def setUp(self):
self.cases = []
# Test 1
A = poisson((5000,), format="csr")
Ai = A + 1.0j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((Ai, None, 0.12, "symmetric", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((Ai, None, 0.12, "symmetric", ("energy", {"krylov": "gmres"})))
# Test 2
A = poisson((71, 71), format="csr")
Ai = A + (0.625 / 0.01) * 1j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((Ai, None, 1e-3, "symmetric", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((Ai, None, 1e-3, "symmetric", ("energy", {"krylov": "cgnr"})))
# Test 3
A = poisson((60, 60), format="csr")
Ai = 1.0j * A
self.cases.append((Ai, None, 0.3, "symmetric", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((Ai, None, 0.6, "symmetric", ("energy", {"krylov": "cgnr", "maxiter": 8})))
self.cases.append((Ai, None, 0.6, "symmetric", ("energy", {"krylov": "gmres", "maxiter": 8})))
# Test 4
# Use an "inherently" imaginary problem, the Gauge Laplacian in 2D from
# Quantum Chromodynamics,
A = gauge_laplacian(70, spacing=1.0, beta=0.41)
self.cases.append((A, None, 0.4, "hermitian", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((A, None, 0.4, "hermitian", ("energy", {"krylov": "cg"})))
def test_coarse_grid_solver(self):
cases = []
cases.append(csr_matrix(diag(arange(1, 5, dtype=float))))
cases.append(poisson((4,), format='csr'))
cases.append(poisson((4, 4), format='csr'))
from pyamg.krylov import cg
def fn(A, b):
return cg(A, b)[0]
# method should be almost exact for small matrices
for A in cases:
for solver in ['splu', 'pinv', 'pinv2', 'lu', 'cholesky',
'cg', fn]:
s = coarse_grid_solver(solver)
b = arange(A.shape[0], dtype=A.dtype)
x = s(A, b)
assert_almost_equal(A*x, b)
# subsequent calls use cached data
x = s(A, b)
assert_almost_equal(A*x, b)
def setUp(self):
self.cases = []
A = poisson((5000,), format='csr')
self.cases.append((A, None, 0.4, 'symmetric',
('jacobi', {'omega': 4.0 / 3.0})))
self.cases.append((A, None, 0.4, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, None, 0.5, 'symmetric',
('energy', {'krylov': 'gmres'})))
A = poisson((60, 60), format='csr')
self.cases.append((A, None, 0.42, 'symmetric',
('jacobi', {'omega': 4.0 / 3.0})))
self.cases.append((A, None, 0.42, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, None, 0.42, 'symmetric',
('energy', {'krylov': 'cgnr'})))
A, B = linear_elasticity((50, 50), format='bsr')
self.cases.append((A, B, 0.32, 'symmetric',
('jacobi', {'omega': 4.0 / 3.0})))
self.cases.append((A, B, 0.22, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, B, 0.42, 'symmetric',
('energy', {'krylov': 'cgnr'})))
self.cases.append((A, B, 0.42, 'symmetric',
('energy', {'krylov': 'gmres'})))
def setUp(self):
self.cases = []
# Test 1
A = poisson((5000,), format='csr')
Ai = A + 1.0j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((Ai, None, 0.12, 'symmetric',
('energy', {'krylov': 'gmres'})))
# Test 2
A = poisson((71, 71), format='csr')
Ai = A + (0.625 / 0.01) * 1.0j *\
scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((Ai, None, 1e-3, 'symmetric',
('energy', {'krylov': 'cgnr'})))
# Test 3
A = poisson((60, 60), format='csr')
Ai = 1.0j * A
self.cases.append((Ai, None, 0.35, 'symmetric',
('energy', {'krylov': 'cgnr', 'maxiter': 8})))
self.cases.append((Ai, None, 0.35, 'symmetric',
('energy', {'krylov': 'gmres', 'maxiter': 8})))
# Test 4 Use an "inherently" imaginary problem, the Gauge Laplacian in
# 2D from Quantum Chromodynamics,
A = gauge_laplacian(70, spacing=1.0, beta=0.41)
self.cases.append((A, None, 0.25, 'hermitian',
('energy', {'krylov': 'cg'})))
def setUp(self):
self.cases = []
# Consider "Helmholtz" like problems with an imaginary shift so that the operator
# should still be SPD in a sense and SA should perform well.
# There are better near nullspace vectors than the default,
# but a constant should give a convergent solver, nonetheless.
A = poisson( (100,), format='csr'); A = A + 1.0j*scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((A, None))
A = poisson( (10,10), format='csr'); A = A + 1.0j*scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((A, None))
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
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 = []
# Consider "Helmholtz" like problems with an imaginary shift so that
# the operator should still be SPD in a sense and SA should perform
# well. There are better near nullspace vectors than the default, but
# a constant should give a convergent solver, nonetheless.
A = poisson((100,), format='csr')
A = A + 1.0j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((A, None))
A = poisson((10, 10), format='csr')
A = A + 1.0j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((A, None))
def setUp(self):
self.cases = []
np.random.seed(2828777142)
# Poisson problems in 1D and 2D
N = 20
self.cases.append((poisson(
(2 * N, ), format='csr'), np.random.rand(2 * N, ))) # 0
self.cases.append((poisson(
(N, N), format='csr'), np.random.rand(N * N, ))) # 1
# Boxed examples
A = load_example('recirc_flow')['A'].tocsr() # 2
self.cases.append((A, np.random.rand(A.shape[0], )))
A = load_example('bar')['A'].tobsr(blocksize=(3, 3)) # 3
self.cases.append((A, np.random.rand(A.shape[0], )))
def setUp(self):
self.cases = []
A = poisson((5000,), format="csr")
self.cases.append((A, None, 0.4, "symmetric", ("energy", {"krylov": "cg"})))
self.cases.append((A, None, 0.4, "symmetric", ("energy", {"krylov": "gmres"})))
A = poisson((75, 75), format="csr")
self.cases.append((A, None, 0.26, "symmetric", ("energy", {"krylov": "cg"})))
self.cases.append((A, None, 0.30, "symmetric", ("energy", {"krylov": "cgnr"})))
A, B = linear_elasticity((50, 50), format="bsr")
self.cases.append((A, B, 0.3, "symmetric", ("energy", {"krylov": "cg"})))
self.cases.append((A, B, 0.3, "symmetric", ("energy", {"krylov": "cgnr"})))
self.cases.append((A, B, 0.3, "symmetric", ("energy", {"krylov": "gmres"})))
def setUp(self):
self.cases = []
# Test 1
A = poisson((5000, ), format='csr')
Ai = A + 1.0j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((Ai, None, 0.12, 'symmetric', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((Ai, None, 0.12, 'symmetric', ('energy', {
'krylov': 'gmres'
})))
# Test 2
A = poisson((71, 71), format='csr')
Ai = A + (0.625 / 0.01) * 1j * scipy.sparse.eye(A.shape[0], A.shape[1])
self.cases.append((Ai, None, 1e-3, 'symmetric', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((Ai, None, 1e-3, 'symmetric', ('energy', {
'krylov': 'cgnr'
})))
# Test 3
A = poisson((60, 60), format='csr')
Ai = 1.0j * A
self.cases.append((Ai, None, 0.3, 'symmetric', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((Ai, None, 0.6, 'symmetric', ('energy', {
'krylov': 'cgnr',
'maxiter': 8
})))
self.cases.append((Ai, None, 0.6, 'symmetric', ('energy', {
'krylov': 'gmres',
'maxiter': 8
})))
# Test 4
# Use an "inherently" imaginary problem, the Gauge Laplacian in 2D from
# Quantum Chromodynamics,
A = gauge_laplacian(70, spacing=1.0, beta=0.41)
self.cases.append((A, None, 0.4, 'hermitian', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((A, None, 0.4, 'hermitian', ('energy', {
'krylov': 'cg'
})))
def test_symmetry(self):
# Test that a basic V-cycle yields a symmetric linear operator. Common
# reasons for failure are problems with using the same rho for the
# pres/post-smoothers and using the same block_D_inv for
# pre/post-smoothers.
n = 500
A = poisson((n,), format="csr")
smoothers = [
("gauss_seidel", {"sweep": "symmetric"}),
("schwarz", {"sweep": "symmetric"}),
("block_gauss_seidel", {"sweep": "symmetric"}),
"jacobi",
"block_jacobi",
]
rng = arange(1, n + 1, dtype="float").reshape(-1, 1)
Bs = [ones((n, 1)), hstack((ones((n, 1)), rng))]
for smoother in smoothers:
for B in Bs:
ml = smoothed_aggregation_solver(A, B, max_coarse=10, presmoother=smoother, postsmoother=smoother)
P = ml.aspreconditioner()
x = rand(n)
y = rand(n)
assert_approx_equal(dot(P * x, y), dot(x, P * y))
def test_DAD(self):
A = poisson((50, 50), format='csr')
x = sp.rand(A.shape[0])
b = sp.rand(A.shape[0])
D = diag_sparse(1.0 /
np.sqrt(10**(12 * sp.rand(A.shape[0]) - 6))).tocsr()
D_inv = diag_sparse(1.0 / D.data)
# DAD = D * A * D
B = np.ones((A.shape[0], 1))
# TODO force 2 level method and check that result is the same
kwargs = {'max_coarse': 1, 'max_levels': 2, 'coarse_solver': 'splu'}
sa = smoothed_aggregation_solver(D * A * D, D_inv * B, **kwargs)
residuals = []
x_sol = sa.solve(b, x0=x, maxiter=10, tol=1e-12, residuals=residuals)
del x_sol
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Diagonal Scaling Test: %1.3e, %1.3e" %
# (avg_convergence_ratio, 0.25)
assert (avg_convergence_ratio < 0.25)
def test_improve_candidates(self):
##
# test improve_candidates for the Poisson problem and elasticity, where rho_scale is
# the amount that each successive improve_candidates option should improve convergence
# over the previous improve_candidates option.
improve_candidates_list = [None, [('block_gauss_seidel', {'iterations' : 4, 'sweep':'symmetric'})] ]
# make tests repeatable
numpy.random.seed(0)
cases = []
A_elas,B_elas = linear_elasticity( (60,60), format='bsr')
# Matrix Candidates rho_scale
cases.append( (poisson( (61,61), format='csr'), ones((61*61,1)), 0.9 ) )
cases.append( (A_elas, B_elas, 0.9 ) )
for (A,B,rho_scale) in cases:
last_rho = -1.0
x0 = rand(A.shape[0],1)
b = rand(A.shape[0],1)
for improve_candidates in improve_candidates_list:
ml = smoothed_aggregation_solver(A, B, max_coarse=10, improve_candidates=improve_candidates)
residuals=[]
x_sol = ml.solve(b,x0=x0,maxiter=20,tol=1e-10, residuals=residuals)
rho = (residuals[-1]/residuals[0])**(1.0/len(residuals))
if last_rho == -1.0:
last_rho = rho
else:
# each successive improve_candidates option should be an improvement on the previous
# print "\nimprove_candidates Test: %1.3e, %1.3e, %d\n"%(rho,rho_scale*last_rho,A.shape[0])
assert(rho < rho_scale*last_rho)
last_rho = rho
def setUp(self):
self.cases = []
# Poisson problems in 2D
for N in [2,3,5,7,8]:
A = poisson( (N,N), format='csr'); A.data = A.data + 0.001j*rand(A.nnz)
self.cases.append(A)
def test_solver_parameters(self):
A = poisson((50, 50), format='csr')
for method in methods:
# method = ('richardson', {'omega':4.0/3.0})
ml = smoothed_aggregation_solver(A,
presmoother=method,
postsmoother=method,
max_coarse=10)
residuals = profile_solver(ml)
assert ((residuals[-1] / residuals[0])**(1.0 / len(residuals)) <
0.95)
assert (ml.symmetric_smoothing)
for method in methods2:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
residuals = profile_solver(ml)
assert ((residuals[-1] / residuals[0])**(1.0 / len(residuals)) <
0.95)
assert (not ml.symmetric_smoothing)
for method in methods3:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
assert (ml.symmetric_smoothing)
for method in methods4:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
assert (not ml.symmetric_smoothing)
def test_accel(self):
from pyamg import smoothed_aggregation_solver
from pyamg.krylov import cg, bicgstab
A = poisson((50, 50), format='csr')
b = rand(A.shape[0])
ml = smoothed_aggregation_solver(A)
# cg halts based on the preconditioner norm
for accel in ['cg', cg]:
x = ml.solve(b, maxiter=30, tol=1e-8, accel=accel)
assert(precon_norm(b - A*x, ml) < 1e-8*precon_norm(b, ml))
residuals = []
x = ml.solve(b, maxiter=30, tol=1e-8, residuals=residuals,
accel=accel)
assert(precon_norm(b - A*x, ml) < 1e-8*precon_norm(b, ml))
# print residuals
assert_almost_equal(precon_norm(b - A*x, ml), residuals[-1])
# cgs and bicgstab use the Euclidean norm
for accel in ['bicgstab', 'cgs', bicgstab]:
x = ml.solve(b, maxiter=30, tol=1e-8, accel=accel)
assert(norm(b - A*x) < 1e-8*norm(b))
residuals = []
x = ml.solve(b, maxiter=30, tol=1e-8, residuals=residuals,
accel=accel)
assert(norm(b - A*x) < 1e-8*norm(b))
# print residuals
assert_almost_equal(norm(b - A*x), residuals[-1])
def test_symmetry(self):
# Test that a basic V-cycle yields a symmetric linear operator. Common
# reasons for failure are problems with using the same rho for the
# pres/post-smoothers and using the same block_D_inv for
# pre/post-smoothers.
n = 500
A = poisson((n,), format='csr')
smoothers = [('gauss_seidel', {'sweep': 'symmetric'}),
('schwarz', {'sweep': 'symmetric'}),
('block_gauss_seidel', {'sweep': 'symmetric'}),
'jacobi', 'block_jacobi']
rng = np.arange(1, n + 1, dtype='float').reshape(-1, 1)
Bs = [np.ones((n, 1)), sp.hstack((np.ones((n, 1)), rng))]
# TODO:
# why does python 3 require significant=6 while python 2 passes
# why does python 3 yield a different dot() below than python 2
# only for: ('gauss_seidel', {'sweep': 'symmetric'})
for smoother in smoothers:
for B in Bs:
ml = smoothed_aggregation_solver(A, B, max_coarse=10,
presmoother=smoother,
postsmoother=smoother)
P = ml.aspreconditioner()
np.random.seed(0)
x = sp.rand(n,)
y = sp.rand(n,)
out = (np.dot(P * x, y), np.dot(x, P * y))
# print("smoother = %s %g %g" % (smoother, out[0], out[1]))
assert_approx_equal(out[0], out[1])
def test_solver_parameters(self):
A = poisson((50, 50), format='csr')
for method in methods:
#method = ('richardson', {'omega':4.0/3.0})
ml = smoothed_aggregation_solver(A,
presmoother=method,
postsmoother=method,
max_coarse=10)
residuals = profile_solver(ml)
#print "method",method
#print "residuals",residuals
#print "convergence rate:",(residuals[-1]/residuals[0])**(1.0/len(residuals))
assert ((residuals[-1] / residuals[0])**(1.0 / len(residuals)) <
0.95)
for method in methods2:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
residuals = profile_solver(ml)
#print "method",method
#print "residuals",residuals
#print "convergence rate:",(residuals[-1]/residuals[0])**(1.0/len(residuals))
assert ((residuals[-1] / residuals[0])**(1.0 / len(residuals)) <
0.95)
def test_poisson(self):
cases = []
cases.append((500,))
cases.append((250, 250))
cases.append((25, 25, 25))
for case in cases:
A = poisson(case, format='csr')
np.random.seed(0) # make tests repeatable
x = sp.rand(A.shape[0])
b = A*sp.rand(A.shape[0]) # zeros_like(x)
ml = ruge_stuben_solver(A, max_coarse=50)
res = []
x_sol = ml.solve(b, x0=x, maxiter=20, tol=1e-12,
residuals=res)
del x_sol
avg_convergence_ratio = (res[-1]/res[0])**(1.0/len(res))
assert(avg_convergence_ratio < 0.20)
def test_solver_parameters(self):
A = poisson((50, 50), format='csr')
for method in methods:
# method = ('richardson', {'omega':4.0/3.0})
ml = smoothed_aggregation_solver(A, presmoother=method,
postsmoother=method,
max_coarse=10)
residuals = profile_solver(ml)
assert((residuals[-1]/residuals[0])**(1.0/len(residuals)) < 0.95)
assert(ml.symmetric_smoothing)
for method in methods2:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
residuals = profile_solver(ml)
assert((residuals[-1]/residuals[0])**(1.0/len(residuals)) < 0.95)
assert(not ml.symmetric_smoothing)
for method in methods3:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
assert(ml.symmetric_smoothing)
for method in methods4:
ml = smoothed_aggregation_solver(A, max_coarse=10)
change_smoothers(ml, presmoother=method[0], postsmoother=method[1])
assert(not ml.symmetric_smoothing)
def test_DAD(self):
A = poisson((50, 50), format="csr")
x = rand(A.shape[0])
b = rand(A.shape[0])
D = diag_sparse(1.0 / sqrt(10 ** (12 * rand(A.shape[0]) - 6))).tocsr()
D_inv = diag_sparse(1.0 / D.data)
DAD = D * A * D
B = ones((A.shape[0], 1))
# TODO force 2 level method and check that result is the same
kwargs = {"max_coarse": 1, "max_levels": 2, "coarse_solver": "splu"}
sa = smoothed_aggregation_solver(D * A * D, D_inv * B, **kwargs)
residuals = []
x_sol = sa.solve(b, x0=x, maxiter=10, tol=1e-12, residuals=residuals)
avg_convergence_ratio = (residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Diagonal Scaling Test: %1.3e, %1.3e" %
# (avg_convergence_ratio, 0.25)
assert avg_convergence_ratio < 0.25
def test_coarse_solver_opts(self):
# these tests are meant to test whether coarse solvers are correctly
# passed parameters
A = poisson((30, 30), format='csr')
b = sp.rand(A.shape[0], 1)
# for each pair, the first entry should yield an SA solver that
# converges in fewer iterations for a basic Poisson problem
coarse_solver_pairs = [(('jacobi', {'iterations': 30}), 'jacobi')]
coarse_solver_pairs.append((('gauss_seidel', {
'iterations': 30
}), 'gauss_seidel'))
coarse_solver_pairs.append(('gauss_seidel', 'jacobi'))
coarse_solver_pairs.append(('cg', ('cg', {'tol': 10.0})))
coarse_solver_pairs.append(('pinv2', ('pinv2', {'cond': 1.0})))
for coarse1, coarse2 in coarse_solver_pairs:
r1 = []
r2 = []
sa1 = smoothed_aggregation_solver(A,
coarse_solver=coarse1,
max_coarse=500)
sa2 = smoothed_aggregation_solver(A,
coarse_solver=coarse2,
max_coarse=500)
x1 = sa1.solve(b, residuals=r1)
x2 = sa2.solve(b, residuals=r2)
del x1, x2
assert ((len(r1) + 5) < len(r2))
def test_symmetry(self):
# Test that a basic V-cycle yields a symmetric linear operator. Common
# reasons for failure are problems with using the same rho for the
# pres/post-smoothers and using the same block_D_inv for
# pre/post-smoothers.
n = 500
A = poisson((n,), format='csr')
smoothers = [('gauss_seidel', {'sweep': 'symmetric'}),
('schwarz', {'sweep': 'symmetric'}),
('block_gauss_seidel', {'sweep': 'symmetric'}),
'jacobi', 'block_jacobi']
rng = np.arange(1, n + 1, dtype='float').reshape(-1, 1)
Bs = [np.ones((n, 1)), np.hstack((np.ones((n, 1)), rng))]
# TODO:
# why does python 3 require significant=6 while python 2 passes
# why does python 3 yield a different dot() below than python 2
# only for: ('gauss_seidel', {'sweep': 'symmetric'})
for smoother in smoothers:
for B in Bs:
ml = smoothed_aggregation_solver(A, B, max_coarse=10,
presmoother=smoother,
postsmoother=smoother)
P = ml.aspreconditioner()
np.random.seed(3849986793)
x = np.random.rand(n,)
y = np.random.rand(n,)
out = (np.dot(P * x, y), np.dot(x, P * y))
# print("smoother = %s %g %g" % (smoother, out[0], out[1]))
assert_approx_equal(out[0], out[1])
def test_DAD(self):
A = poisson((50, 50), format='csr')
x = rand(A.shape[0])
b = rand(A.shape[0])
D = diag_sparse(1.0 / sqrt(10 ** (12 * rand(A.shape[0]) - 6))).tocsr()
D_inv = diag_sparse(1.0 / D.data)
DAD = D * A * D
B = ones((A.shape[0], 1))
# TODO force 2 level method and check that result is the same
kwargs = {'max_coarse': 1, 'max_levels': 2, 'coarse_solver': 'splu'}
sa = rootnode_solver(D * A * D, D_inv * B, **kwargs)
residuals = []
x_sol = sa.solve(b, x0=x, maxiter=10, tol=1e-12, residuals=residuals)
avg_convergence_ratio =\
(residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
# print "Diagonal Scaling Test: %1.3e, %1.3e" %
# (avg_convergence_ratio, 0.4)
assert(avg_convergence_ratio < 0.4)
def test_symmetry(self):
# Test that a basic V-cycle yields a symmetric linear operator. Common
# reasons for failure are problems with using the same rho for the
# pres/post-smoothers and using the same block_D_inv for
# pre/post-smoothers.
n = 500
A = poisson((n,), format='csr')
smoothers = [('gauss_seidel', {'sweep': 'symmetric'}),
('schwarz', {'sweep': 'symmetric'}),
('block_gauss_seidel', {'sweep': 'symmetric'}),
'jacobi', 'block_jacobi']
Bs = [np.ones((n, 1)),
sp.hstack((np.ones((n, 1)),
np.arange(1, n + 1, dtype='float').reshape(-1, 1)))]
for smoother in smoothers:
for B in Bs:
ml = rootnode_solver(A, B, max_coarse=10,
presmoother=smoother,
postsmoother=smoother)
P = ml.aspreconditioner()
x = sp.rand(n,)
y = sp.rand(n,)
assert_approx_equal(np.dot(P * x, y), np.dot(x, P * y))
def test_coarse_solver_opts(self):
# these tests are meant to test whether coarse solvers are correctly
# passed parameters
A = poisson((30, 30), format='csr')
b = sp.rand(A.shape[0], 1)
# for each pair, the first entry should yield an SA solver that
# converges in fewer iterations for a basic Poisson problem
coarse_solver_pairs = [(('jacobi', {'iterations': 30}), 'jacobi')]
coarse_solver_pairs.append((('gauss_seidel',
{'iterations': 30}), 'gauss_seidel'))
coarse_solver_pairs.append(('gauss_seidel', 'jacobi'))
coarse_solver_pairs.append(('cg', ('cg', {'tol': 10.0})))
coarse_solver_pairs.append(('pinv2', ('pinv2', {'cond': 1.0})))
for coarse1, coarse2 in coarse_solver_pairs:
r1 = []
r2 = []
sa1 = rootnode_solver(A, coarse_solver=coarse1)
sa2 = rootnode_solver(A, coarse_solver=coarse2)
x1 = sa1.solve(b, residuals=r1)
x2 = sa2.solve(b, residuals=r2)
del x1, x2
assert((len(r1) + 5) < len(r2))
def test_coarse_solver_opts(self):
# these tests are meant to test whether coarse solvers are correctly
# passed parameters
A = poisson((30, 30), format='csr')
b = np.random.rand(A.shape[0], 1)
# for each pair, the first entry should yield an SA solver that
# converges in fewer iterations for a basic Poisson problem
coarse_solver_pairs = [(('jacobi', {'iterations': 30}), 'jacobi')]
coarse_solver_pairs.append((('gauss_seidel',
{'iterations': 30}), 'gauss_seidel'))
coarse_solver_pairs.append(('gauss_seidel', 'jacobi'))
coarse_solver_pairs.append(('cg', ('cg', {'tol': 10.0})))
# scipy >= 1.7: pinv takes 'rtol'
# scipy < 1.7: pinv takes 'cond'
kword = 'rtol'
if kword not in sla.pinv.__code__.co_varnames:
kword = 'cond'
coarse_solver_pairs.append(('pinv', ('pinv', {kword: 1.0})))
for coarse1, coarse2 in coarse_solver_pairs:
r1 = []
r2 = []
sa1 = rootnode_solver(A, coarse_solver=coarse1, max_coarse=500)
sa2 = rootnode_solver(A, coarse_solver=coarse2, max_coarse=500)
x1 = sa1.solve(b, residuals=r1)
x2 = sa2.solve(b, residuals=r2)
del x1, x2
assert((len(r1) + 5) < len(r2))
def test_symmetry(self):
# Test that a basic V-cycle yields a symmetric linear operator. Common
# reasons for failure are problems with using the same rho for the
# pres/post-smoothers and using the same block_D_inv for
# pre/post-smoothers.
n = 500
A = poisson((n,), format='csr')
smoothers = [('gauss_seidel', {'sweep': 'symmetric'}),
('schwarz', {'sweep': 'symmetric'}),
('block_gauss_seidel', {'sweep': 'symmetric'}),
'jacobi', 'block_jacobi']
Bs = [ones((n, 1)),
hstack((ones((n, 1)),
arange(1, n + 1, dtype='float').reshape(-1, 1)))]
for smoother in smoothers:
for B in Bs:
ml = rootnode_solver(A, B, max_coarse=10,
presmoother=smoother,
postsmoother=smoother)
P = ml.aspreconditioner()
x = rand(n,)
y = rand(n,)
assert_approx_equal(dot(P * x, y), dot(x, P * y))
def test_improve_candidates(self):
# test improve_candidates for the Poisson problem and elasticity, where
# rho_scale is the amount that each successive improve_candidates
# option should improve convergence over the previous
# improve_candidates option.
improve_candidates_list = [None, [("block_gauss_seidel", {"iterations": 4, "sweep": "symmetric"})]]
# make tests repeatable
numpy.random.seed(0)
cases = []
A_elas, B_elas = linear_elasticity((60, 60), format="bsr")
# Matrix Candidates rho_scale
cases.append((poisson((75, 75), format="csr"), ones((75 * 75, 1)), 0.9))
cases.append((A_elas, B_elas, 0.9))
for (A, B, rho_scale) in cases:
last_rho = -1.0
x0 = rand(A.shape[0], 1)
b = rand(A.shape[0], 1)
for improve_candidates in improve_candidates_list:
ml = rootnode_solver(A, B, max_coarse=10, improve_candidates=improve_candidates)
residuals = []
x_sol = ml.solve(b, x0=x0, maxiter=20, tol=1e-10, residuals=residuals)
del x_sol
rho = (residuals[-1] / residuals[0]) ** (1.0 / len(residuals))
if last_rho == -1.0:
last_rho = rho
else:
# each successive improve_candidates option should be an
# improvement on the previous print "\nimprove_candidates
# Test: %1.3e, %1.3e,
# %d\n"%(rho,rho_scale*last_rho,A.shape[0])
assert rho < rho_scale * last_rho
last_rho = rho
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 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 setUp(self):
self.cases = []
# random matrices
np.random.seed(0)
for N in [2, 3, 5]:
self.cases.append(csr_matrix(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, 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 = []
A = poisson((5000,), format="csr")
self.cases.append((A, None, 0.4, "symmetric", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((A, None, 0.4, "symmetric", ("energy", {"krylov": "cg"})))
self.cases.append((A, None, 0.5, "symmetric", ("energy", {"krylov": "gmres"})))
A = poisson((60, 60), format="csr")
self.cases.append((A, None, 0.42, "symmetric", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((A, None, 0.42, "symmetric", ("energy", {"krylov": "cg"})))
self.cases.append((A, None, 0.42, "symmetric", ("energy", {"krylov": "cgnr"})))
A, B = linear_elasticity((50, 50), format="bsr")
self.cases.append((A, B, 0.32, "symmetric", ("jacobi", {"omega": 4.0 / 3.0})))
self.cases.append((A, B, 0.22, "symmetric", ("energy", {"krylov": "cg"})))
self.cases.append((A, B, 0.42, "symmetric", ("energy", {"krylov": "cgnr"})))
self.cases.append((A, B, 0.42, "symmetric", ("energy", {"krylov": "gmres"})))
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 test_precision(self):
"""Check single precision.
Test that x_32 == x_64 up to single precision tolerance
"""
np.random.seed(3158637515) # make tests repeatable
A = poisson((10, 10), dtype=np.float64, format='csr')
b = np.random.rand(A.shape[0]).astype(A.dtype)
ml = smoothed_aggregation_solver(A)
x = np.random.rand(A.shape[0]).astype(A.dtype)
x32 = ml.solve(b, x0=x, maxiter=1)
np.random.seed(3158637515) # make tests repeatable
A = poisson((10, 10), dtype=np.float32, format='csr')
b = np.random.rand(A.shape[0]).astype(A.dtype)
ml = smoothed_aggregation_solver(A)
x = np.random.rand(A.shape[0]).astype(A.dtype)
x64 = ml.solve(b, x0=x, maxiter=1)
assert_array_almost_equal(x32, x64, decimal=5)
def setUp(self):
self.cases = []
# Poisson problems in 1D and 2D
for N in [10, 11, 19, 26]:
A = poisson((N,), format='csr')
tempAgg = standard_aggregation(A)
T, B = fit_candidates(tempAgg, np.ones((A.shape[0],)) )
self.cases.append( {'A': A, 'T': T, 'B': B} )
for N in [5, 7, 9]:
A = poisson((N,N), format='csr')
tempAgg = standard_aggregation(A)
T, B = fit_candidates(tempAgg, np.ones((A.shape[0],)) )
self.cases.append( {'A': A, 'T': T, 'B': B} )
for name in ['knot', 'airfoil', 'bar']:
A = ex['A'].tocsr()
tempAgg = standard_aggregation(A)
T, B = fit_candidates(tempAgg, np.ones((A.shape[0],)) )
self.cases.append( {'A': A, 'T': T, 'B': B} )
def setUp(self):
self.cases = []
# Poisson problems in 1D and 2D
for N in [10, 11, 19, 26]:
A = poisson((N, ), format='csr')
tempAgg = standard_aggregation(A)
T, B = fit_candidates(tempAgg, np.ones((A.shape[0], )))
self.cases.append({'A': A, 'T': T, 'B': B})
for N in [5, 7, 9]:
A = poisson((N, N), format='csr')
tempAgg = standard_aggregation(A)
T, B = fit_candidates(tempAgg, np.ones((A.shape[0], )))
self.cases.append({'A': A, 'T': T, 'B': B})
for name in ['knot', 'airfoil', 'bar']:
A = ex['A'].tocsr()
tempAgg = standard_aggregation(A)
T, B = fit_candidates(tempAgg, np.ones((A.shape[0], )))
self.cases.append({'A': A, 'T': T, 'B': B})
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 = []
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_matrix_formats(self):
# Do dense, csr, bsr and csc versions of A all yield the same solver
A = poisson((7, 7), format="csr")
cases = [A.tobsr(blocksize=(1, 1))]
cases.append(A.tocsc())
cases.append(A.todense())
sa_old = rootnode_solver(A, max_coarse=10)
for AA in cases:
sa_new = rootnode_solver(AA, max_coarse=10)
assert abs(ravel(sa_old.levels[-1].A.todense() - sa_new.levels[-1].A.todense())).max() < 0.01
sa_old = sa_new
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):
self.cases = []
A = poisson((5000,), format='csr')
self.cases.append((A, None, 0.4, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, None, 0.4, 'symmetric',
('energy', {'krylov': 'gmres'})))
A = poisson((75, 75), format='csr')
self.cases.append((A, None, 0.26, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, None, 0.30, 'symmetric',
('energy', {'krylov': 'cgnr'})))
A, B = linear_elasticity((50, 50), format='bsr')
self.cases.append((A, B, 0.3, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, B, 0.3, 'symmetric',
('energy', {'krylov': 'cgnr'})))
self.cases.append((A, B, 0.3, 'symmetric',
('energy', {'krylov': 'gmres'})))
def setUp(self):
self.cases = []
A = poisson((5000,), format='csr')
self.cases.append((A, None, 0.4, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, None, 0.4, 'symmetric',
('energy', {'krylov': 'gmres'})))
A = poisson((75, 75), format='csr')
self.cases.append((A, None, 0.26, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, None, 0.30, 'symmetric',
('energy', {'krylov': 'cgnr'})))
A, B = linear_elasticity((50, 50), format='bsr')
self.cases.append((A, B, 0.3, 'symmetric',
('energy', {'krylov': 'cg'})))
self.cases.append((A, B, 0.3, 'symmetric',
('energy', {'krylov': 'cgnr'})))
self.cases.append((A, B, 0.3, 'symmetric',
('energy', {'krylov': 'gmres'})))
def test_matrix_formats(self):
# Do dense, csr, bsr and csc versions of A all yield the same solver
A = poisson((7, 7), format='csr')
cases = [A.tobsr(blocksize=(1, 1))]
cases.append(A.tocsc())
cases.append(A.toarray())
rs_old = ruge_stuben_solver(A, max_coarse=10)
for AA in cases:
rs_new = ruge_stuben_solver(AA, max_coarse=10)
assert(np.abs(np.ravel(rs_old.levels[-1].A.toarray() - rs_new.levels[-1].A.toarray())).max() < 0.01)
rs_old = rs_new
def demo():
"""Outline basic demo."""
A = poisson((100, 100), format='csr') # 2D FD Poisson problem
B = None # no near-null spaces guesses for SA
b = sp.rand(A.shape[0], 1) # a random right-hand side
# use AMG based on Smoothed Aggregation (SA) and display info
mls = smoothed_aggregation_solver(A, B=B)
print(mls)
# Solve Ax=b with no acceleration ('standalone' solver)
standalone_residuals = []
x = mls.solve(b, tol=1e-10, accel=None, residuals=standalone_residuals)
# Solve Ax=b with Conjugate Gradient (AMG as a preconditioner to CG)
accelerated_residuals = []
x = mls.solve(b, tol=1e-10, accel='cg', residuals=accelerated_residuals)
del x
# Compute relative residuals
standalone_residuals = \
np.array(standalone_residuals) / standalone_residuals[0]
accelerated_residuals = \
np.array(accelerated_residuals) / accelerated_residuals[0]
# Compute (geometric) convergence factors
factor1 = standalone_residuals[-1]**(1.0 / len(standalone_residuals))
factor2 = accelerated_residuals[-1]**(1.0 / len(accelerated_residuals))
print(" MG convergence factor: %g" % (factor1))
print("MG with CG acceleration convergence factor: %g" % (factor2))
# Plot convergence history
try:
import matplotlib.pyplot as plt
plt.figure()
plt.title('Convergence History')
plt.xlabel('Iteration')
plt.ylabel('Relative Residual')
plt.semilogy(standalone_residuals,
label='Standalone',
linestyle='-',
marker='o')
plt.semilogy(accelerated_residuals,
label='Accelerated',
linestyle='-',
marker='s')
plt.legend()
plt.show()
except ImportError:
print("\n\nNote: pylab not available on your system.")
def setUp(self):
self.cases = []
A = poisson((5000, ), format='csr')
self.cases.append((A, None, 0.4, 'symmetric', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((A, None, 0.4, 'symmetric', ('energy', {
'krylov': 'cg'
})))
self.cases.append((A, None, 0.5, 'symmetric', ('energy', {
'krylov': 'gmres'
})))
A = poisson((60, 60), format='csr')
self.cases.append((A, None, 0.42, 'symmetric', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((A, None, 0.42, 'symmetric', ('energy', {
'krylov': 'cg'
})))
self.cases.append((A, None, 0.42, 'symmetric', ('energy', {
'krylov': 'cgnr'
})))
A, B = linear_elasticity((50, 50), format='bsr')
self.cases.append((A, B, 0.32, 'symmetric', ('jacobi', {
'omega': 4.0 / 3.0
})))
self.cases.append((A, B, 0.22, 'symmetric', ('energy', {
'krylov': 'cg'
})))
self.cases.append((A, B, 0.42, 'symmetric', ('energy', {
'krylov': 'cgnr'
})))
self.cases.append((A, B, 0.42, 'symmetric', ('energy', {
'krylov': 'gmres'
})))
def test_matrix_formats(self):
# Do dense, csr, bsr and csc versions of A all yield the same solver
A = poisson((7, 7), format='csr')
cases = [A.tobsr(blocksize=(1, 1))]
cases.append(A.tocsc())
cases.append(A.todense())
sa_old = rootnode_solver(A, max_coarse=10)
for AA in cases:
sa_new = rootnode_solver(AA, max_coarse=10)
dff = sa_old.levels[-1].A.todense() - sa_new.levels[-1].A.todense()
assert(np.abs(np.ravel(dff)).max() < 0.01)
sa_old = sa_new
def test_matrix_formats(self):
warnings.simplefilter('ignore', SparseEfficiencyWarning)
# Do dense, csr, bsr and csc versions of A all yield the same solver
A = poisson((7, 7), format='csr')
cases = [A.tobsr(blocksize=(1, 1))]
cases.append(A.tocsc())
cases.append(A.toarray())
sa_old = smoothed_aggregation_solver(A, max_coarse=10)
for AA in cases:
sa_new = smoothed_aggregation_solver(AA, max_coarse=10)
assert(np.abs(np.ravel(sa_old.levels[-1].A.toarray() - sa_new.levels[-1].A.toarray())).max() < 0.01)
sa_old = sa_new
def setUp(self):
self.cases = []
# random matrices
np.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 test_coarse_grid_solver(self):
cases = []
cases.append(csr_matrix(diag(arange(1, 5))))
cases.append(poisson((4, ), format='csr'))
cases.append(poisson((4, 4), format='csr'))
# Make cases complex
cases = [G + 1e-5j * G for G in cases]
cases = [0.5 * (G + G.H) for G in cases]
# method should be almost exact for small matrices
for A in cases:
for solver in ['splu', 'pinv', 'pinv2', 'lu', 'cholesky', 'cg']:
s = coarse_grid_solver(solver)
b = arange(A.shape[0], dtype=A.dtype)
x = s(A, b)
assert_almost_equal(A * x, b)
# subsequent calls use cached data
x = s(A, b)
assert_almost_equal(A * x, b)
def test_improve_candidates(self):
# test improve_candidates for the Poisson problem and elasticity, where
# rho_scale is the amount that each successive improve_candidates
# option should improve convergence over the previous
# improve_candidates option.
improve_candidates_list = [
None,
[('block_gauss_seidel', {
'iterations': 4,
'sweep': 'symmetric'
})]
]
# make tests repeatable
np.random.seed(0)
cases = []
A_elas, B_elas = linear_elasticity((60, 60), format='bsr')
# Matrix, Candidates, rho_scale
cases.append((poisson((61, 61), format='csr'), np.ones(
(61 * 61, 1)), 0.9))
cases.append((A_elas, B_elas, 0.9))
for (A, B, rho_scale) in cases:
last_rho = -1.0
x0 = sp.rand(A.shape[0], 1)
b = sp.rand(A.shape[0], 1)
for ic in improve_candidates_list:
ml = smoothed_aggregation_solver(A,
B,
max_coarse=10,
improve_candidates=ic)
residuals = []
x_sol = ml.solve(b,
x0=x0,
maxiter=20,
tol=1e-10,
residuals=residuals)
del x_sol
rho = (residuals[-1] / residuals[0])**(1.0 / len(residuals))
if last_rho == -1.0:
last_rho = rho
else:
# each successive improve_candidates option should be an
# improvement on the previous print "\nimprove_candidates
# Test: %1.3e, %1.3e,
# %d\n"%(rho,rho_scale*last_rho,A.shape[0])
assert (rho < rho_scale * last_rho)
last_rho = rho
def topyamg(self):
r""" Create a `pyamg` stencil matrix to be used in pyamg
This allows retrieving the grid matrix equivalent of the real-space grid.
Subsequently the returned matrix may be used in pyamg for solutions etc.
The `pyamg` suite is it-self a rather complicated code with many options.
For details we refer to `pyamg <pyamg https://github.com/pyamg/pyamg/>`_.
Returns
-------
A : scipy.sparse.csr_matrix which contains the grid stencil for a `pyamg` solver.
b : numpy.ndarray containing RHS of the linear system of equations.
Examples
--------
This example proves the best method for a variety of cases in regards of the 3D Poisson problem:
>>> grid = Grid(0.01)
>>> A, b = grid.topyamg() # automatically setups the current boundary conditions
>>> # add terms etc. to A and/or b
>>> import pyamg
>>> from scipy.sparse.linalg import cg
>>> ml = pyamg.aggregation.smoothed_aggregation_solver(A, max_levels=1000)
>>> M = ml.aspreconditioner(cycle='W') # pre-conditioner
>>> x, info = cg(A, b, tol=1e-12, M=M)
See Also
--------
pyamg_index : convert grid indices into the sparse matrix indices for ``A``
pyamg_fix : fixes stencil for indices and fixes the source for the RHS matrix (uses `pyamg_source`)
pyamg_source : fix the RHS matrix ``b`` to a constant value
pyamg_boundary_condition : setup the sparse matrix ``A`` to given boundary conditions (called in this routine)
"""
from pyamg.gallery import poisson
# Initially create the CSR matrix
A = poisson(self.shape, dtype=self.dtype, format='csr')
b = np.zeros(A.shape[0], dtype=A.dtype)
# Now apply the boundary conditions
self.pyamg_boundary_condition(A, b)
return A, b
def test_matrix_formats(self):
warnings.filterwarnings('ignore', category=SparseEfficiencyWarning)
# Do dense, csr, bsr and csc versions of A all yield the same solver
A = poisson((7, 7), format='csr')
cases = [A.tobsr(blocksize=(1, 1))]
cases.append(A.tocsc())
cases.append(A.toarray())
np.random.seed(111908910)
sa_old = adaptive_sa_solver(A, initial_candidates=np.ones((49, 1)),
max_coarse=10)[0]
for AA in cases:
np.random.seed(111908910)
sa_new = adaptive_sa_solver(AA,
initial_candidates=np.ones((49, 1)),
max_coarse=10)[0]
assert(abs(np.ravel(sa_old.levels[-1].A.toarray()
- sa_new.levels[-1].A.toarray())).max() < 0.01)
sa_old = sa_new
def test_poisson(self):
cases = []
# perturbed Laplacian
A = poisson((50, 50), format='csr')
Ai = A.copy()
Ai.data = Ai.data + 1e-5j * np.random.rand(Ai.nnz)
cases.append((Ai, 0.25))
# imaginary Laplacian
Ai = 1.0j * A
cases.append((Ai, 0.25))
# JBS: Not sure if this is a valid test case
# imaginary shift
# Ai = A + 1.1j*sparse.eye(A.shape[0], A.shape[1])
# cases.append((Ai,0.8))
for A, rratio in cases:
[asa, work] = adaptive_sa_solver(A,
num_candidates=1,
symmetry='symmetric')
# sa = smoothed_aggregation_solver(A, B = np.ones((A.shape[0],1)) )
b = np.zeros((A.shape[0], ))
x0 = (np.random.rand(A.shape[0], ) +
1.0j * np.random.rand(A.shape[0], ))
residuals0 = []
sol0 = asa.solve(b,
x0=x0,
maxiter=20,
tol=1e-10,
residuals=residuals0)
del sol0
conv_asa = \
(residuals0[-1] / residuals0[0]) ** (1.0 / len(residuals0))
assert (conv_asa < rratio)
def test_poisson(self):
A = poisson((50, 50), format='csr')
[asa, work] = adaptive_sa_solver(A, num_candidates=1)
sa = smoothed_aggregation_solver(A, B=np.ones((A.shape[0], 1)))
b = sp.rand(A.shape[0])
residuals0 = []
residuals1 = []
sol0 = asa.solve(b, maxiter=20, tol=1e-10, residuals=residuals0)
sol1 = sa.solve(b, maxiter=20, tol=1e-10, residuals=residuals1)
del sol0, sol1
conv_asa = (residuals0[-1] / residuals0[0])**(1.0 / len(residuals0))
conv_sa = (residuals1[-1] / residuals1[0])**(1.0 / len(residuals1))
# print "ASA convergence (Poisson)",conv_asa
# print "SA convergence (Poisson)",conv_sa
assert (conv_asa < 1.2 * conv_sa)
def test_aspreconditioner(self):
from pyamg import smoothed_aggregation_solver
from scipy.sparse.linalg import cg
from pyamg.krylov import fgmres
A = poisson((50, 50), format='csr')
b = rand(A.shape[0])
ml = smoothed_aggregation_solver(A)
for cycle in ['V', 'W', 'F']:
M = ml.aspreconditioner(cycle=cycle)
x, info = cg(A, b, tol=1e-8, maxiter=30, M=M)
# cg satisfies convergence in the preconditioner norm
assert (precon_norm(b - A * x, ml) < 1e-8 * precon_norm(b, ml))
for cycle in ['AMLI']:
M = ml.aspreconditioner(cycle=cycle)
x, info = fgmres(A, b, tol=1e-8, maxiter=30, M=M)
# fgmres satisfies convergence in the 2-norm
assert (norm(b - A * x) < 1e-8 * norm(b))