def setUp(self):
self.cases = []
# Poisson problems in 1D and 2D
N = 20
self.cases.append((poisson((2*N,), format='csr'), rand(2*N,))) # 0
self.cases.append((poisson((N, N), format='csr'), rand(N*N,))) # 1
# Boxed examples
A = load_example('recirc_flow')['A'].tocsr() # 2
self.cases.append((A, rand(A.shape[0],)))
A = load_example('bar')['A'].tobsr(blocksize=(3, 3)) # 3
self.cases.append((A, rand(A.shape[0],)))
def setUp(self):
self.cases = []
# Poisson problems in 1D and 2D
N = 20
self.cases.append((poisson((2*N,), format='csr'), rand(2*N,))) # 0
self.cases.append((poisson((N, N), format='csr'), rand(N*N,))) # 1
# Boxed examples
A = load_example('recirc_flow')['A'].tocsr() # 2
self.cases.append((A, rand(A.shape[0],)))
A = load_example('bar')['A'].tobsr(blocksize=(3, 3)) # 3
self.cases.append((A, rand(A.shape[0],)))
def test_nonhermitian(self):
# problem data
data = load_example('helmholtz_2D')
A = data['A'].tocsr()
B = data['B']
np.random.seed(28082572)
x0 = np.random.rand(A.shape[0]) + 1.0j * np.random.rand(A.shape[0])
b = A * np.random.rand(A.shape[0]) + 1.0j * (A * np.random.rand(A.shape[0]))
# solver parameters
smooth = ('energy', {'krylov': 'gmres'})
SA_build_args = {'max_coarse': 25, 'coarse_solver': 'pinv',
'symmetry': 'symmetric'}
SA_solve_args = {'cycle': 'V', 'maxiter': 20, 'tol': 1e-8}
strength = [('evolution', {'k': 2, 'epsilon': 2.0})]
smoother = ('gauss_seidel_nr', {'sweep': 'symmetric', 'iterations': 1})
# Construct solver with nonsymmetric parameters
sa = smoothed_aggregation_solver(A, B=B, smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
**SA_build_args)
residuals = []
# stand-alone solve
x = sa.solve(b, x0=x0, residuals=residuals, **SA_solve_args)
residuals = np.array(residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
assert(avg_convergence_ratio < 0.85)
# accelerated solve
residuals = []
x = sa.solve(b, x0=x0, residuals=residuals, accel='gmres',
**SA_solve_args)
del x
residuals = np.array(residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
assert(avg_convergence_ratio < 0.7)
# test that nonsymmetric parameters give the same result as symmetric
# parameters for the complex-symmetric matrix A
strength = 'symmetric'
SA_build_args['symmetry'] = 'nonsymmetric'
sa_nonsymm = smoothed_aggregation_solver(A, B=np.ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args)
SA_build_args['symmetry'] = 'symmetric'
sa_symm = smoothed_aggregation_solver(A, B=np.ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args)
for (symm_lvl, nonsymm_lvl) in zip(sa_nonsymm.levels, sa_symm.levels):
assert_array_almost_equal(symm_lvl.A.toarray(),
nonsymm_lvl.A.toarray())
def setUp(self):
self.cases = []
# random matrices
np.random.seed(954619597)
for N in [2, 3, 5]:
self.cases.append(
sparse.csr_matrix(np.random.rand(N, N)) +
sparse.csr_matrix(1.0j * np.random.rand(N, N)))
# Poisson problems in 1D and 2D
for N in [2, 3, 5, 7, 10, 11, 19]:
A = poisson((N, ), format='csr')
A.data = A.data + 1.0j * A.data
self.cases.append(A)
for N in [2, 3, 7, 9]:
A = poisson((N, N), format='csr')
A.data = A.data + 1.0j * np.random.rand(A.data.shape[0], )
self.cases.append(A)
for name in ['knot', 'airfoil', 'bar']:
ex = load_example(name)
A = ex['A'].tocsr()
A.data = A.data + 0.5j * np.random.rand(A.data.shape[0], )
self.cases.append(A)
def test_nonhermitian(self):
# problem data
data = load_example('helmholtz_2D')
A = data['A'].tocsr()
B = data['B']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0]) + 1.0j * scipy.rand(A.shape[0])
b = A * scipy.rand(A.shape[0]) + 1.0j * (A * scipy.rand(A.shape[0]))
# solver parameters
smooth = ('energy', {'krylov': 'gmres'})
SA_build_args = {'max_coarse': 25, 'coarse_solver': 'pinv2',
'symmetry': 'symmetric'}
SA_solve_args = {'cycle': 'V', 'maxiter': 20, 'tol': 1e-8}
strength = [('evolution', {'k': 2, 'epsilon': 2.0})]
smoother = ('gauss_seidel_nr', {'sweep': 'symmetric', 'iterations': 1})
# Construct solver with nonsymmetric parameters
sa = smoothed_aggregation_solver(A, B=B, smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
**SA_build_args)
residuals = []
# stand-alone solve
x = sa.solve(b, x0=x0, residuals=residuals, **SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
assert(avg_convergence_ratio < 0.85)
# accelerated solve
residuals = []
x = sa.solve(b, x0=x0, residuals=residuals, accel='gmres',
**SA_solve_args)
del x
residuals = array(residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
assert(avg_convergence_ratio < 0.6)
# test that nonsymmetric parameters give the same result as symmetric
# parameters for the complex-symmetric matrix A
strength = 'symmetric'
SA_build_args['symmetry'] = 'nonsymmetric'
sa_nonsymm = smoothed_aggregation_solver(A, B=ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args)
SA_build_args['symmetry'] = 'symmetric'
sa_symm = smoothed_aggregation_solver(A, B=ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args)
for (symm_lvl, nonsymm_lvl) in zip(sa_nonsymm.levels, sa_symm.levels):
assert_array_almost_equal(symm_lvl.A.todense(),
nonsymm_lvl.A.todense())
def test_distance(self):
data = load_example('airfoil')
cases = []
cases.append((data['A'].tocsr(), data['vertices']))
for (A, V) in cases:
dim = V.shape[1]
for theta in [1.5, 2.0, 2.5]:
cost = [0]
lower_bound = 3 * dim + float(A.shape[0]) / A.nnz
upper_bound = 3 * dim + 3
distance_strength_of_connection(A,
V,
theta=theta,
relative_drop=True,
cost=cost)
assert (cost[0] >= lower_bound)
assert (cost[0] <= upper_bound)
for (A, V) in cases:
for theta in [0.5, 1.0, 1.5]:
cost = [0]
lower_bound = 3 * dim + float(A.shape[0]) / A.nnz
upper_bound = 3 * dim + 3
distance_strength_of_connection(A,
V,
theta=theta,
relative_drop=False,
cost=cost)
assert (cost[0] >= lower_bound)
assert (cost[0] <= upper_bound)
def test_nonsymmetric(self):
# problem data
data = load_example('recirc_flow')
A = data['A'].tocsr()
B = data['B']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = A * scipy.rand(A.shape[0])
# solver parameters
smooth = ('energy', {'krylov': 'gmres'})
SA_build_args = {
'max_coarse': 25,
'coarse_solver': 'pinv2',
'symmetry': 'nonsymmetric'
}
SA_solve_args = {'cycle': 'V', 'maxiter': 20, 'tol': 1e-8}
strength = [('evolution', {'k': 2, 'epsilon': 8.0})]
smoother = ('gauss_seidel_nr', {'sweep': 'symmetric', 'iterations': 1})
improve_candidates = [('gauss_seidel_nr', {
'sweep': 'symmetric',
'iterations': 4
}), None]
# Construct solver with nonsymmetric parameters
sa = rootnode_solver(A, B=B, smooth=smooth, improve_candidates=improve_candidates, \
strength=strength, presmoother=smoother, postsmoother=smoother, **SA_build_args)
residuals = []
# stand-alone solve
x = sa.solve(b, x0=x0, residuals=residuals, **SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio = (residuals[-1] /
residuals[0])**(1.0 / len(residuals))
#print "Test 1 %1.3e, %1.3e" % (avg_convergence_ratio, 0.7)
assert (avg_convergence_ratio < 0.7)
# accelerated solve
residuals = []
x = sa.solve(b,
x0=x0,
residuals=residuals,
accel='gmres',
**SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio = (residuals[-1] /
residuals[0])**(1.0 / len(residuals))
#print "Test 2 %1.3e, %1.3e" % (avg_convergence_ratio, 0.45)
assert (avg_convergence_ratio < 0.45)
# test that nonsymmetric parameters give the same result as symmetric parameters
# for Poisson problem
A = poisson((15, 15), format='csr')
strength = 'symmetric'
SA_build_args['symmetry'] = 'nonsymmetric'
sa_nonsymm = rootnode_solver(A, B=ones((A.shape[0],1)), smooth=smooth, \
strength=strength, presmoother=smoother, postsmoother=smoother, improve_candidates=None,**SA_build_args)
SA_build_args['symmetry'] = 'symmetric'
sa_symm = rootnode_solver(A, B=ones((A.shape[0],1)), smooth=smooth, \
strength=strength, presmoother=smoother, postsmoother=smoother, improve_candidates=None,**SA_build_args)
for (symm_lvl, nonsymm_lvl) in zip(sa_nonsymm.levels, sa_symm.levels):
assert_array_almost_equal(symm_lvl.A.todense(),
nonsymm_lvl.A.todense())
def test_nonsymmetric(self):
# problem data
data = load_example('recirc_flow')
A = data['A'].tocsr()
B = data['B']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = A * scipy.rand(A.shape[0])
# solver parameters
smooth = ('energy', {'krylov': 'gmres'})
SA_build_args = {'max_coarse': 25, 'coarse_solver': 'pinv2',
'symmetry': 'nonsymmetric'}
SA_solve_args = {'cycle': 'V', 'maxiter': 20, 'tol': 1e-8}
strength = [('evolution', {'k': 2, 'epsilon': 8.0})]
smoother = ('gauss_seidel_nr', {'sweep': 'symmetric', 'iterations': 1})
improve_candidates = [('gauss_seidel_nr', {'sweep': 'symmetric',
'iterations': 4}), None]
# Construct solver with nonsymmetric parameters
sa = rootnode_solver(A, B=B, smooth=smooth,
improve_candidates=improve_candidates,
strength=strength,
presmoother=smoother,
postsmoother=smoother, **SA_build_args)
residuals = []
# stand-alone solve
x = sa.solve(b, x0=x0, residuals=residuals, **SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Test 1 %1.3e, %1.3e" % (avg_convergence_ratio, 0.7)
assert(avg_convergence_ratio < 0.7)
# accelerated solve
residuals = []
x = sa.solve(b, x0=x0, residuals=residuals, accel='gmres',
**SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Test 2 %1.3e, %1.3e" % (avg_convergence_ratio, 0.45)
assert(avg_convergence_ratio < 0.45)
# test that nonsymmetric parameters give the same result as symmetric
# parameters for Poisson problem
A = poisson((15, 15), format='csr')
strength = 'symmetric'
SA_build_args['symmetry'] = 'nonsymmetric'
sa_nonsymm = rootnode_solver(A, B=ones((A.shape[0], 1)), smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None, **SA_build_args)
SA_build_args['symmetry'] = 'symmetric'
sa_symm = rootnode_solver(A, B=ones((A.shape[0], 1)), smooth=smooth,
strength=strength, presmoother=smoother,
postsmoother=smoother,
improve_candidates=None, **SA_build_args)
for (symm_lvl, nonsymm_lvl) in zip(sa_nonsymm.levels, sa_symm.levels):
assert_array_almost_equal(symm_lvl.A.todense(),
nonsymm_lvl.A.todense())
def test_nonhermitian(self):
# problem data
data = load_example("helmholtz_2D")
A = data["A"].tocsr()
B = data["B"]
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0]) + 1.0j * scipy.rand(A.shape[0])
b = A * scipy.rand(A.shape[0]) + 1.0j * (A * scipy.rand(A.shape[0]))
# solver parameters
smooth = ("energy", {"krylov": "gmres"})
SA_build_args = {"max_coarse": 25, "coarse_solver": "pinv2", "symmetry": "symmetric"}
SA_solve_args = {"cycle": "V", "maxiter": 20, "tol": 1e-8}
strength = [("evolution", {"k": 2, "epsilon": 2.0})]
smoother = ("gauss_seidel_nr", {"sweep": "symmetric", "iterations": 1})
# Construct solver with nonsymmetric parameters
sa = smoothed_aggregation_solver(
A, B=B, smooth=smooth, strength=strength, presmoother=smoother, postsmoother=smoother, **SA_build_args
)
residuals = []
# stand-alone solve
x = sa.solve(b, x0=x0, residuals=residuals, **SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio = (residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
assert avg_convergence_ratio < 0.85
# accelerated solve
residuals = []
x = sa.solve(b, x0=x0, residuals=residuals, accel="gmres", **SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio = (residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
assert avg_convergence_ratio < 0.6
# test that nonsymmetric parameters give the same result as symmetric
# parameters for the complex-symmetric matrix A
strength = "symmetric"
SA_build_args["symmetry"] = "nonsymmetric"
sa_nonsymm = smoothed_aggregation_solver(
A,
B=ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args
)
SA_build_args["symmetry"] = "symmetric"
sa_symm = smoothed_aggregation_solver(
A,
B=ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args
)
for (symm_lvl, nonsymm_lvl) in zip(sa_nonsymm.levels, sa_symm.levels):
assert_array_almost_equal(symm_lvl.A.todense(), nonsymm_lvl.A.todense())
def setUp(self):
self.cases = []
# Poisson problems in 1D and 2D
for N in [2, 3, 5, 7, 10, 11, 19]:
self.cases.append(poisson((N,), format='csr'))
for N in [2, 3, 7, 9]:
self.cases.append(poisson((N, N), format='csr'))
for name in ['knot', 'airfoil', 'bar']:
ex = load_example(name)
self.cases.append(ex['A'].tocsr())
def setUp(self):
self.cases = []
# Poisson problems in 1D and 2D
for N in [2, 3, 5, 7, 10, 11, 19]:
self.cases.append(poisson((N, ), format='csr'))
for N in [2, 3, 7, 9]:
self.cases.append(poisson((N, N), format='csr'))
for name in ['knot', 'airfoil', 'bar']:
ex = load_example(name)
self.cases.append(ex['A'].tocsr())
def test_prefilter(self):
"""Check that using prefilter reduces NNZ in P"""
np.random.seed(0) # make tests repeatable
cases = []
# Simple, real-valued diffusion problems
X = load_example('airfoil')
A = X['A'].tocsr()
B = X['B']
cases.append((A, B,
('energy', {'krylov': 'cg', 'degree': 2, 'maxiter': 3}),
{'theta': 0.05}))
cases.append((A, B,
('energy', {'krylov': 'gmres', 'degree': 2,
'maxiter': 3}),
{'k': 3}))
cases.append((A.tobsr(blocksize=(2, 2)),
np.hstack((B, np.random.rand(B.shape[0], 1))),
('energy', {'krylov': 'cg', 'degree': 2,
'maxiter': 3}),
{'theta': 0.1}))
# Simple, imaginary-valued problems
iA = 1.0j * A
iB = 1.0 + np.random.rand(iA.shape[0], 2)\
+ 1.0j * (1.0 + np.random.rand(iA.shape[0], 2))
cases.append((iA, B,
('energy', {'krylov': 'cg', 'degree': 2, 'maxiter': 3}),
{'theta': 0.05}))
cases.append((iA, iB,
('energy', {'krylov': 'gmres', 'degree': 2,
'maxiter': 3}),
{'k': 3}))
cases.append((A.tobsr(blocksize=(2, 2)),
np.hstack((B, np.random.rand(B.shape[0], 1))),
('energy', {'krylov': 'cg', 'degree': 2, 'maxiter': 3}),
{'theta': 0.1}))
for A, B, smooth, prefilter in cases:
ml_nofilter = rootnode_solver(A, B=B, max_coarse=1, max_levels=2,
smooth=smooth, keep=True)
smooth[1]['prefilter'] = prefilter
ml_filter = rootnode_solver(A, B=B, max_coarse=1, max_levels=2,
smooth=smooth, keep=True)
assert_equal(ml_nofilter.levels[0].P.nnz >
ml_filter.levels[0].P.nnz, True)
def test_prefilter(self):
"""Check that using prefilter reduces NNZ in P"""
np.random.seed(0) # make tests repeatable
cases = []
# Simple, real-valued diffusion problems
X = load_example('airfoil')
A = X['A'].tocsr()
B = X['B']
cases.append((A, B,
('energy', {'krylov': 'cg', 'degree': 2, 'maxiter': 3}),
{'theta': 0.05}))
cases.append((A, B,
('energy', {'krylov': 'gmres', 'degree': 2,
'maxiter': 3}),
{'k': 3}))
cases.append((A.tobsr(blocksize=(2, 2)),
np.hstack((B, np.random.rand(B.shape[0], 1))),
('energy', {'krylov': 'cg', 'degree': 2,
'maxiter': 3}),
{'theta': 0.1}))
# Simple, imaginary-valued problems
iA = 1.0j * A
iB = 1.0 + np.random.rand(iA.shape[0], 2)\
+ 1.0j * (1.0 + np.random.rand(iA.shape[0], 2))
cases.append((iA, B,
('energy', {'krylov': 'cg', 'degree': 2, 'maxiter': 3}),
{'theta': 0.05}))
cases.append((iA, iB,
('energy', {'krylov': 'gmres', 'degree': 2,
'maxiter': 3}),
{'k': 3}))
cases.append((A.tobsr(blocksize=(2, 2)),
np.hstack((B, np.random.rand(B.shape[0], 1))),
('energy', {'krylov': 'cg', 'degree': 2, 'maxiter': 3}),
{'theta': 0.1}))
for A, B, smooth, prefilter in cases:
ml_nofilter = rootnode_solver(A, B=B, max_coarse=1, max_levels=2,
smooth=smooth, keep=True)
smooth[1]['prefilter'] = prefilter
ml_filter = rootnode_solver(A, B=B, max_coarse=1, max_levels=2,
smooth=smooth, keep=True)
assert_equal(ml_nofilter.levels[0].P.nnz >
ml_filter.levels[0].P.nnz, True)
def setUp(self):
self.cases = []
# random matrices
np.random.seed(0)
for N in [2, 3, 5]:
self.cases.append(csr_matrix(sp.rand(N, N)))
# Poisson problems in 1D and 2D
for N in [2, 3, 5, 7, 10, 11, 19]:
self.cases.append(poisson((N, ), format='csr'))
for N in [2, 3, 5, 7, 10, 11]:
self.cases.append(poisson((N, N), format='csr'))
for name in ['knot', 'airfoil', 'bar']:
ex = load_example(name)
self.cases.append(ex['A'].tocsr())
def setUp(self):
self.cases = []
# random matrices
np.random.seed(0)
for N in [2, 3, 5]:
self.cases.append(csr_matrix(np.random.rand(N, N)))
# Poisson problems in 1D and 2D
for N in [2, 3, 5, 7, 10, 11, 19]:
self.cases.append(poisson((N,), format='csr'))
for N in [2, 3, 5, 7, 8]:
self.cases.append(poisson((N, N), format='csr'))
for name in ['knot', 'airfoil', 'bar']:
ex = load_example(name)
self.cases.append(ex['A'].tocsr())
def test_prefilter(self):
"""Check that using prefilter reduces NNZ in P"""
np.random.seed(0) # make tests repeatable
cases = []
# Simple, real-valued diffusion problems
X = load_example("airfoil")
A = X["A"].tocsr()
B = X["B"]
cases.append((A, B, ("energy", {"krylov": "cg", "degree": 2, "maxiter": 3}), {"theta": 0.05}))
cases.append((A, B, ("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3}), {"k": 3}))
cases.append(
(
A.tobsr(blocksize=(2, 2)),
np.hstack((B, np.random.rand(B.shape[0], 1))),
("energy", {"krylov": "cg", "degree": 2, "maxiter": 3}),
{"theta": 0.1},
)
)
# Simple, imaginary-valued problems
iA = 1.0j * A
iB = 1.0 + np.random.rand(iA.shape[0], 2) + 1.0j * (1.0 + np.random.rand(iA.shape[0], 2))
cases.append((iA, B, ("energy", {"krylov": "cg", "degree": 2, "maxiter": 3}), {"theta": 0.05}))
cases.append((iA, iB, ("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3}), {"k": 3}))
cases.append(
(
A.tobsr(blocksize=(2, 2)),
np.hstack((B, np.random.rand(B.shape[0], 1))),
("energy", {"krylov": "cg", "degree": 2, "maxiter": 3}),
{"theta": 0.1},
)
)
for A, B, smooth, prefilter in cases:
ml_nofilter = rootnode_solver(A, B=B, max_coarse=1, max_levels=2, smooth=smooth, keep=True)
smooth[1]["prefilter"] = prefilter
ml_filter = rootnode_solver(A, B=B, max_coarse=1, max_levels=2, smooth=smooth, keep=True)
assert_equal(ml_nofilter.levels[0].P.nnz > ml_filter.levels[0].P.nnz, True)
def test_distance_strength_of_connection(self):
data = load_example('airfoil')
cases = []
cases.append((data['A'].tocsr(), data['vertices']))
for (A, V) in cases:
for theta in [1.5, 2.0, 2.5]:
result = distance_soc(A, V, theta=theta)
expected = reference_distance_soc(A, V, theta=theta)
assert_equal(result.nnz, expected.nnz)
assert_array_almost_equal(result.todense(), expected.todense())
for (A, V) in cases:
for theta in [0.5, 1.0, 1.5]:
result = distance_soc(A, V, theta=theta, relative_drop=False)
expected = reference_distance_soc(A, V, theta=theta,
relative_drop=False)
assert_equal(result.nnz, expected.nnz)
assert_array_almost_equal(result.todense(), expected.todense())
def setUp(self):
self.cases = []
# random matrices
numpy.random.seed(0)
for N in [2,3,5]:
self.cases.append( csr_matrix(rand(N,N)) + csr_matrix(1.0j*rand(N,N)))
# Poisson problems in 1D and 2D
for N in [2,3,5,7,10,11,19]:
A = poisson( (N,), format='csr'); A.data = A.data + 1.0j*A.data;
self.cases.append(A)
for N in [2,3,7,9]:
A = poisson( (N,N), format='csr'); A.data = A.data + 1.0j*rand(A.data.shape[0],);
self.cases.append(A)
for name in ['knot','airfoil','bar']:
ex = load_example(name)
A = ex['A'].tocsr(); A.data = A.data + 0.5j*rand(A.data.shape[0],);
self.cases.append(A)
def setUp(self):
self.cases = []
#
# Random matrices, cases 0-2
np.random.seed(0)
for N in [2, 3, 5]:
self.cases.append(csr_matrix(np.random.rand(N, N)))
# Poisson problems in 1D, cases 3-9
for N in [2, 3, 5, 7, 10, 11, 19]:
self.cases.append(poisson((N,), format="csr"))
# Poisson problems in 2D, cases 10-15
for N in [2, 3, 5, 7, 10, 11]:
self.cases.append(poisson((N, N), format="csr"))
for name in ["knot", "airfoil", "bar"]:
ex = load_example(name)
self.cases.append(ex["A"].tocsr())
def setUp(self):
cases = []
seed(0)
for i in range(5):
A = rand(8, 8) > 0.5
cases.append(canonical_graph(A + A.T).astype(float))
cases.append(zeros((1, 1)))
cases.append(zeros((2, 2)))
cases.append(zeros((8, 8)))
cases.append(ones((2, 2)) - eye(2))
cases.append(poisson((5,)))
cases.append(poisson((5, 5)))
cases.append(poisson((11, 11)))
cases.append(poisson((5, 5, 5)))
for name in ['airfoil', 'bar', 'knot']:
cases.append(load_example(name)['A'])
cases = [canonical_graph(G) for G in cases]
self.cases = cases
def setUp(self):
cases = []
np.random.seed(651978631)
for i in range(5):
A = np.random.rand(8, 8) > 0.5
cases.append(canonical_graph(A + A.T).astype(float))
cases.append(np.zeros((1, 1)))
cases.append(np.zeros((2, 2)))
cases.append(np.zeros((8, 8)))
cases.append(np.ones((2, 2)) - np.eye(2))
cases.append(poisson((5, )))
cases.append(poisson((5, 5)))
cases.append(poisson((11, 11)))
cases.append(poisson((5, 5, 5)))
for name in ['airfoil', 'bar', 'knot']:
cases.append(load_example(name)['A'])
cases = [canonical_graph(G) for G in cases]
self.cases = cases
def test_distance(self):
data = load_example('airfoil')
cases = []
cases.append((data['A'].tocsr(), data['vertices']))
for (A, V) in cases:
dim = V.shape[1]
for theta in [1.5, 2.0, 2.5]:
cost = [0]
lower_bound = 3*dim + float(A.shape[0]) / A.nnz
upper_bound = 3*dim + 3
distance_strength_of_connection(A, V, theta=theta, relative_drop=True, cost=cost)
assert(cost[0] >= lower_bound)
assert(cost[0] <= upper_bound)
for (A, V) in cases:
for theta in [0.5, 1.0, 1.5]:
cost = [0]
lower_bound = 3*dim + float(A.shape[0]) / A.nnz
upper_bound = 3*dim + 3
distance_strength_of_connection(A, V, theta=theta, relative_drop=False, cost=cost)
assert(cost[0] >= lower_bound)
assert(cost[0] <= upper_bound)
def test_range(self):
"""Check that P*R=B"""
np.random.seed(0) # make tests repeatable
cases = []
# Simple, real-valued diffusion problems
X = load_example("airfoil")
A = X["A"].tocsr()
B = X["B"]
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "local"})))
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((A, B, ("energy", {"maxiter": 3})))
cases.append((A, B, ("energy", {"krylov": "cgnr"})))
cases.append((A, B, ("energy", {"krylov": "gmres", "degree": 2})))
A = poisson((10, 10), format="csr")
B = np.ones((A.shape[0], 1))
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "diagonal"})))
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "local"})))
cases.append((A, B, "energy"))
cases.append((A, B, ("energy", {"degree": 2})))
cases.append((A, B, ("energy", {"krylov": "cgnr", "degree": 2})))
cases.append((A, B, ("energy", {"krylov": "gmres"})))
# Simple, imaginary-valued problems
iA = 1.0j * A
iB = 1.0 + np.random.rand(iA.shape[0], 2) + 1.0j * (1.0 + np.random.rand(iA.shape[0], 2))
cases.append((iA, B, ("jacobi", {"filter": True, "weighting": "diagonal"})))
cases.append((iA, B, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA, iB, ("jacobi", {"filter": True, "weighting": "local"})))
cases.append((iA, iB, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA.tobsr(blocksize=(5, 5)), B, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA.tobsr(blocksize=(5, 5)), iB, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA, B, ("energy", {"krylov": "cgnr", "degree": 2})))
cases.append((iA, iB, ("energy", {"krylov": "cgnr"})))
cases.append(
(
iA.tobsr(blocksize=(5, 5)),
B,
("energy", {"krylov": "cgnr", "degree": 2, "maxiter": 3, "postfilter": {"theta": 0.05}}),
)
)
cases.append(
(
iA.tobsr(blocksize=(5, 5)),
B,
("energy", {"krylov": "cgnr", "degree": 2, "maxiter": 3, "prefilter": {"theta": 0.05}}),
)
)
cases.append((iA.tobsr(blocksize=(5, 5)), B, ("energy", {"krylov": "cgnr", "degree": 2, "maxiter": 3})))
cases.append((iA.tobsr(blocksize=(5, 5)), iB, ("energy", {"krylov": "cgnr"})))
cases.append((iA, B, ("energy", {"krylov": "gmres"})))
cases.append((iA, iB, ("energy", {"krylov": "gmres", "degree": 2})))
cases.append((iA.tobsr(blocksize=(5, 5)), B, ("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3})))
cases.append((iA.tobsr(blocksize=(5, 5)), iB, ("energy", {"krylov": "gmres"})))
# Simple, imaginary-valued problems
iA = A + 1.0j * scipy.sparse.eye(A.shape[0], A.shape[1])
cases.append((iA, B, ("jacobi", {"filter": True, "weighting": "local"})))
cases.append((iA, B, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA, iB, ("jacobi", {"filter": True, "weighting": "diagonal"})))
cases.append((iA, iB, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA.tobsr(blocksize=(4, 4)), iB, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((iA, B, ("energy", {"krylov": "cgnr"})))
cases.append((iA.tobsr(blocksize=(4, 4)), iB, ("energy", {"krylov": "cgnr"})))
cases.append((iA, B, ("energy", {"krylov": "gmres"})))
cases.append((iA.tobsr(blocksize=(4, 4)), iB, ("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3})))
cases.append(
(
iA.tobsr(blocksize=(4, 4)),
iB,
("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3, "postfilter": {"theta": 0.05}}),
)
)
cases.append(
(
iA.tobsr(blocksize=(4, 4)),
iB,
("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3, "prefilter": {"theta": 0.05}}),
)
)
A = gauge_laplacian(10, spacing=1.0, beta=0.21)
B = np.ones((A.shape[0], 1))
cases.append((A, iB, ("jacobi", {"filter": True, "weighting": "diagonal"})))
cases.append((A, iB, ("jacobi", {"filter": True, "weighting": "local"})))
cases.append((A, B, ("energy", {"krylov": "cg"})))
cases.append((A, iB, ("energy", {"krylov": "cgnr"})))
cases.append((A, iB, ("energy", {"krylov": "gmres"})))
cases.append(
(
A.tobsr(blocksize=(2, 2)),
B,
("energy", {"krylov": "cgnr", "degree": 2, "maxiter": 3, "postfilter": {"theta": 0.05}}),
)
)
cases.append(
(
A.tobsr(blocksize=(2, 2)),
B,
("energy", {"krylov": "cgnr", "degree": 2, "maxiter": 3, "prefilter": {"theta": 0.05}}),
)
)
cases.append((A.tobsr(blocksize=(2, 2)), B, ("energy", {"krylov": "cgnr", "degree": 2, "maxiter": 3})))
cases.append((A.tobsr(blocksize=(2, 2)), iB, ("energy", {"krylov": "cg"})))
cases.append((A.tobsr(blocksize=(2, 2)), B, ("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3})))
cases.append(
(
A.tobsr(blocksize=(2, 2)),
B,
("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3, "postfilter": {"theta": 0.05}}),
)
)
cases.append(
(
A.tobsr(blocksize=(2, 2)),
B,
("energy", {"krylov": "gmres", "degree": 2, "maxiter": 3, "prefilter": {"theta": 0.05}}),
)
)
#
A, B = linear_elasticity((10, 10))
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "diagonal"})))
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "local"})))
cases.append((A, B, ("jacobi", {"filter": True, "weighting": "block"})))
cases.append((A, B, ("energy", {"degree": 2})))
cases.append((A, B, ("energy", {"degree": 3, "postfilter": {"theta": 0.05}})))
cases.append((A, B, ("energy", {"degree": 3, "prefilter": {"theta": 0.05}})))
cases.append((A, B, ("energy", {"krylov": "cgnr"})))
cases.append((A, B, ("energy", {"krylov": "gmres", "degree": 2})))
# Classic SA cases
for A, B, smooth in cases:
ml = smoothed_aggregation_solver(A, B=B, max_coarse=1, max_levels=2, smooth=smooth)
P = ml.levels[0].P
B = ml.levels[0].B
R = ml.levels[1].B
assert_almost_equal(P * R, B)
def blocksize(A):
# Helper Function: return the blocksize of a matrix
if isspmatrix_bsr(A):
return A.blocksize[0]
else:
return 1
# Root-node cases
counter = 0
for A, B, smooth in cases:
counter += 1
if isinstance(smooth, tuple):
smoother = smooth[0]
else:
smoother = smooth
if smoother == "energy" and (B.shape[1] >= blocksize(A)):
ic = [("gauss_seidel_nr", {"sweep": "symmetric", "iterations": 4}), None]
ml = rootnode_solver(
A,
B=B,
max_coarse=1,
max_levels=2,
smooth=smooth,
improve_candidates=ic,
keep=True,
symmetry="nonsymmetric",
)
T = ml.levels[0].T.tocsr()
Cpts = ml.levels[0].Cpts
Bf = ml.levels[0].B
Bf_H = ml.levels[0].BH
Bc = ml.levels[1].B
P = ml.levels[0].P.tocsr()
# P should preserve B in its range, wherever P
# has enough nonzeros
mask = (P.indptr[1:] - P.indptr[:-1]) >= B.shape[1]
assert_almost_equal((P * Bc)[mask, :], Bf[mask, :])
assert_almost_equal((P * Bc)[mask, :], Bf_H[mask, :])
# P should be the identity at Cpts
I1 = eye(T.shape[1], T.shape[1], format="csr", dtype=T.dtype)
I2 = P[Cpts, :]
assert_almost_equal(I1.data, I2.data)
assert_equal(I1.indptr, I2.indptr)
assert_equal(I1.indices, I2.indices)
# T should be the identity at Cpts
I2 = T[Cpts, :]
assert_almost_equal(I1.data, I2.data)
assert_equal(I1.indptr, I2.indptr)
assert_equal(I1.indices, I2.indices)
def setUp(self):
cases = [None for i in range(5)]
cluster_node_incidence_input = [None for i in range(5)]
cluster_node_incidence_output = [None for i in range(5)]
cluster_center_input = [None for i in range(5)]
cluster_center_output = [None for i in range(5)]
bellman_ford_input = [None for i in range(5)]
bellman_ford_output = [None for i in range(5)]
# bellman_ford_balanced_input = [None for i in range(5)]
# bellman_ford_balanced_output = [None for i in range(5)]
lloyd_cluster_input = [None for i in range(5)]
lloyd_cluster_output = [None for i in range(5)]
lloyd_cluster_exact_input = [None for i in range(5)]
lloyd_cluster_exact_output = [None for i in range(5)]
# (0) 6 node undirected, unit length
# (1) 12 node undirected, unit length
# (2) 16 node undirected, random length
# (3) 16 node directed, random length
# (4) 191 node unstructured finite element matrix
# (0) 6 node undirected, unit length
#
# [3] ---- [4] ---- [5]
# | \ / | \ / |
# | / \ | / \ |
# [0] ---- [1] ---- [2]
xy = np.array([[0, 0], [1, 0], [2, 0], [0, 1], [1, 1], [2, 1]])
del xy
G = np.zeros((6, 6))
G[0, [1, 3, 4]] = 1
G[1, [0, 2, 3, 4, 5]] = 1
G[2, [1, 4, 5]] = 1
G[3, [0, 1, 4]] = 1
G[4, [0, 1, 2, 3, 5]] = 1
G[5, [1, 2, 4]] = 1
G[[0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]] = [3, 5, 3, 3, 5, 3]
G = sparse.csr_matrix(G)
cases[0] = (G)
cm = np.array([0, 1, 1, 0, 0, 1], dtype=np.int32)
ICp = np.array([0, 3, 6], dtype=np.int32)
ICi = np.array([0, 3, 4, 1, 2, 5], dtype=np.int32)
L = np.array([0, 0, 1, 1, 2, 2], dtype=np.int32)
cluster_node_incidence_input[0] = {'num_clusters': 2, 'cm': cm}
cluster_node_incidence_output[0] = {'ICp': ICp, 'ICi': ICi, 'L': L}
cluster_center_input[0] = {
'a': [0, 1],
'num_clusters': 2,
'cm': np.array([0, 1, 1, 0, 0, 1], dtype=np.int32),
'ICp': np.array([0, 3, 6], dtype=np.int32),
'ICi': np.array([0, 3, 4, 1, 2, 5], dtype=np.int32),
'L': np.array([0, 0, 1, 1, 2, 2], dtype=np.int32)
}
cluster_center_output[0] = [0, 1]
bellman_ford_input[0] = {'seeds': [0, 5]}
bellman_ford_output[0] = {
'cm': np.array([0, 0, 1, 0, 0, 1], dtype=np.int32),
'd': np.array([0., 1., 1., 1., 1., 0.], dtype=G.dtype)
}
lloyd_cluster_input[0] = {'seeds': np.array([0, 5], dtype=np.int32)}
lloyd_cluster_output[0] = {
'cm': np.array([0, 0, 1, 0, 0, 1], dtype=np.int32),
'd': np.array([1., 0., 0., 1., 0., 0.], dtype=G.dtype),
'c': np.array([0, 5], dtype=np.int32)
}
lloyd_cluster_exact_input[0] = {
'seeds': np.array([0, 5], dtype=np.int32)
}
lloyd_cluster_exact_output[0] = {
'cm': np.array([0, 0, 1, 0, 1, 1], dtype=np.int32),
'd': np.array([0, 1, 1, 1, 1, 0], dtype=G.dtype),
'c': np.array([0, 2], dtype=np.int32)
}
# (1) 12 node undirected, unit length
#
# _[1] ---- [7]
# / | |
# / | |
# [0] ---- [2] ---- [6] ---- [8]
# | | |
# | | |
# [3] ---- [5] ---- [9] _
# | | \
# | | \
# [4] ---- [10]---- [11]
xy = np.array([[0, 2], [1, 3], [1, 2], [1, 1], [2, 0], [2, 1], [2, 2],
[2, 3], [3, 2], [3, 1], [3, 0], [4, 0]])
del xy
G = np.zeros((12, 12))
G[0, [1, 2]] = 1
G[1, [0, 2, 7]] = 1
G[2, [0, 1, 3, 6]] = 1
G[3, [2, 5]] = 1
G[4, [5, 10]] = 1
G[5, [3, 4, 6, 9]] = 1
G[6, [2, 5, 7, 8]] = 1
G[7, [1, 6]] = 1
G[8, [6, 9]] = 1
G[9, [5, 8, 10, 11]] = 1
G[10, [4, 9, 11]] = 1
G[11, [9, 10]] = 1
G[np.arange(12), np.arange(12)] = [2, 3, 4, 2, 2, 4, 4, 2, 2, 4, 3, 2]
G = sparse.csr_matrix(G)
cases.append(G)
cm = np.array([0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1], dtype=np.int32)
ICp = np.array([0, 6, 12], dtype=np.int32)
ICi = np.array([0, 1, 2, 3, 6, 7, 4, 5, 8, 9, 10, 11], dtype=np.int32)
L = np.array([0, 1, 2, 3, 0, 1, 4, 5, 2, 3, 4, 5], dtype=np.int32)
cluster_node_incidence_input.append({'num_clusters': 2, 'cm': cm})
cluster_node_incidence_output.append({'ICp': ICp, 'ICi': ICi, 'L': L})
cluster_center_input[0] = {
'a': [0, 1],
'num_clusters': 2,
'cm': np.array([0, 1, 1, 0, 0, 1], dtype=np.int32),
'ICp': np.array([0, 3, 6], dtype=np.int32),
'ICi': np.array([0, 3, 4, 1, 2, 5], dtype=np.int32),
'L': np.array([0, 0, 1, 1, 2, 2], dtype=np.int32)
}
cluster_center_output[0] = [0, 1]
# (2) 16 node undirected, random length (0,2)
np.random.seed(2244369509)
G.data[:] = np.random.rand(len(G.data)) * 2
cases.append(G)
# (3) 16 node directed, random length
# >[1] ---> [7]
# / | |
# / v v
# [0] <--- [2] <--- [6] <--- [8]
# | ^ ^
# v | |
# [3] ---> [5] <--- [9] <
# | ^ \
# v | \
# [4] ---> [10]---> [11]
xy = np.array([[0, 2], [1, 3], [1, 2], [1, 1], [2, 0], [2, 1], [2, 2],
[2, 3], [3, 2], [3, 1], [3, 0], [4, 0]])
del xy
G = np.zeros((12, 12))
G[0, [1]] = 1
G[1, [2, 7]] = 1
G[2, [0, 3]] = 1
G[3, [5]] = 1
G[4, [10]] = 1
G[5, [4, 6]] = 1
G[6, [2]] = 1
G[7, [6]] = 1
G[8, [6]] = 1
G[9, [5, 8]] = 1
G[10, [9, 11]] = 1
G[11, [9]] = 1
G = sparse.csr_matrix(G)
np.random.seed(1664236979)
G.data[:] = np.random.rand(len(G.data)) * 2
cases.append(G)
# (4) 191 node unstructured finite element matrix
cases.append(load_example('unit_square')['A'])
self.cases = cases
self.cluster_node_incidence_input = cluster_node_incidence_input
self.cluster_node_incidence_output = cluster_node_incidence_output
self.cluster_center_input = cluster_center_input
self.cluster_center_output = cluster_center_output
self.bellman_ford_input = bellman_ford_input
self.bellman_ford_output = bellman_ford_output
# self.bellman_ford_balanced_input = bellman_ford_balanced_input
# self.bellman_ford_balanced_output = bellman_ford_balanced_output
self.lloyd_cluster_input = lloyd_cluster_input
self.lloyd_cluster_output = lloyd_cluster_output
self.lloyd_cluster_exact_input = lloyd_cluster_exact_input
self.lloyd_cluster_exact_output = lloyd_cluster_exact_output
def test_range(self):
"""Check that P*R=B."""
warnings.filterwarnings('ignore', category=UserWarning,
message='Having less target vectors')
np.random.seed(18410243) # make tests repeatable
cases = []
# Simple, real-valued diffusion problems
name = 'airfoil'
X = load_example('airfoil')
A = X['A'].tocsr()
B = X['B']
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'local'}), name))
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'block'}), name))
cases.append((A, B, ('energy', {'maxiter': 3}), name))
cases.append((A, B, ('energy', {'krylov': 'cgnr', 'weighting': 'diagonal'}), name))
cases.append((A, B, ('energy', {'krylov': 'gmres', 'degree': 2}), name))
name = 'poisson'
A = poisson((10, 10), format='csr')
B = np.ones((A.shape[0], 1))
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'diagonal'}), name))
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'local'}), name))
cases.append((A, B, 'energy', name))
cases.append((A, B, ('energy', {'degree': 2}), name))
cases.append((A, B, ('energy', {'krylov': 'cgnr', 'degree': 2,
'weighting': 'diagonal'}), name))
cases.append((A, B, ('energy', {'krylov': 'gmres'}), name))
# Simple, imaginary-valued problems
name = 'random imaginary'
iA = 1.0j * A
iB = 1.0 + np.random.rand(iA.shape[0], 2)\
+ 1.0j * (1.0 + np.random.rand(iA.shape[0], 2))
cases.append((iA, B, ('jacobi',
{'filter_entries': True, 'weighting': 'diagonal'}), name))
cases.append((iA, B, ('jacobi',
{'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA, iB, ('jacobi',
{'filter_entries': True, 'weighting': 'local'}), name))
cases.append((iA, iB, ('jacobi',
{'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), B,
('jacobi', {'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), iB,
('jacobi', {'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA, B, ('energy', {'krylov': 'cgnr', 'degree': 2,
'weighting': 'diagonal'}), name))
cases.append((iA, iB, ('energy', {'krylov': 'cgnr',
'weighting': 'diagonal'}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), B,
('energy',
{'krylov': 'cgnr', 'degree': 2, 'maxiter': 3,
'weighting': 'diagonal', 'postfilter': {'theta': 0.05}}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), B,
('energy',
{'krylov': 'cgnr', 'degree': 2, 'maxiter': 3,
'weighting': 'diagonal',
'prefilter': {'theta': 0.05}}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), B,
('energy',
{'krylov': 'cgnr', 'degree': 2,
'weighting': 'diagonal', 'maxiter': 3}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), iB,
('energy', {'krylov': 'cgnr', 'weighting': 'diagonal'}), name))
cases.append((iA, B, ('energy', {'krylov': 'gmres'}), name))
cases.append((iA, iB, ('energy', {'krylov': 'gmres', 'degree': 2}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), B,
('energy',
{'krylov': 'gmres', 'degree': 2, 'maxiter': 3}), name))
cases.append((iA.tobsr(blocksize=(5, 5)), iB,
('energy', {'krylov': 'gmres'}), name))
# Simple, imaginary-valued problems
name = 'random imaginary + I'
iA = A + 1.0j * sparse.eye(A.shape[0], A.shape[1])
cases.append((iA, B, ('jacobi',
{'filter_entries': True, 'weighting': 'local'}), name))
cases.append((iA, B, ('jacobi',
{'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA, iB, ('jacobi',
{'filter_entries': True, 'weighting': 'diagonal'}), name))
cases.append((iA, iB, ('jacobi',
{'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA.tobsr(blocksize=(4, 4)), iB,
('jacobi', {'filter_entries': True, 'weighting': 'block'}), name))
cases.append((iA, B, ('energy', {'krylov': 'cgnr', 'weighting': 'diagonal'}), name))
cases.append((iA.tobsr(blocksize=(4, 4)), iB,
('energy', {'krylov': 'cgnr', 'weighting': 'diagonal'}), name))
cases.append((iA, B, ('energy', {'krylov': 'gmres'}), name))
cases.append((iA.tobsr(blocksize=(4, 4)), iB,
('energy',
{'krylov': 'gmres', 'degree': 2, 'maxiter': 3}), name))
cases.append((iA.tobsr(blocksize=(4, 4)), iB,
('energy',
{'krylov': 'gmres', 'degree': 2, 'maxiter': 3,
'postfilter': {'theta': 0.05}}), name))
cases.append((iA.tobsr(blocksize=(4, 4)), iB,
('energy',
{'krylov': 'gmres', 'degree': 2, 'maxiter': 3,
'prefilter': {'theta': 0.05}}), name))
name = 'gauge laplacian'
A = gauge_laplacian(10, spacing=1.0, beta=0.21)
B = np.ones((A.shape[0], 1))
cases.append((A, iB, ('jacobi',
{'filter_entries': True, 'weighting': 'diagonal'}), name))
cases.append((A, iB, ('jacobi',
{'filter_entries': True, 'weighting': 'local'}), name))
cases.append((A, B, ('energy', {'krylov': 'cg'}), name))
cases.append((A, iB, ('energy', {'krylov': 'cgnr', 'weighting': 'diagonal'}), name))
cases.append((A, iB, ('energy', {'krylov': 'gmres'}), name))
name = 'gauge laplacian bsr'
cases.append((A.tobsr(blocksize=(2, 2)), B,
('energy', {'krylov': 'cgnr', 'degree': 2, 'weighting': 'diagonal',
'maxiter': 3, 'postfilter': {'theta': 0.05}}),
name))
cases.append((A.tobsr(blocksize=(2, 2)), B,
('energy', {'krylov': 'cgnr', 'degree': 2, 'weighting': 'diagonal',
'maxiter': 3, 'prefilter': {'theta': 0.05}}),
name))
cases.append((A.tobsr(blocksize=(2, 2)), B,
('energy', {'krylov': 'cgnr', 'degree': 2, 'maxiter': 3,
'weighting': 'diagonal'}),
name))
cases.append((A.tobsr(blocksize=(2, 2)), iB,
('energy', {'krylov': 'cg'}), name))
cases.append((A.tobsr(blocksize=(2, 2)), B,
('energy', {'krylov': 'gmres', 'degree': 2, 'maxiter': 3}),
name))
cases.append((A.tobsr(blocksize=(2, 2)), B,
('energy', {'krylov': 'gmres', 'degree': 2,
'maxiter': 3, 'postfilter': {'theta': 0.05}}),
name))
cases.append((A.tobsr(blocksize=(2, 2)), B,
('energy', {'krylov': 'gmres', 'degree': 2,
'maxiter': 3, 'prefilter': {'theta': 0.05}}),
name))
#
name = 'linear elasticity'
A, B = linear_elasticity((10, 10))
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'diagonal'}), name))
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'local'}), name))
cases.append((A, B, ('jacobi',
{'filter_entries': True, 'weighting': 'block'}), name))
cases.append((A, B, ('energy', {'degree': 2}), name))
cases.append((A, B, ('energy', {'degree': 3, 'postfilter': {'theta': 0.05}}), name))
cases.append((A, B, ('energy', {'degree': 3, 'prefilter': {'theta': 0.05}}), name))
cases.append((A, B, ('energy', {'krylov': 'cgnr', 'weighting': 'diagonal'}), name))
cases.append((A, B, ('energy', {'krylov': 'gmres', 'degree': 2}), name))
# Classic SA cases
for A, B, smooth, _name in cases:
ml = smoothed_aggregation_solver(A, B=B, max_coarse=1,
max_levels=2, smooth=smooth)
P = ml.levels[0].P
B = ml.levels[0].B
R = ml.levels[1].B
assert_almost_equal(P * R, B)
def _get_blocksize(A):
# Helper Function: return the blocksize of a matrix
if sparse.isspmatrix_bsr(A):
return A.blocksize[0]
return 1
# Root-node cases
counter = 0
for A, B, smooth, _name in cases:
counter += 1
if isinstance(smooth, tuple):
smoother = smooth[0]
else:
smoother = smooth
if smoother == 'energy' and (B.shape[1] >= _get_blocksize(A)):
ic = [('gauss_seidel_nr',
{'sweep': 'symmetric', 'iterations': 4}), None]
ml = rootnode_solver(A, B=B, max_coarse=1, max_levels=2,
smooth=smooth,
improve_candidates=ic,
keep=True, symmetry='nonsymmetric')
T = ml.levels[0].T.tocsr()
Cpts = ml.levels[0].Cpts
Bf = ml.levels[0].B
Bf_H = ml.levels[0].BH
Bc = ml.levels[1].B
P = ml.levels[0].P.tocsr()
T.eliminate_zeros()
P.eliminate_zeros()
# P should preserve B in its range, wherever P
# has enough nonzeros
mask = ((P.indptr[1:] - P.indptr[:-1]) >= B.shape[1])
assert_almost_equal((P*Bc)[mask, :], Bf[mask, :])
assert_almost_equal((P*Bc)[mask, :], Bf_H[mask, :])
# P should be the identity at Cpts
I1 = sparse.eye(T.shape[1], T.shape[1], format='csr', dtype=T.dtype)
I2 = P[Cpts, :]
assert_almost_equal(I1.data, I2.data)
assert_equal(I1.indptr, I2.indptr)
assert_equal(I1.indices, I2.indices)
# T should be the identity at Cpts
I2 = T[Cpts, :]
assert_almost_equal(I1.data, I2.data)
assert_equal(I1.indptr, I2.indptr)
assert_equal(I1.indices, I2.indices)
# 3D example of viewing aggregates from SA using VTK
from pyamg.aggregation import standard_aggregation
from pyamg.vis import vis_coarse, vtk_writer
from pyamg.gallery import load_example
# retrieve the problem
data = load_example('unit_cube')
A = data['A'].tocsr()
V = data['vertices']
E2V = data['elements']
# perform smoothed aggregation
Agg, rootnodes = standard_aggregation(A)
# create the vtk file of aggregates
vis_coarse.vis_aggregate_groups(Verts=V, E2V=E2V, Agg=Agg, \
mesh_type='tet', output='vtk', \
fname='output_aggs.vtu')
# create the vtk file for a mesh
vtk_writer.write_basic_mesh(Verts=V, E2V=E2V, \
mesh_type='tet', \
fname='output_mesh.vtu')
# to use Paraview:
# start Paraview: Paraview --data=output_mesh.vtu
# apply
# under display in the object inspector:
# select wireframe representation
# select a better solid color
# selecting surface with edges and low opacity also helps
from scipy.linalg import svd
import pylab
from smoothed_aggregation_helmholtz_solver import smoothed_aggregation_helmholtz_solver, \
planewaves
from pyamg.util.linalg import norm
from pyamg import gallery
from convergence_tools import *
from my_vis import my_vis, shrink_elmts
if __name__ == '__main__':
# Retrieve 2-D Helmholtz Operator and problem data.
# This is operator was discretized with a local
# discontinuous Galerkin method.
data = gallery.load_example('helmholtz_2D')
A = data['A'].tocsr()
omega = data['omega']
h = data['h']
ppw = data['ppw']
elements = data['elements']
vertices = data['vertices']
print "\nRunning 2D Helmholtz Example"
print "-- %1.2f Points-per-wavelength" % ppw
print "-- %1.2e = h, %1.2f = omega" % (h, omega)
print "-- Discretized with a local discontinuous Galerkin method\n on annulus-shaped domain"
# random initial guess for zero right-hand-side
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
# 2D example of viewing aggregates from SA using VTK
from pyamg.aggregation import standard_aggregation
from pyamg.vis import vis_coarse, vtk_writer
from pyamg.gallery import load_example
# retrieve the problem
data = load_example('unit_square')
A = data['A'].tocsr()
V = data['vertices']
E2V = data['elements']
# perform smoothed aggregation
Agg, rootnodes = standard_aggregation(A)
# create the vtk file of aggregates
vis_coarse.vis_aggregate_groups(Verts=V, E2V=E2V, Agg=Agg, \
mesh_type='tri', output='vtk', \
fname='output_aggs.vtu')
# create the vtk file for a mesh
vtk_writer.write_basic_mesh(Verts=V, E2V=E2V, \
mesh_type='tri', \
fname='output_mesh.vtu')
# to use Paraview:
# start Paraview: Paraview --data=output_mesh.vtu
# apply
# under display in the object inspector:
# select wireframe representation
# select a better solid color
# open file: output_aggs.vtu
# 2D example of viewing aggregates from SA using VTK
from pyamg.aggregation import standard_aggregation
from pyamg.vis import vis_coarse, vtk_writer
from pyamg.gallery import load_example
from pyamg import *
from scipy import *
# retrieve the problem
data = load_example('unit_square')
A = data['A'].tocsr()
V = data['vertices']
E2V = data['elements']
# perform smoothed aggregation
ml = smoothed_aggregation_solver(A,keep=True,max_coarse=10)
b = sin(pi*V[:,0])*sin(pi*V[:,1])
x = ml.solve(b)
# create the vtk file of aggregates
vis_coarse.vis_aggregate_groups(Verts=V, E2V=E2V,
Agg=ml.levels[0].AggOp, mesh_type='tri',
output='vtk', fname='output_aggs.vtu')
# create the vtk file for mesh and solution
vtk_writer.write_basic_mesh(Verts=V, E2V=E2V,
pdata = x,
mesh_type='tri',
fname='output_mesh.vtu')
# to use Paraview:
# start Paraview: Paraview --data=output_mesh.vtu
# 2D example of viewing aggregates from SA using VTK
from pyamg.aggregation import standard_aggregation
from pyamg.vis import vis_coarse, vtk_writer
from pyamg.gallery import load_example
# retrieve the problem
data = load_example("unit_square")
A = data["A"].tocsr()
V = data["vertices"]
E2V = data["elements"]
# perform smoothed aggregation
Agg, rootnodes = standard_aggregation(A)
# create the vtk file of aggregates
vis_coarse.vis_aggregate_groups(Verts=V, E2V=E2V, Agg=Agg, mesh_type="tri", output="vtk", fname="output_aggs.vtu")
# create the vtk file for a mesh
vtk_writer.write_basic_mesh(Verts=V, E2V=E2V, mesh_type="tri", fname="output_mesh.vtu")
# to use Paraview:
# start Paraview: Paraview --data=output_mesh.vtu
# apply
# under display in the object inspector:
# select wireframe representation
# select a better solid color
# open file: output_aggs.vtu
# under display in the object inspector:
# select surface with edges representation
# select a better solid color
# increase line width and point size to see these aggs (if present)
def test_range(self):
"""Check that P*R=B"""
numpy.random.seed(0) #make tests repeatable
cases = []
##
# Simple, real-valued diffusion problems
X = load_example('airfoil')
A = X['A'].tocsr(); B = X['B']
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'local'}) ))
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((A,B,('energy', {'maxiter' : 3}) ))
cases.append((A,B,('energy', {'krylov' : 'cgnr'}) ))
cases.append((A,B,('energy', {'krylov' : 'gmres', 'degree' : 2}) ))
A = poisson((10,10), format='csr')
B = ones((A.shape[0],1))
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'diagonal'}) ))
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'local'}) ))
cases.append((A,B,'energy'))
cases.append((A,B,('energy', {'degree' : 2}) ))
cases.append((A,B,('energy', {'krylov' : 'cgnr', 'degree' : 2}) ))
cases.append((A,B,('energy', {'krylov' : 'gmres'}) ))
##
# Simple, imaginary-valued problems
iA = 1.0j*A
iB = 1.0 + rand(iA.shape[0],2) + 1.0j*(1.0 + rand(iA.shape[0],2))
cases.append((iA, B,('jacobi', {'filter' : True, 'weighting' : 'diagonal'}) ))
cases.append((iA, B,('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA,iB,('jacobi', {'filter' : True, 'weighting' : 'local'}) ))
cases.append((iA,iB,('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA.tobsr(blocksize=(5,5)), B, ('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA.tobsr(blocksize=(5,5)), iB, ('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA,B, ('energy', {'krylov' : 'cgnr', 'degree' : 2}) ))
cases.append((iA,iB,('energy', {'krylov' : 'cgnr'}) ))
cases.append((iA.tobsr(blocksize=(5,5)),B, ('energy', {'krylov' : 'cgnr', 'degree' : 2, 'maxiter' : 3}) ))
cases.append((iA.tobsr(blocksize=(5,5)),iB,('energy', {'krylov' : 'cgnr'}) ))
cases.append((iA,B, ('energy', {'krylov' : 'gmres'}) ))
cases.append((iA,iB,('energy', {'krylov' : 'gmres', 'degree' : 2}) ))
cases.append((iA.tobsr(blocksize=(5,5)),B, ('energy', {'krylov' : 'gmres', 'degree' : 2, 'maxiter' : 3}) ))
cases.append((iA.tobsr(blocksize=(5,5)),iB,('energy', {'krylov' : 'gmres'}) ))
##
#
# Simple, imaginary-valued problems
iA = A + 1.0j*scipy.sparse.eye(A.shape[0], A.shape[1])
cases.append((iA,B, ('jacobi', {'filter' : True, 'weighting' : 'local'}) ))
cases.append((iA,B, ('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA,iB,('jacobi', {'filter' : True, 'weighting' : 'diagonal'}) ))
cases.append((iA,iB,('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA.tobsr(blocksize=(4,4)), iB, ('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((iA,B, ('energy', {'krylov' : 'cgnr'}) ))
cases.append((iA.tobsr(blocksize=(4,4)),iB,('energy', {'krylov' : 'cgnr'}) ))
cases.append((iA,B, ('energy', {'krylov' : 'gmres'}) ))
cases.append((iA.tobsr(blocksize=(4,4)),iB, ('energy', {'krylov' : 'gmres', 'degree' : 2, 'maxiter' : 3}) ))
##
#
A = gauge_laplacian(10, spacing=1.0, beta=0.21)
B = ones((A.shape[0],1))
cases.append((A,iB,('jacobi', {'filter' : True, 'weighting' : 'diagonal'}) ))
cases.append((A,iB,('jacobi', {'filter' : True, 'weighting' : 'local'}) ))
cases.append((A,B, ('energy', {'krylov' : 'cg'}) ))
cases.append((A,iB, ('energy', {'krylov' : 'cgnr'}) ))
cases.append((A,iB, ('energy', {'krylov' : 'gmres'}) ))
cases.append((A.tobsr(blocksize=(2,2)),B, ('energy', {'krylov' : 'cgnr', 'degree' : 2, 'maxiter' : 3}) ))
cases.append((A.tobsr(blocksize=(2,2)),iB,('energy', {'krylov' : 'cg'}) ))
cases.append((A.tobsr(blocksize=(2,2)),B, ('energy', {'krylov' : 'gmres', 'degree' : 2, 'maxiter' : 3}) ))
##
#
A,B = linear_elasticity((10,10))
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'diagonal'}) ))
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'local'}) ))
cases.append((A,B,('jacobi', {'filter' : True, 'weighting' : 'block'}) ))
cases.append((A,B,('energy', {'degree' : 2}) ))
cases.append((A,B,('energy', {'krylov' : 'cgnr'}) ))
cases.append((A,B,('energy', {'krylov' : 'gmres', 'degree' : 2}) ))
##
# Classic SA cases
for A,B,smooth in cases:
ml = smoothed_aggregation_solver(A, B=B, max_coarse=1, max_levels=2, smooth=smooth )
P = ml.levels[0].P
B = ml.levels[0].B
R = ml.levels[1].B
assert_almost_equal(P*R, B)
def blocksize(A):
# Helper Function: return the blocksize of a matrix
if isspmatrix_bsr(A):
return A.blocksize[0]
else:
return 1
##
# Root-node cases
counter = 0
for A,B,smooth in cases:
counter += 1
if isinstance( smooth, tuple):
smoother = smooth[0]
else:
smoother = smooth
if smoother == 'energy' and (B.shape[1] >= blocksize(A)):
ml = rootnode_solver(A, B=B, max_coarse=1, max_levels=2, smooth=smooth,
improve_candidates =[('gauss_seidel_nr', {'sweep': 'symmetric', 'iterations': 4}), None],
keep=True, symmetry='nonsymmetric')
T = ml.levels[0].T.tocsr()
Cpts = ml.levels[0].Cpts
Bf = ml.levels[0].B
Bf_H = ml.levels[0].BH
Bc = ml.levels[1].B
P = ml.levels[0].P.tocsr()
##
# P should preserve B in its range, wherever P
# has enough nonzeros
mask = ((P.indptr[1:] - P.indptr[:-1]) >= B.shape[1])
assert_almost_equal( (P*Bc)[mask,:], Bf[mask,:])
assert_almost_equal( (P*Bc)[mask,:], Bf_H[mask,:])
##
# P should be the identity at Cpts
I = eye(T.shape[1], T.shape[1], format='csr', dtype=T.dtype)
I2 = P[Cpts,:]
assert_almost_equal(I.data, I2.data)
assert_equal(I.indptr, I2.indptr)
assert_equal(I.indices, I2.indices)
##
# T should be the identity at Cpts
I2 = T[Cpts,:]
assert_almost_equal(I.data, I2.data)
assert_equal(I.indptr, I2.indptr)
assert_equal(I.indices, I2.indices)
from pyamg.gallery import load_example
from pyamg import smoothed_aggregation_solver, rootnode_solver
from convergence_tools import print_cycle_history
if __name__ == '__main__':
print "Test convergence of a small recirculating flow problem " + \
"that generates a nonsymmetric matrix "
choice = input('\n Input Choice:\n' + \
'1: Run smoothed_aggregation_solver\n' + \
'2: Run rootnode_solver\n' )
# Recirculating flow, nonsymmetric matrix
data = load_example('recirc_flow')
A = data['A'].tocsr()
B = data['B']
elements = data['elements']
vertice = data['vertices']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = A*scipy.rand(A.shape[0])
##
# For demonstration, show that a solver constructed for a symmetric
# operator fails for this matrix.
smooth=('energy', {'krylov' : 'cg'})
SA_build_args={'max_levels':10, 'max_coarse':25, 'coarse_solver':'pinv2', \
'symmetry':'hermitian'}
# 3D example of viewing aggregates from SA using VTK
from pyamg.aggregation import standard_aggregation
from pyamg.vis import vis_coarse, vtk_writer
from pyamg.gallery import load_example
# retrieve the problem
data = load_example('unit_cube')
A = data['A'].tocsr()
V = data['vertices']
E2V = data['elements']
# perform smoothed aggregation
Agg, rootnodes = standard_aggregation(A)
# create the vtk file of aggregates
vis_coarse.vis_aggregate_groups(Verts=V, E2V=E2V, Agg=Agg, \
mesh_type='tet', output='vtk', \
fname='output_aggs.vtu')
# create the vtk file for a mesh
vtk_writer.write_basic_mesh(Verts=V, E2V=E2V, \
mesh_type='tet', \
fname='output_mesh.vtu')
# to use Paraview:
# start Paraview: Paraview --data=output_mesh.vtu
# apply
# under display in the object inspector:
# select wireframe representation
# select a better solid color
# selecting surface with edges and low opacity also helps
def test_evolution_strength_of_connection(self):
# Params: A, B, epsilon=4.0, k=2, proj_type="l2"
cases = []
# Ensure that isotropic diffusion results in isotropic strength stencil
for N in [3, 5, 7, 10]:
A = poisson((N, ), format='csr')
B = np.ones((A.shape[0], 1))
cases.append({
'A': A.copy(),
'B': B.copy(),
'epsilon': 4.0,
'k': 2,
'proj': 'l2'
})
# Ensure that anisotropic diffusion results in an anisotropic
# strength stencil
for N in [3, 6, 7]:
u = np.ones(N * N)
A = spdiags([-u, -0.001 * u, 2.002 * u, -0.001 * u, -u],
[-N, -1, 0, 1, N],
N * N,
N * N,
format='csr')
B = np.ones((A.shape[0], 1))
cases.append({
'A': A.copy(),
'B': B.copy(),
'epsilon': 4.0,
'k': 2,
'proj': 'l2'
})
# Ensure that isotropic elasticity results in an isotropic stencil
for N in [3, 6, 7]:
(A, B) = linear_elasticity((N, N), format='bsr')
cases.append({
'A': A.copy(),
'B': B.copy(),
'epsilon': 32.0,
'k': 8,
'proj': 'D_A'
})
# Run an example with a non-uniform stencil
ex = load_example('airfoil')
A = ex['A'].tocsr()
B = np.ones((A.shape[0], 1))
cases.append({
'A': A.copy(),
'B': B.copy(),
'epsilon': 8.0,
'k': 4,
'proj': 'D_A'
})
Absr = A.tobsr(blocksize=(5, 5))
cases.append({
'A': Absr.copy(),
'B': B.copy(),
'epsilon': 8.0,
'k': 4,
'proj': 'D_A'
})
# Different B
B = arange(1, 2 * A.shape[0] + 1, dtype=float).reshape(-1, 2)
cases.append({
'A': A.copy(),
'B': B.copy(),
'epsilon': 4.0,
'k': 2,
'proj': 'l2'
})
cases.append({
'A': Absr.copy(),
'B': B.copy(),
'epsilon': 4.0,
'k': 2,
'proj': 'l2'
})
# Zero row and column
A.data[A.indptr[4]:A.indptr[5]] = 0.0
A = A.tocsc()
A.data[A.indptr[4]:A.indptr[5]] = 0.0
A.eliminate_zeros()
A = A.tocsr()
A.sort_indices()
cases.append({
'A': A.copy(),
'B': B.copy(),
'epsilon': 4.0,
'k': 2,
'proj': 'l2'
})
Absr = A.tobsr(blocksize=(5, 5))
cases.append({
'A': Absr.copy(),
'B': B.copy(),
'epsilon': 4.0,
'k': 2,
'proj': 'l2'
})
for ca in cases:
scipy.random.seed(0) # make results deterministic
result = evolution_soc(ca['A'],
ca['B'],
epsilon=ca['epsilon'],
k=ca['k'],
proj_type=ca['proj'],
symmetrize_measure=False)
scipy.random.seed(0) # make results deterministic
expected = reference_evolution_soc(ca['A'],
ca['B'],
epsilon=ca['epsilon'],
k=ca['k'],
proj_type=ca['proj'])
assert_array_almost_equal(result.todense(),
expected.todense(),
decimal=4)
scipy.random.seed(0) # make results deterministic
result = evolution_soc(ca['A'],
ca['B'],
epsilon=ca['epsilon'],
k=ca['k'],
proj_type=ca['proj'],
symmetrize_measure=False,
weighting='local')
scipy.random.seed(0) # make results deterministic
expected = reference_evolution_soc(ca['A'],
ca['B'],
epsilon=ca['epsilon'],
k=ca['k'],
proj_type=ca['proj'],
weighting='local')
assert_array_almost_equal(result.todense(),
expected.todense(),
decimal=4)
# Test Scale Invariance for multiple near nullspace candidates
(A, B) = linear_elasticity((5, 5), format='bsr')
scipy.random.seed(0) # make results deterministic
result_unscaled = evolution_soc(A,
B,
epsilon=4.0,
k=2,
proj_type="D_A",
symmetrize_measure=False)
# create scaled A
D = spdiags([arange(A.shape[0], 2 * A.shape[0], dtype=float)], [0],
A.shape[0],
A.shape[0],
format='csr')
Dinv = spdiags([1.0 / arange(A.shape[0], 2 * A.shape[0], dtype=float)],
[0],
A.shape[0],
A.shape[0],
format='csr')
scipy.random.seed(0) # make results deterministic
result_scaled = evolution_soc((D * A * D).tobsr(blocksize=(2, 2)),
Dinv * B,
epsilon=4.0,
k=2,
proj_type="D_A",
symmetrize_measure=False)
assert_array_almost_equal(result_scaled.todense(),
result_unscaled.todense(),
decimal=2)
from scipy.linalg import svd
import pylab
from smoothed_aggregation_helmholtz_solver import smoothed_aggregation_helmholtz_solver, \
planewaves
from pyamg.util.linalg import norm
from pyamg import gallery
from convergence_tools import *
from my_vis import my_vis, shrink_elmts
if __name__ == '__main__':
# Retrieve 2-D Helmholtz Operator and problem data.
# This is operator was discretized with a local
# discontinuous Galerkin method.
data = gallery.load_example('helmholtz_2D')
A = data['A'].tocsr()
omega = data['omega']
h = data['h']
ppw = data['ppw']
elements = data['elements']
vertices = data['vertices']
print "\nRunning 2D Helmholtz Example"
print "-- %1.2f Points-per-wavelength"%ppw
print "-- %1.2e = h, %1.2f = omega"%(h,omega)
print "-- Discretized with a local discontinuous Galerkin method\n on annulus-shaped domain"
# random initial guess for zero right-hand-side
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
"""
import numpy
import scipy
from pyamg.gallery import load_example
from pyamg import smoothed_aggregation_solver
from convergence_tools import print_cycle_history
if __name__ == '__main__':
print "\nDiffusion problem discretized with p=5 and the local\n" + \
"discontinuous Galerkin method."
# Discontinuous Galerkin Diffusion Problem
data = load_example('local_disc_galerkin_diffusion')
A = data['A'].tocsr()
B = data['B']
elements = data['elements']
vertices = data['vertices']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = numpy.zeros_like(x0)
##
# For demonstration, show that a naive SA solver
# yields unsatisfactory convergence
smooth = ('jacobi', {'filter': True})
strength = ('symmetric', {'theta': 0.1})
SA_solve_args = {'cycle': 'W', 'maxiter': 20, 'tol': 1e-8, 'accel': 'cg'}
SA_build_args={'max_levels':10, 'max_coarse':25, 'coarse_solver':'pinv2', \
##
# To run the best solver found above, uncomment next two lines
#from elas_diagnostic import elas_diagnostic
#elas_diagnostic(A)
if choice == 4:
##
# Try a basic nonsymmetric recirculating flow problem
# --> Only use V-cycles by specifying cycle_list
# --> Don't specify symmetry and definiteness and allow for auto-detection
# --> Specify the maximum coarse size and coarse grid solver with coarse_size_list
# --> Try two different Krylov wrappers and set the maximum number of iterations
# and halting tolerance with krylov_list.
A = gallery.load_example('recirc_flow')['A'].tocsr()
solver_diagnostics(A,
fname='recirc_flow_diagnostic',
cycle_list=['V'],
coarse_size_list=[(15, 'pinv')],
krylov_list=[('gmres', {
'tol': 1e-12,
'maxiter': 100
}), ('bicgstab', {
'tol': 1e-12,
'maxiter': 100
})])
##
# To run the best solver found above, uncomment next two lines
#from recirc_flow_diagnostic import recirc_flow_diagnostic
#recirc_flow_diagnostic(A)
def test_evolution_strength_of_connection(self):
# Params: A, B, epsilon=4.0, k=2, proj_type="l2"
cases = []
# Ensure that isotropic diffusion results in isotropic strength stencil
for N in [3, 5, 7, 10]:
A = poisson((N,), format='csr')
B = np.ones((A.shape[0], 1))
cases.append({'A': A.copy(), 'B': B.copy(), 'epsilon': 4.0,
'k': 2, 'proj': 'l2'})
# Ensure that anisotropic diffusion results in an anisotropic
# strength stencil
for N in [3, 6, 7]:
u = np.ones(N*N)
A = spdiags([-u, -0.001*u, 2.002*u, -0.001*u, -u],
[-N, -1, 0, 1, N], N*N, N*N, format='csr')
B = np.ones((A.shape[0], 1))
cases.append({'A': A.copy(), 'B': B.copy(), 'epsilon': 4.0,
'k': 2, 'proj': 'l2'})
# Ensure that isotropic elasticity results in an isotropic stencil
for N in [3, 6, 7]:
(A, B) = linear_elasticity((N, N), format='bsr')
cases.append({'A': A.copy(), 'B': B.copy(), 'epsilon': 32.0,
'k': 8, 'proj': 'D_A'})
# Run an example with a non-uniform stencil
ex = load_example('airfoil')
A = ex['A'].tocsr()
B = np.ones((A.shape[0], 1))
cases.append({'A': A.copy(), 'B': B.copy(), 'epsilon': 8.0, 'k': 4,
'proj': 'D_A'})
Absr = A.tobsr(blocksize=(5, 5))
cases.append({'A': Absr.copy(), 'B': B.copy(), 'epsilon': 8.0, 'k': 4,
'proj': 'D_A'})
# Different B
B = arange(1, 2*A.shape[0]+1, dtype=float).reshape(-1, 2)
cases.append({'A': A.copy(), 'B': B.copy(), 'epsilon': 4.0, 'k': 2,
'proj': 'l2'})
cases.append({'A': Absr.copy(), 'B': B.copy(), 'epsilon': 4.0, 'k': 2,
'proj': 'l2'})
# Zero row and column
A.data[A.indptr[4]:A.indptr[5]] = 0.0
A = A.tocsc()
A.data[A.indptr[4]:A.indptr[5]] = 0.0
A.eliminate_zeros()
A = A.tocsr()
A.sort_indices()
cases.append({'A': A.copy(), 'B': B.copy(), 'epsilon': 4.0, 'k': 2,
'proj': 'l2'})
Absr = A.tobsr(blocksize=(5, 5))
cases.append({'A': Absr.copy(), 'B': B.copy(), 'epsilon': 4.0, 'k': 2,
'proj': 'l2'})
for ca in cases:
scipy.random.seed(0) # make results deterministic
result = evolution_soc(ca['A'], ca['B'], epsilon=ca['epsilon'],
k=ca['k'], proj_type=ca['proj'],
symmetrize_measure=False)
scipy.random.seed(0) # make results deterministic
expected = reference_evolution_soc(ca['A'], ca['B'],
epsilon=ca['epsilon'],
k=ca['k'], proj_type=ca['proj'])
assert_array_almost_equal(result.todense(), expected.todense(),
decimal=4)
# Test Scale Invariance for multiple near nullspace candidates
(A, B) = linear_elasticity((5, 5), format='bsr')
scipy.random.seed(0) # make results deterministic
result_unscaled = evolution_soc(A, B, epsilon=4.0,
k=2, proj_type="D_A",
symmetrize_measure=False)
# create scaled A
D = spdiags([arange(A.shape[0], 2*A.shape[0], dtype=float)],
[0], A.shape[0], A.shape[0], format='csr')
Dinv = spdiags([1.0/arange(A.shape[0], 2*A.shape[0], dtype=float)],
[0], A.shape[0], A.shape[0], format='csr')
scipy.random.seed(0) # make results deterministic
result_scaled = evolution_soc((D*A*D).tobsr(blocksize=(2, 2)),
Dinv*B, epsilon=4.0, k=2,
proj_type="D_A",
symmetrize_measure=False)
assert_array_almost_equal(result_scaled.todense(),
result_unscaled.todense(), decimal=2)
from pyamg.gallery import load_example
from pyamg import smoothed_aggregation_solver, rootnode_solver
from convergence_tools import print_cycle_history
if __name__ == '__main__':
print "Test convergence of a small recirculating flow problem " + \
"that generates a nonsymmetric matrix "
choice = input('\n Input Choice:\n' + \
'1: Run smoothed_aggregation_solver\n' + \
'2: Run rootnode_solver\n' )
# Recirculating flow, nonsymmetric matrix
data = load_example('recirc_flow')
A = data['A'].tocsr()
B = data['B']
elements = data['elements']
vertice = data['vertices']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = A * scipy.rand(A.shape[0])
##
# For demonstration, show that a solver constructed for a symmetric
# operator fails for this matrix.
smooth = ('energy', {'krylov': 'cg'})
SA_build_args={'max_levels':10, 'max_coarse':25, 'coarse_solver':'pinv2', \
'symmetry':'hermitian'}
"""
import numpy
import scipy
from pyamg.gallery import load_example
from pyamg import smoothed_aggregation_solver
from convergence_tools import print_cycle_history
if __name__ == '__main__':
print "\nDiffusion problem discretized with p=5 and the local\n" + \
"discontinuous Galerkin method."
# Discontinuous Galerkin Diffusion Problem
data = load_example('local_disc_galerkin_diffusion')
A = data['A'].tocsr()
B = data['B']
elements = data['elements']
vertices = data['vertices']
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = numpy.zeros_like(x0)
##
# For demonstration, show that a naive SA solver
# yields unsatisfactory convergence
smooth=('jacobi', {'filter' : True})
strength=('symmetric', {'theta' : 0.1})
SA_solve_args={'cycle':'W', 'maxiter':20, 'tol':1e-8, 'accel' : 'cg'}
SA_build_args={'max_levels':10, 'max_coarse':25, 'coarse_solver':'pinv2', \
def test_nonsymmetric(self):
# problem data
data = load_example("recirc_flow")
A = data["A"].tocsr()
B = data["B"]
numpy.random.seed(625)
x0 = scipy.rand(A.shape[0])
b = A * scipy.rand(A.shape[0])
# solver parameters
smooth = ("energy", {"krylov": "gmres"})
SA_build_args = {"max_coarse": 25, "coarse_solver": "pinv2", "symmetry": "nonsymmetric"}
SA_solve_args = {"cycle": "V", "maxiter": 20, "tol": 1e-8}
strength = [("evolution", {"k": 2, "epsilon": 8.0})]
smoother = ("gauss_seidel_nr", {"sweep": "symmetric", "iterations": 1})
improve_candidates = [("gauss_seidel_nr", {"sweep": "symmetric", "iterations": 4}), None]
# Construct solver with nonsymmetric parameters
sa = rootnode_solver(
A,
B=B,
smooth=smooth,
improve_candidates=improve_candidates,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
**SA_build_args
)
residuals = []
# stand-alone solve
x = sa.solve(b, x0=x0, residuals=residuals, **SA_solve_args)
residuals = array(residuals)
avg_convergence_ratio = (residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Test 1 %1.3e, %1.3e" % (avg_convergence_ratio, 0.7)
assert avg_convergence_ratio < 0.7
# accelerated solve
residuals = []
x = sa.solve(b, x0=x0, residuals=residuals, accel="gmres", **SA_solve_args)
del x
residuals = array(residuals)
avg_convergence_ratio = (residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Test 2 %1.3e, %1.3e" % (avg_convergence_ratio, 0.45)
assert avg_convergence_ratio < 0.45
# test that nonsymmetric parameters give the same result as symmetric
# parameters for Poisson problem
A = poisson((15, 15), format="csr")
strength = "symmetric"
SA_build_args["symmetry"] = "nonsymmetric"
sa_nonsymm = rootnode_solver(
A,
B=ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args
)
SA_build_args["symmetry"] = "symmetric"
sa_symm = rootnode_solver(
A,
B=ones((A.shape[0], 1)),
smooth=smooth,
strength=strength,
presmoother=smoother,
postsmoother=smoother,
improve_candidates=None,
**SA_build_args
)
for (symm_lvl, nonsymm_lvl) in zip(sa_nonsymm.levels, sa_symm.levels):
assert_array_almost_equal(symm_lvl.A.todense(), nonsymm_lvl.A.todense())