
# Generate the C/F splitting
fn, kwargs = unpack_arg(CF)
if fn == 'RS':
splitting = split.RS(C)
elif fn == 'PMIS':
splitting = split.PMIS(C)
elif fn == 'PMISc':
splitting = split.PMISc(C)
elif fn == 'CLJP':
splitting = split.CLJP(C)
elif fn == 'CLJPc':
splitting = split.CLJPc(C)
else:
raise ValueError('unknown C/F splitting method (%s)' % CF)
# Generate the interpolation matrix that maps from the coarse-grid to the
# fine-grid
P = direct_interpolation(A, C, splitting)
# Generate the restriction matrix that maps from the fine-grid to the
# coarse-grid
R = P.T.tocsr()
# Store relevant information for this level
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].splitting = splitting # C/F splitting
levels[-1].P = P # prolongation operator
levels[-1].R = R # restriction operator
levels.append(multilevel_solver.level())
# Form next level through Galerkin product
A = R * A * P
levels[-1].A = A
def extend_hierarchy(levels, strength, aggregate, smooth, improve_candidates,
diagonal_dominance=False, keep=True, test_ind=0):
"""Service routine to implement the strength of connection, aggregation,
tentative prolongation construction, and prolongation smoothing. Called by
smoothed_aggregation_solver.
"""
def unpack_arg(v):
if isinstance(v, tuple):
return v[0], v[1]
else:
return v, {}
A = levels[-1].A
B = levels[-1].B
if A.symmetry == "nonsymmetric":
AH = A.H.asformat(A.format)
BH = levels[-1].BH
# Improve near nullspace candidates by relaxing on A B = 0
fn, kwargs = unpack_arg(improve_candidates[len(levels)-1])
if fn is not None:
b = np.zeros((A.shape[0], 1), dtype=A.dtype)
B = relaxation_as_linear_operator((fn, kwargs), A, b) * B
levels[-1].B = B
if A.symmetry == "nonsymmetric":
BH = relaxation_as_linear_operator((fn, kwargs), AH, b) * BH
levels[-1].BH = BH
# Compute the 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 is None:
C = A.tocsr()
else:
raise ValueError('unrecognized strength of connection method: %s' %
str(fn))
# Avoid coarsening diagonally dominant rows
flag, kwargs = unpack_arg(diagonal_dominance)
if flag:
C = eliminate_diag_dom_nodes(A, C, **kwargs)
# Compute the aggregation matrix AggOp (i.e., the nodal coarsening of A).
# AggOp is a boolean matrix, where the sparsity pattern for the k-th column
# denotes the fine-grid nodes agglomerated into k-th coarse-grid node.
fn, kwargs = unpack_arg(aggregate[len(levels)-1])
if fn == 'standard':
AggOp, Cnodes = standard_aggregation(C, **kwargs)
elif fn == 'naive':
AggOp, Cnodes = naive_aggregation(C, **kwargs)
elif fn == 'lloyd':
AggOp, Cnodes = lloyd_aggregation(C, **kwargs)
elif fn == 'pairwise':
AggOp, Cnodes = pairwise_aggregation(A, B, **kwargs)
elif fn == 'predefined':
AggOp = kwargs['AggOp'].tocsr()
Cnodes = kwargs['Cnodes']
else:
raise ValueError('unrecognized aggregation method %s' % str(fn))
# ----------------------------------------------------------------------------- #
# ------------------- New ideal interpolation constructed -------------------- #
# ----------------------------------------------------------------------------- #
# pdb.set_trace()
# splitting = CR(A)
# Cpts = [i for i in range(0,AggOp.shape[0]) if splitting[i]==1]
# Compute prolongation operator.
if test_ind==0:
T = new_ideal_interpolation(A=A, AggOp=AggOp, Cnodes=Cnodes, B=B[:, 0:blocksize(A)], SOC=C)
else:
T = py_ideal_interpolation(A=A, AggOp=AggOp, Cnodes=Cnodes, B=B[:, 0:blocksize(A)], SOC=C)
print "\nSize of sparsity pattern - ", T.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))
P = T
# ----------------------------------------------------------------------------- #
# ----------------------------------------------------------------------------- #
# 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':
# symmetrically scale out the diagonal, include scaling in P, R
A = P.H * A * P
[dum, Dinv, dum] = symmetric_rescaling(A,copy=False)
P = bsr_matrix(P * diags(Dinv,offsets=0,format='csr'), blocksize=A.blocksize)
del dum
R = P.H
elif symmetry == 'symmetric':
# symmetrically scale out the diagonal, include scaling in P, R
A = P.T * A * P
[dum, Dinv, dum] = symmetric_rescaling(A,copy=False)
P = bsr_matrix(P * diags(Dinv,offsets=0,format='csr'), blocksize=A.blocksize)
del dum
R = P.T
elif symmetry == 'nonsymmetric':
raise TypeError('New ideal interpolation not implemented for non-symmetric matrix.')
if keep:
levels[-1].C = C # strength of connection matrix
levels[-1].AggOp = AggOp # aggregation operator
levels[-1].Fpts = [i for i in range(0,AggOp.shape[0]) if i not in Cnodes]
levels[-1].P = P # smoothed prolongator
levels[-1].R = R # restriction operator
levels[-1].Cpts = Cnodes # Cpts (i.e., rootnodes)
levels.append(multilevel_solver.level())
A.symmetry = symmetry
levels[-1].A = A
levels[-1].B = R*B # right near nullspace candidates
test = A.tocsr()
print "\nSize of coarse operator - ", test.nnz
if A.symmetry == "nonsymmetric":
levels[-1].BH = BH # left near nullspace candidates
