def test_newton(self):
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
class QuadraticModelEvaluator:
'''Simple model evalator for f(x) = x^2 - 2.'''
def __init__(self):
self.dtype = float
def compute_f(self, x):
return x**2 - 2.0
def get_jacobian(self, x0):
return 2.0 * x0
def get_preconditioner(self, x0):
return None
def get_preconditioner_inverse(self, x0):
return None
def inner_product(self, x, y):
return np.dot(x.T.conj(), y)
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
qmodeleval = QuadraticModelEvaluator()
x0 = np.array([[1.0]])
tol = 1.0e-10
out = nm.newton(x0, qmodeleval, nonlinear_tol=tol)
self.assertEqual(out['info'], 0)
self.assertAlmostEqual(out['x'][0, 0], np.sqrt(2.0), delta=tol)
def my_newton(args, modeleval, psi0, g, mu, yaml_emitter=None, debug=True):
"""Solve with Newton.
"""
recycling_solver_kwargs = {
"explicit_residual": args.resexp,
"maxiter": 200
}
if args.krylov_method == "cg":
RecyclingSolver = krypy.recycling.RecyclingCg
elif args.krylov_method == "minres":
RecyclingSolver = krypy.recycling.RecyclingMinres
elif args.krylov_method == "minresfo":
RecyclingSolver = krypy.recycling.RecyclingMinres
recycling_solver_kwargs["ortho"] = "mgs"
elif args.krylov_method == "gmres":
RecyclingSolver = krypy.recycling.RecyclingGmres
else:
raise ValueError("Unknown Krylov solver "
"%s"
"." % args.krylov_method)
def vector_factory_generator(x):
"""Generates the vector factory with i*x if requested."""
if not args.defl_include_ix and args.defl_num_ritz_vectors == 0:
# no deflation
return None
else:
ritz_factory = krypy.recycling.factories.RitzFactorySimple(
n_vectors=args.defl_num_ritz_vectors, which="sm")
if args.defl_include_ix:
return krypy.recycling.factories.UnionFactory(
[ritz_factory, IxFactory(x)])
return ritz_factory
# perform newton iteration
yaml_emitter.add_key("Newton results")
newton_out = nm.newton(
psi0,
modeleval,
RecyclingSolver=RecyclingSolver,
recycling_solver_kwargs=recycling_solver_kwargs,
vector_factory_generator=vector_factory_generator,
nonlinear_tol=1.0e-10,
eta0=args.eta,
forcing_term="constant",
compute_f_extra_args={
"g": g,
"mu": mu
},
debug=debug,
yaml_emitter=yaml_emitter,
newton_maxiter=30,
)
yaml_emitter.add_comment("done.")
# assert( newton_out['info'] == 0 )
return newton_out
def my_newton(args, modeleval, psi0, g, mu, yaml_emitter=None, debug=True):
'''Solve with Newton.
'''
recycling_solver_kwargs = {
'explicit_residual': args.resexp,
'maxiter': 200
}
if args.krylov_method == 'cg':
RecyclingSolver = krypy.recycling.RecyclingCg
elif args.krylov_method == 'minres':
RecyclingSolver = krypy.recycling.RecyclingMinres
elif args.krylov_method == 'minresfo':
RecyclingSolver = krypy.recycling.RecyclingMinres
recycling_solver_kwargs['ortho'] = 'mgs'
elif args.krylov_method == 'gmres':
RecyclingSolver = krypy.recycling.RecyclingGmres
else:
raise ValueError('Unknown Krylov solver ''%s''.' % args.krylov_method)
def vector_factory_generator(x):
'''Generates the vector factory with i*x if requested.'''
if not args.defl_include_ix and args.defl_num_ritz_vectors == 0:
# no deflation
return None
else:
ritz_factory = krypy.recycling.factories.RitzFactorySimple(
n_vectors=args.defl_num_ritz_vectors,
which='sm'
)
if args.defl_include_ix:
return krypy.recycling.factories.UnionFactory(
[ritz_factory, IxFactory(x)])
return ritz_factory
# perform newton iteration
yaml_emitter.add_key('Newton results')
newton_out = nm.newton(psi0,
modeleval,
RecyclingSolver=RecyclingSolver,
recycling_solver_kwargs=recycling_solver_kwargs,
vector_factory_generator=vector_factory_generator,
nonlinear_tol=1.0e-10,
eta0=args.eta,
forcing_term='constant',
compute_f_extra_args={'g': g, 'mu': mu},
debug=debug,
yaml_emitter=yaml_emitter,
newton_maxiter=30
)
yaml_emitter.add_comment('done.')
#assert( newton_out['info'] == 0 )
return newton_out
def _main():
"""Main function.
"""
args = _parse_input_arguments()
# read the mesh
print("Reading the mesh...", end=" ")
pynoshmesh, psi, A, field_data = mesh.mesh_io.read_mesh(args.filename)
print("done.")
# build the model evaluator
mu = 8.0e-2
ginla_modelval = gm.ModelEvaluator(pynoshmesh, A, mu)
# initial guess
num_nodes = len(pynoshmesh.nodes)
psi0 = np.ones((num_nodes, 1), dtype=complex)
nums_deflation_vectors = list(range(50))
last_resvecs = []
for num_deflation_vectors in nums_deflation_vectors:
print(
(
"Performing Newton iteration with %d deflation vectors..."
% num_deflation_vectors
)
)
# perform newton iteration
newton_out = nm.newton(
psi0,
ginla_modelval,
linear_solver=nm.minres,
nonlinear_tol=1.0e-10,
eta0=1.0e-13,
forcing_term="constant",
use_preconditioner=True,
deflate_ix=True,
num_deflation_vectors=num_deflation_vectors,
)
print(" done.")
assert newton_out[1] == 0, "Newton did not converge."
last_resvecs.append(newton_out[3][-1])
multiplot_data_series(last_resvecs)
pp.title(
"Residual curves for the last Newton step for %s. Darker=More deflation vectors."
% args.filename
)
pp.show()
# matplotlib2tikz.save('w-defl.tex')
return
def my_newton(args, modeleval, psi0, g, mu, yaml_emitter=None, debug=True):
"""Solve with Newton.
"""
recycling_solver_kwargs = {"explicit_residual": args.resexp, "maxiter": 200}
if args.krylov_method == "cg":
RecyclingSolver = krypy.recycling.RecyclingCg
elif args.krylov_method == "minres":
RecyclingSolver = krypy.recycling.RecyclingMinres
elif args.krylov_method == "minresfo":
RecyclingSolver = krypy.recycling.RecyclingMinres
recycling_solver_kwargs["ortho"] = "mgs"
elif args.krylov_method == "gmres":
RecyclingSolver = krypy.recycling.RecyclingGmres
else:
raise ValueError("Unknown Krylov solver " "%s" "." % args.krylov_method)
def vector_factory_generator(x):
"""Generates the vector factory with i*x if requested."""
if not args.defl_include_ix and args.defl_num_ritz_vectors == 0:
# no deflation
return None
else:
ritz_factory = krypy.recycling.factories.RitzFactorySimple(
n_vectors=args.defl_num_ritz_vectors, which="sm"
)
if args.defl_include_ix:
return krypy.recycling.factories.UnionFactory(
[ritz_factory, IxFactory(x)]
)
return ritz_factory
# perform newton iteration
yaml_emitter.add_key("Newton results")
newton_out = nm.newton(
psi0,
modeleval,
RecyclingSolver=RecyclingSolver,
recycling_solver_kwargs=recycling_solver_kwargs,
vector_factory_generator=vector_factory_generator,
nonlinear_tol=1.0e-10,
eta0=args.eta,
forcing_term="constant",
compute_f_extra_args={"g": g, "mu": mu},
debug=debug,
yaml_emitter=yaml_emitter,
newton_maxiter=30,
)
yaml_emitter.add_comment("done.")
# assert( newton_out['info'] == 0 )
return newton_out
def _main():
"""Main function.
"""
args = _parse_input_arguments()
# read the mesh
print("Reading the mesh...", end=" ")
pynoshmesh, psi, A, field_data = mesh.mesh_io.read_mesh(args.filename)
print("done.")
# build the model evaluator
mu = 8.0e-2
ginla_modelval = gm.ModelEvaluator(pynoshmesh, A, mu)
# initial guess
num_nodes = len(pynoshmesh.nodes)
psi0 = np.ones((num_nodes, 1), dtype=complex)
nums_deflation_vectors = list(range(50))
last_resvecs = []
for num_deflation_vectors in nums_deflation_vectors:
print(("Performing Newton iteration with %d deflation vectors..." %
num_deflation_vectors))
# perform newton iteration
newton_out = nm.newton(
psi0,
ginla_modelval,
linear_solver=nm.minres,
nonlinear_tol=1.0e-10,
eta0=1.0e-13,
forcing_term="constant",
use_preconditioner=True,
deflate_ix=True,
num_deflation_vectors=num_deflation_vectors,
)
print(" done.")
assert newton_out[1] == 0, "Newton did not converge."
last_resvecs.append(newton_out[3][-1])
multiplot_data_series(last_resvecs)
pp.title(
"Residual curves for the last Newton step for %s. Darker=More deflation vectors."
% args.filename)
pp.show()
# matplotlib2tikz.save('w-defl.tex')
return
def test_newton(self):
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
class QuadraticModelEvaluator:
'''Simple model evalator for f(x) = x^2 - 2.'''
def __init__(self):
self.dtype = float
def compute_f(self, x):
return x**2 - 2.0
def get_jacobian(self, x0):
return 2.0 * x0
def get_preconditioner(self, x0):
return None
def get_preconditioner_inverse(self, x0):
return None
def inner_product(self, x, y):
return np.dot(x.T.conj(), y)
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
qmodeleval = QuadraticModelEvaluator()
x0 = np.array( [[1.0]] )
tol = 1.0e-10
out = nm.newton(x0, qmodeleval, nonlinear_tol=tol)
self.assertEqual(out['info'], 0)
self.assertAlmostEqual(out['x'][0,0], np.sqrt(2.0), delta=tol)
def find_beautiful_states(modeleval,
param_name,
param_range,
forcing_term,
save_doubles=True
):
'''Loop through a set of parameters/initial states and try to find
starting points that (quickly) lead to "interesting looking" solutions.
Such solutions are filtered out only by their energy at the moment.
'''
# Define search space.
# Don't use Mu=0 as the preconditioner is singular for mu=0, psi=0.
Alpha = np.linspace(0.2, 1.0, 5)
Frequencies = [0.0, 0.5, 1.0, 2.0]
# Compile the search space.
# If all nodes sit in x-y-plane, the frequency loop in z-direction can be
# omitted.
if modeleval.mesh.node_coords.shape[1] == 2:
search_space_k = [(a, b) for a in Frequencies for b in Frequencies]
elif modeleval.mesh.node_coords.shape[1] == 3:
search_space_k = [(a, b, c)
for a in Frequencies
for b in Frequencies
for c in Frequencies
]
search_space = [(a, k) for a in reversed(Alpha) for k in search_space_k]
solution_id = 0
for p in param_range:
# Reset the solutions each time the problem parameters change.
found_solutions = []
# Loop over initial states.
for alpha, k in search_space:
modeleval.set_parameter(param_name, p)
print '%s = %g; alpha = %g; k = %s' % (param_name, p, alpha, k,)
# Set the intitial guess for Newton.
if len(k) == 2:
psi0 = alpha \
* np.cos(k[0] * np.pi * modeleval.mesh.node_coords[:, 0]) \
* np.cos(k[1] * np.pi * modeleval.mesh.node_coords[:, 1]) \
+ 1j * 0
elif len(k) == 3:
psi0 = alpha \
* np.cos(k[0] * np.pi * modeleval.mesh.node_coords[:, 0]) \
* np.cos(k[1] * np.pi * modeleval.mesh.node_coords[:, 1]) \
* np.cos(k[2] * np.pi * modeleval.mesh.node_coords[:, 2]) \
+ 1j * 0
else:
raise RuntimeError('Illegal k.')
print 'Performing Newton iteration...'
linsolve_maxiter = 500 # 2*len(psi0)
# perform newton iteration
newton_out = nm.newton(psi0[:, None],
modeleval,
linear_solver=nm.minres,
linear_solver_maxiter=linsolve_maxiter,
linear_solver_extra_args={},
nonlinear_tol=1.0e-10,
forcing_term=forcing_term,
eta0=1.0e-10,
use_preconditioner=True,
deflation_generators=[lambda x: 1j*x],
num_deflation_vectors=0,
debug=True,
newton_maxiter=50
)
print ' done.'
num_krylov_iters = \
[len(resvec) for resvec in newton_out['linear relresvecs']]
print 'Num Krylov iterations:', num_krylov_iters
print 'Newton residuals:', newton_out['Newton residuals']
if newton_out['info'] == 0:
num_newton_iters = len(newton_out['linear relresvecs'])
psi = newton_out['x']
# Use the energy as a measure for ruling out boring states
# such as psi==0 or psi==1 overall.
energy = modeleval.energy(psi)
print 'Energy of solution state: %g.' % energy
if energy > -0.999 and energy < -0.001:
# Store the file as VTU such that ParaView can loop through
# and display them at once. For this, also be sure to keep
# the file name in the format
# 'interesting<-krylovfails03>-01414.vtu'.
filename = 'interesting-'
num_krylov_fails = num_krylov_iters.count(linsolve_maxiter)
if num_krylov_fails > 0:
filename += 'krylovfails%s-' \
% repr(num_krylov_fails).rjust(2, '0')
filename += repr(num_newton_iters).rjust(2, '0') \
+ repr(solution_id).rjust(3, '0') \
+ '.vtu'
print('Interesting state found for %s=%g!'
% (param_name, p),
)
# Check if we already stored that one.
already_found = False
for state in found_solutions:
# Synchronize the complex argument.
state *= np.exp(1j *
(np.angle(psi[0]) - np.angle(state[0]))
)
diff = psi - state
if modeleval.inner_product(diff, diff) < 1.0e-10:
already_found = True
break
if already_found and not save_doubles:
print '-- But we already have that one.'
else:
found_solutions.append(psi)
print 'Storing in %s.' % filename
if len(k) == 2:
function_string = 'psi0(X) = %g * cos(%g*pi*x) * cos(%g*pi*y)' % (alpha, k[0], k[1])
elif len(k) == 3:
function_string = 'psi0(X) = %g * cos(%g*pi*x) * cos(%g*pi*y) * cos(%g*pi*z)' % (alpha, k[0], k[1], k[2])
else:
raise RuntimeError('Illegal k.')
modeleval.mesh.write(
filename,
point_data={'psi': psi,
'psi0': psi0,
'V': modeleval._V,
'A': modeleval._raw_magnetic_vector_potential
},
field_data={'g': modeleval._g,
'mu': modeleval.mu,
'psi0(X)': function_string
}
)
solution_id += 1
print
return
def find_beautiful_states(modeleval,
mu_range,
forcing_term,
save_doubles=True):
"""Loop through a set of parameters/initial states and try to find
starting points that (quickly) lead to "interesting looking" solutions.
Such solutions are filtered out only by their energy at the moment.
"""
# Define search space.
# Don't use Mu=0 as the preconditioner is singular for mu=0, psi=0.
Alpha = np.linspace(0.2, 1.0, 5)
Frequencies = [0.0, 0.5, 1.0, 2.0]
# Compile the search space.
# If all nodes sit in x-y-plane, the frequency loop in z-direction can be omitted.
if modeleval.mesh.node_coords.shape[1] == 2 or np.all(
np.abs(modeleval.mesh.node_coords[:, 2]) < 1.0e-13):
search_space_k = [(a, b) for a in Frequencies for b in Frequencies]
elif modeleval.mesh.node_coords.shape[1] == 3:
search_space_k = [(a, b, c) for a in Frequencies for b in Frequencies
for c in Frequencies]
search_space = [(a, k) for a in reversed(Alpha) for k in search_space_k]
solution_id = 0
for mu in mu_range:
# Reset the solutions each time the problem parameters change.
found_solutions = []
# Loop over initial states.
for alpha, k in search_space:
print("mu = {}; alpha = {}; k = {}".format(mu, alpha, k))
# Set the intitial guess for Newton.
if len(k) == 2:
psi0 = (
alpha *
np.cos(k[0] * np.pi * modeleval.mesh.node_coords[:, 0]) *
np.cos(k[1] * np.pi * modeleval.mesh.node_coords[:, 1]) +
1j * 0)
elif len(k) == 3:
psi0 = (
alpha *
np.cos(k[0] * np.pi * modeleval.mesh.node_coords[:, 0]) *
np.cos(k[1] * np.pi * modeleval.mesh.node_coords[:, 1]) *
np.cos(k[2] * np.pi * modeleval.mesh.node_coords[:, 2]) +
1j * 0)
else:
raise RuntimeError("Illegal k.")
print("Performing Newton iteration...")
linsolve_maxiter = 500 # 2*len(psi0)
try:
newton_out = nm.newton(
psi0[:, None],
modeleval,
nonlinear_tol=1.0e-10,
newton_maxiter=50,
compute_f_extra_args={
"mu": mu,
"g": 1.0
},
eta0=1.0e-10,
forcing_term=forcing_term,
)
except krypy.utils.ConvergenceError:
print("Krylov convergence failure. Skip.\n")
continue
print(" done.")
num_krylov_iters = [
len(resvec) for resvec in newton_out["linear relresvecs"]
]
print("Num Krylov iterations:", num_krylov_iters)
print("Newton residuals:", newton_out["Newton residuals"])
if newton_out["info"] == 0:
num_newton_iters = len(newton_out["linear relresvecs"])
psi = newton_out["x"]
# Use the energy as a measure for ruling out boring states such as
# psi==0 or psi==1 overall.
energy = modeleval.energy(psi)
print("Energy of solution state: %g." % energy)
if energy > -0.999 and energy < -0.001:
# Store the file as VTU such that ParaView can loop through and
# display them at once. For this, also be sure to keep the file name
# in the format 'interesting<-krylovfails03>-01414.vtu'.
filename = "interesting-"
num_krylov_fails = num_krylov_iters.count(linsolve_maxiter)
if num_krylov_fails > 0:
filename += "krylovfails{:02d}-".format(
num_krylov_fails)
filename += "{:02d}{:03d}.vtu".format(
num_newton_iters, solution_id)
print("Interesting state found for mu={}!".format(mu))
# Check if we already stored that one.
already_found = False
for state in found_solutions:
# Synchronize the complex argument.
state *= np.exp(
1j * (np.angle(psi[0]) - np.angle(state[0])))
diff = psi - state
if modeleval.inner_product(diff, diff) < 1.0e-10:
already_found = True
break
if already_found and not save_doubles:
print("-- But we already have that one.")
else:
found_solutions.append(psi)
print("Storing in {}.".format(filename))
# if len(k) == 2:
# function_string = (
# "psi0(X) = %g * cos(%g*pi*x) * cos(%g*pi*y)"
# % (alpha, k[0], k[1])
# )
# elif len(k) == 3:
# function_string = (
# "psi0(X) = %g * cos(%g*pi*x) * cos(%g*pi*y) * cos(%g*pi*z)"
# % (alpha, k[0], k[1], k[2])
# )
# else:
# raise RuntimeError("Illegal k.")
modeleval.mesh.write(
filename,
point_data={
"psi": np.column_stack([psi.real, psi.imag]),
"psi0": np.column_stack([psi0.real,
psi0.imag]),
"V": modeleval._V,
"A": modeleval._raw_magnetic_vector_potential,
},
field_data={
"g": np.array(1.0),
"mu": np.array(mu),
# "psi0(X)": function_string,
},
)
solution_id += 1
print()
return
def find_beautiful_states(modeleval, mu_range, forcing_term, save_doubles=True):
"""Loop through a set of parameters/initial states and try to find
starting points that (quickly) lead to "interesting looking" solutions.
Such solutions are filtered out only by their energy at the moment.
"""
# Define search space.
# Don't use Mu=0 as the preconditioner is singular for mu=0, psi=0.
Alpha = np.linspace(0.2, 1.0, 5)
Frequencies = [0.0, 0.5, 1.0, 2.0]
# Compile the search space.
# If all nodes sit in x-y-plane, the frequency loop in z-direction can be omitted.
if modeleval.mesh.node_coords.shape[1] == 2 or np.all(np.abs(modeleval.mesh.node_coords[:, 2]) < 1.0e-13):
search_space_k = [(a, b) for a in Frequencies for b in Frequencies]
elif modeleval.mesh.node_coords.shape[1] == 3:
search_space_k = [
(a, b, c) for a in Frequencies for b in Frequencies for c in Frequencies
]
search_space = [(a, k) for a in reversed(Alpha) for k in search_space_k]
solution_id = 0
for mu in mu_range:
# Reset the solutions each time the problem parameters change.
found_solutions = []
# Loop over initial states.
for alpha, k in search_space:
print("mu = {}; alpha = {}; k = {}".format(mu, alpha, k))
# Set the intitial guess for Newton.
if len(k) == 2:
psi0 = (
alpha
* np.cos(k[0] * np.pi * modeleval.mesh.node_coords[:, 0])
* np.cos(k[1] * np.pi * modeleval.mesh.node_coords[:, 1])
+ 1j * 0
)
elif len(k) == 3:
psi0 = (
alpha
* np.cos(k[0] * np.pi * modeleval.mesh.node_coords[:, 0])
* np.cos(k[1] * np.pi * modeleval.mesh.node_coords[:, 1])
* np.cos(k[2] * np.pi * modeleval.mesh.node_coords[:, 2])
+ 1j * 0
)
else:
raise RuntimeError("Illegal k.")
print("Performing Newton iteration...")
linsolve_maxiter = 500 # 2*len(psi0)
try:
newton_out = nm.newton(
psi0[:, None],
modeleval,
nonlinear_tol=1.0e-10,
newton_maxiter=50,
compute_f_extra_args={"mu": mu, "g": 1.0},
eta0=1.0e-10,
forcing_term=forcing_term,
)
except krypy.utils.ConvergenceError:
print("Krylov convergence failure. Skip.\n")
continue
print(" done.")
num_krylov_iters = [
len(resvec) for resvec in newton_out["linear relresvecs"]
]
print("Num Krylov iterations:", num_krylov_iters)
print("Newton residuals:", newton_out["Newton residuals"])
if newton_out["info"] == 0:
num_newton_iters = len(newton_out["linear relresvecs"])
psi = newton_out["x"]
# Use the energy as a measure for ruling out boring states such as
# psi==0 or psi==1 overall.
energy = modeleval.energy(psi)
print("Energy of solution state: %g." % energy)
if energy > -0.999 and energy < -0.001:
# Store the file as VTU such that ParaView can loop through and
# display them at once. For this, also be sure to keep the file name
# in the format 'interesting<-krylovfails03>-01414.vtu'.
filename = "interesting-"
num_krylov_fails = num_krylov_iters.count(linsolve_maxiter)
if num_krylov_fails > 0:
filename += "krylovfails{:02d}-".format(num_krylov_fails)
filename += "{:02d}{:03d}.vtu".format(num_newton_iters, solution_id)
print("Interesting state found for mu={}!".format(mu))
# Check if we already stored that one.
already_found = False
for state in found_solutions:
# Synchronize the complex argument.
state *= np.exp(1j * (np.angle(psi[0]) - np.angle(state[0])))
diff = psi - state
if modeleval.inner_product(diff, diff) < 1.0e-10:
already_found = True
break
if already_found and not save_doubles:
print("-- But we already have that one.")
else:
found_solutions.append(psi)
print("Storing in {}.".format(filename))
# if len(k) == 2:
# function_string = (
# "psi0(X) = %g * cos(%g*pi*x) * cos(%g*pi*y)"
# % (alpha, k[0], k[1])
# )
# elif len(k) == 3:
# function_string = (
# "psi0(X) = %g * cos(%g*pi*x) * cos(%g*pi*y) * cos(%g*pi*z)"
# % (alpha, k[0], k[1], k[2])
# )
# else:
# raise RuntimeError("Illegal k.")
modeleval.mesh.write(
filename,
point_data={
"psi": np.column_stack([psi.real, psi.imag]),
"psi0": np.column_stack([psi0.real, psi0.imag]),
"V": modeleval._V,
"A": modeleval._raw_magnetic_vector_potential,
},
field_data={
"g": np.array(1.0),
"mu": np.array(mu),
# "psi0(X)": function_string,
},
)
solution_id += 1
print()
return
