def relax(A, x):
fn, kwargs = unpack_arg(prepostsmoother)
if fn == 'gauss_seidel':
gauss_seidel(A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(A, x, numpy.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A, x, numpy.zeros_like(x), iterations=1,
coeffients=[1.0/approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, numpy.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)
if fn == 'gauss_seidel':
gauss_seidel(A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(A, x, numpy.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(A))
elif fn == 'richardson':
polynomial(A, x, numpy.zeros_like(x), iterations=1,
coeffients=[1.0/approximate_spectral_radius(A)])
elif fn == 'gmres':
x[:] = (gmres(A, numpy.zeros_like(x), x0=x,
maxiter=candidate_iters)[0]).reshape(x.shape)
else:
raise TypeError('Unrecognized smoother')
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 = numpy.zeros((bnnz, K+1, K), dtype=T.dtype)
data[:, :-1, :] = T.data
return bsr_matrix((data, T.indices, T.indptr),
shape=((K + 1) * (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 = B_old.shape[0] / nodesize
Bnew = numpy.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 = scipy.rand(levels[0].A.shape[0], 1)
if levels[0].A.dtype == complex:
x = x + 1.0j*scipy.rand(levels[0].A.shape[0], 1)
b = numpy.zeros_like(x)
x = ml.solve(b, x0=x, tol=float(numpy.finfo(numpy.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 = numpy.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(numpy.zeros_like(x), x0=x,
tol=float(numpy.finfo(numpy.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)
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 = numpy.ravel(x).nonzero()[0]
gauss_seidel_indexed(lvl.A, x, numpy.zeros_like(x), indices,
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_ne':
gauss_seidel_ne(lvl.A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'gauss_seidel_nr':
gauss_seidel_nr(lvl.A, x, numpy.zeros_like(x),
iterations=candidate_iters, sweep='symmetric')
elif fn == 'jacobi':
jacobi(lvl.A, x, numpy.zeros_like(x), iterations=1,
omega=1.0 / rho_D_inv_A(lvl.A))
elif fn == 'richardson':
polynomial(lvl.A, x, numpy.zeros_like(x), iterations=1,
coeffients=[1.0/approximate_spectral_radius(lvl.A)])
elif fn == 'gmres':
x[:] = (gmres(lvl.A, numpy.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) * (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 = 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 == 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)
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,
coeffients=[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)
