def test_hadamard_accuracy(self):
systemsize = 4
eigen_decayrate = 2.0
# Create Hadmard Quadratic object
QoI = examples.HadamardQuadratic(systemsize, eigen_decayrate)
mu = np.zeros(systemsize)
std_dev = np.eye(systemsize)
jdist = cp.MvNormal(mu, std_dev)
active_subspace = ActiveSubspace(QoI, n_dominant_dimensions=4,
n_monte_carlo_samples=10000,
use_svd=True, read_rv_samples=False,
write_rv_samples=False)
active_subspace.getDominantDirections(QoI, jdist)
mu_j_analytical = QoI.eval_analytical_QoI_mean(mu, cp.Cov(jdist))
var_j_analytical = QoI.eval_analytical_QoI_variance(mu, cp.Cov(jdist))
# Create reduced collocation object
QoI_dict = {'Hadamard' : {'QoI_func' : QoI.eval_QoI,
'output_dimensions' : 1,
},
}
sc_obj_active = StochasticCollocation2(jdist, 4, 'MvNormal', QoI_dict,
include_derivs=False,
reduced_collocation=True,
dominant_dir=active_subspace.dominant_dir)
sc_obj_active.evaluateQoIs(jdist)
mu_j_active = sc_obj_active.mean(of=['Hadamard'])
var_j_active = sc_obj_active.variance(of=['Hadamard'])
def test_multipleQoI(self):
# This tests for multiple QoIs. We only compute the mean in this test,
# because it is only checking if it can do multiple loops
systemsize = 2
theta = 0
mu = mean_2dim # np.random.randn(systemsize)
std_dev = std_dev_2dim # np.diag(np.random.rand(systemsize))
jdist = cp.MvNormal(mu, std_dev)
QoI1 = examples.Paraboloid2D(systemsize, (theta, ))
QoI2 = examples.PolyRVDV()
QoI_dict = {
'paraboloid2': {
'QoI_func': QoI1.eval_QoI,
'output_dimensions': 1,
},
'PolyRVDV': {
'QoI_func': QoI2.eval_QoI,
'output_dimensions': 1,
}
}
sc_obj = StochasticCollocation2(jdist, 3, 'MvNormal', QoI_dict)
sc_obj.evaluateQoIs(jdist)
mu_js = sc_obj.mean(of=['paraboloid2', 'PolyRVDV'])
# Compare against known values
# 1. Paraboloid2D, we use nested loops
mu_j1_analytical = QoI1.eval_QoI_analyticalmean(mu, cp.Cov(jdist))
err = abs(
(mu_js['paraboloid2'][0] - mu_j1_analytical) / mu_j1_analytical)
self.assertTrue(err < 1.e-15)
def test_nonrv_derivatives(self):
# This test checks the analytical derivative w.r.t complex step
systemsize = 2
n_parameters = 2
mu = mean_2dim # np.random.randn(systemsize)
std_dev = std_dev_2dim # np.diag(np.random.rand(systemsize))
jdist = cp.MvNormal(mu, std_dev)
QoI = examples.PolyRVDV(data_type=complex)
# Create the Stochastic Collocation object
deriv_dict = {
'dv': {
'dQoI_func': QoI.eval_QoIGradient_dv,
'output_dimensions': n_parameters
}
}
QoI_dict = {
'PolyRVDV': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
'deriv_dict': deriv_dict
}
}
dv = np.random.randn(systemsize) + 0j
QoI.set_dv(dv)
sc_obj = StochasticCollocation2(jdist,
3,
'MvNormal',
QoI_dict,
include_derivs=True,
data_type=complex)
sc_obj.evaluateQoIs(jdist, include_derivs=True)
dmu_j = sc_obj.dmean(of=['PolyRVDV'], wrt=['dv'])
dvar_j = sc_obj.dvariance(of=['PolyRVDV'], wrt=['dv'])
dstd_dev = sc_obj.dStdDev(of=['PolyRVDV'], wrt=['dv'])
# Lets do complex step
pert = complex(0, 1e-30)
dmu_j_complex = np.zeros(n_parameters, dtype=complex)
dvar_j_complex = np.zeros(n_parameters, dtype=complex)
dstd_dev_complex = np.zeros(n_parameters, dtype=complex)
for i in range(0, n_parameters):
dv[i] += pert
QoI.set_dv(dv)
sc_obj.evaluateQoIs(jdist, include_derivs=False)
mu_j = sc_obj.mean(of=['PolyRVDV'])
var_j = sc_obj.variance(of=['PolyRVDV'])
std_dev_j = np.sqrt(var_j['PolyRVDV'][0, 0])
dmu_j_complex[i] = mu_j['PolyRVDV'].imag / pert.imag
dvar_j_complex[i] = var_j['PolyRVDV'].imag / pert.imag
dstd_dev_complex[i] = std_dev_j.imag / pert.imag
dv[i] -= pert
err1 = dmu_j['PolyRVDV']['dv'] - dmu_j_complex
self.assertTrue((err1 < 1.e-13).all())
err2 = dvar_j['PolyRVDV']['dv'] - dvar_j_complex
self.assertTrue((err2 < 1.e-13).all())
err3 = dstd_dev['PolyRVDV']['dv'] - dstd_dev_complex
self.assertTrue((err2 < 1.e-13).all())
def run_hadamard(systemsize, eigen_decayrate, std_dev, n_eigenmodes):
n_collocation_pts = 2
# Create Hadmard Quadratic object
QoI = examples.HadamardQuadratic(systemsize, eigen_decayrate)
# # Create stochastic collocation object
# collocation = StochasticCollocation(n_collocation_pts, "Normal")
# Initialize chaospy distribution
x = np.random.rand(QoI.systemsize)
jdist = cp.MvNormal(x, np.diag(std_dev))
threshold_factor = 0.5
dominant_space = DimensionReduction(threshold_factor=threshold_factor,
exact_Hessian=False,
n_arnoldi_sample=71,
min_eigen_accuracy=1.e-2)
dominant_space.getDominantDirections(QoI,
jdist,
max_eigenmodes=n_eigenmodes)
# if systemsize == 64:
# print('x = \n', repr(x))
# print('std_dev = \n', repr(std_dev))
# print('iso_eigenvals = ', dominant_space.iso_eigenvals)
# print("dominant_indices = ", dominant_space.dominant_indices)
# Collocate
# collocation = StochasticCollocation(n_collocation_pts, "Normal")
# mu_j = collocation.normal.reduced_mean(QoI.eval_QoI, jdist, dominant_space)
# print "mu_j = ", mu_j
QoI_dict = {
'Hadamard': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
},
}
sc_obj = StochasticCollocation2(jdist,
n_collocation_pts,
'MvNormal',
QoI_dict,
include_derivs=False,
reduced_collocation=True,
dominant_dir=dominant_space.dominant_dir)
sc_obj.evaluateQoIs(jdist)
mu_j_dict = sc_obj.mean(of=['Hadamard'])
mu_j = mu_j_dict['Hadamard']
# Evaluate the analytical value of the Hadamard Quadratic
covariance = cp.Cov(jdist)
mu_analytic = QoI.eval_analytical_QoI_mean(x, covariance)
# print "mu_analytic = ", mu_analytic
relative_error = np.linalg.norm((mu_j - mu_analytic) / mu_analytic)
# print "relative_error = ", relative_error
return relative_error
def test_reduced_montecarlo(self):
systemsize = 4
eigen_decayrate = 2.0
# Create Hadmard Quadratic object
QoI = HadamardQuadratic(systemsize, eigen_decayrate)
true_eigenvals = np.array([0.08, 0.02, 0.005, 0.00888888888888889])
true_eigenvecs = np.array([[0.5, 0.5, -0.5, -0.5],
[0.5, -0.5, 0.5,
-0.5], [0.5, 0.5, 0.5, 0.5],
[0.5, -0.5, -0.5, 0.5]])
dominant_dir = true_eigenvecs[:, 0:3]
QoI_dict = {
'hadamard4_2': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
}
}
# Create the distribution
mu = np.ones(systemsize)
std_dev = 0.2 * np.eye(systemsize)
jdist = cp.MvNormal(mu, std_dev)
# Create the Monte Carlo object
nsample = 100000
mc_obj = MonteCarlo(nsample,
jdist,
QoI_dict,
reduced_collocation=True,
dominant_dir=dominant_dir,
include_derivs=False)
mc_obj.getSamples(jdist, include_derivs=False)
mu_j_mc = mc_obj.mean(jdist, of=['hadamard4_2'])
# Compare against a reduced stochastic collocation object
sc_obj = StochasticCollocation2(jdist,
3,
'MvNormal',
QoI_dict,
reduced_collocation=True,
dominant_dir=dominant_dir)
sc_obj.evaluateQoIs(jdist)
mu_j_sc = sc_obj.mean(of=['hadamard4_2'])
rel_err = abs((mu_j_mc['hadamard4_2'] - mu_j_sc['hadamard4_2']) /
mu_j_sc['hadamard4_2'])
self.assertTrue(rel_err[0] < 1.e-3)
def test_reduced_normalStochasticCollocation3D(self):
# This is not a very good test because we are comparing the reduced collocation
# against the analytical expected value. The only hting it tells us is that
# the solution is within the ball park of actual value. We still need to
# come up with a better test.
systemsize = 3
mu = mean_3dim # np.random.randn(systemsize)
std_dev = std_dev_3dim # abs(np.diag(np.random.randn(systemsize)))
jdist = cp.MvNormal(mu, std_dev)
# Create QoI Object
QoI = examples.Paraboloid3D(systemsize)
dominant_dir = np.array([[1.0, 0.0], [0.0, 1.0], [0.0, 0.0]],
dtype=np.float)
# Create the Stochastic Collocation object
deriv_dict = {
'xi': {
'dQoI_func': QoI.eval_QoIGradient,
'output_dimensions': systemsize
}
}
QoI_dict = {
'paraboloid': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
'deriv_dict': deriv_dict
}
}
sc_obj = StochasticCollocation2(jdist,
3,
'MvNormal',
QoI_dict,
reduced_collocation=True,
dominant_dir=dominant_dir)
sc_obj.evaluateQoIs(jdist)
mu_js = sc_obj.mean(of=['paraboloid'])
var_js = sc_obj.variance(of=['paraboloid'])
# Analytical mean
mu_j_analytical = QoI.eval_QoI_analyticalmean(mu, cp.Cov(jdist))
err = abs((mu_js['paraboloid'][0] - mu_j_analytical) / mu_j_analytical)
self.assertTrue(err < 1e-1)
# Analytical variance
var_j_analytical = QoI.eval_QoI_analyticalvariance(mu, cp.Cov(jdist))
err = abs(
(var_js['paraboloid'][0, 0] - var_j_analytical) / var_j_analytical)
self.assertTrue(err < 0.01)
def test_normalStochasticCollocation3D(self):
systemsize = 3
mu = mean_3dim # np.random.randn(systemsize)
std_dev = std_dev_3dim # np.diag(np.random.rand(systemsize))
jdist = cp.MvNormal(mu, std_dev)
# Create QoI Object
QoI = examples.Paraboloid3D(systemsize)
# Create the Stochastic Collocation object
deriv_dict = {
'xi': {
'dQoI_func': QoI.eval_QoIGradient,
'output_dimensions': systemsize
}
}
QoI_dict = {
'paraboloid': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
'deriv_dict': deriv_dict
}
}
sc_obj = StochasticCollocation2(jdist, 3, 'MvNormal', QoI_dict)
sc_obj.evaluateQoIs(jdist)
mu_js = sc_obj.mean(of=['paraboloid'])
var_js = sc_obj.variance(of=['paraboloid'])
# Analytical mean
mu_j_analytical = QoI.eval_QoI_analyticalmean(mu, cp.Cov(jdist))
err = abs((mu_js['paraboloid'][0] - mu_j_analytical) / mu_j_analytical)
self.assertTrue(err < 1.e-15)
# Analytical variance
var_j_analytical = QoI.eval_QoI_analyticalvariance(mu, cp.Cov(jdist))
err = abs(
(var_js['paraboloid'][0, 0] - var_j_analytical) / var_j_analytical)
self.assertTrue(err < 1.e-15)
def test_derivatives_scalarQoI(self):
systemsize = 3
mu = mean_3dim # np.random.randn(systemsize)
std_dev = std_dev_3dim # np.diag(np.random.rand(systemsize))
jdist = cp.MvNormal(mu, std_dev)
# Create QoI Object
QoI = examples.Paraboloid3D(systemsize)
# Create the Stochastic Collocation object
deriv_dict = {
'xi': {
'dQoI_func': QoI.eval_QoIGradient,
'output_dimensions': systemsize
}
}
QoI_dict = {
'paraboloid': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
'deriv_dict': deriv_dict
}
}
sc_obj = StochasticCollocation2(jdist,
3,
'MvNormal',
QoI_dict,
include_derivs=True)
sc_obj.evaluateQoIs(jdist, include_derivs=True)
dmu_j = sc_obj.dmean(of=['paraboloid'], wrt=['xi'])
# dvar_j = sc_obj.dvariance(of=['paraboloid'], wrt=['xi'])
# Analytical dmu_j
dmu_j_analytical = np.array([100 * mu[0], 50 * mu[1], 2 * mu[2]])
err = abs(
(dmu_j['paraboloid']['xi'] - dmu_j_analytical) / dmu_j_analytical)
self.assertTrue((err < 1.e-12).all())
def test_nonrv_derivatives_reduced_collocation(self):
# This test checks the analytical derivative w.r.t complex step
systemsize = 2
n_parameters = 2
mu = mean_2dim # np.random.randn(systemsize)
std_dev = std_dev_2dim # abs(np.diag(np.random.randn(systemsize)))
jdist = cp.MvNormal(mu, std_dev)
QoI = examples.PolyRVDV(data_type=complex)
# Create the Stochastic Collocation object
deriv_dict = {
'dv': {
'dQoI_func': QoI.eval_QoIGradient_dv,
'output_dimensions': n_parameters
}
}
QoI_dict = {
'PolyRVDV': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
'deriv_dict': deriv_dict
}
}
dv = np.random.randn(systemsize) + 0j
QoI.set_dv(dv)
# Create dimension reduction object
threshold_factor = 0.9
dominant_space = DimensionReduction(threshold_factor=threshold_factor,
exact_Hessian=True)
# Get the eigenmodes of the Hessian product and the dominant indices
dominant_space.getDominantDirections(QoI, jdist)
dominant_dir = dominant_space.iso_eigenvecs[:, dominant_space.
dominant_indices]
sc_obj = StochasticCollocation2(jdist,
4,
'MvNormal',
QoI_dict,
include_derivs=True,
reduced_collocation=True,
dominant_dir=dominant_dir,
data_type=complex)
sc_obj.evaluateQoIs(jdist, include_derivs=True)
dmu_j = sc_obj.dmean(of=['PolyRVDV'], wrt=['dv'])
dvar_j = sc_obj.dvariance(of=['PolyRVDV'], wrt=['dv'])
dstd_dev = sc_obj.dStdDev(of=['PolyRVDV'], wrt=['dv'])
# Lets do complex step
pert = complex(0, 1e-30)
dmu_j_complex = np.zeros(n_parameters, dtype=complex)
dvar_j_complex = np.zeros(n_parameters, dtype=complex)
dstd_dev_complex = np.zeros(n_parameters, dtype=complex)
for i in range(0, n_parameters):
dv[i] += pert
QoI.set_dv(dv)
sc_obj.evaluateQoIs(jdist, include_derivs=False)
mu_j = sc_obj.mean(of=['PolyRVDV'])
var_j = sc_obj.variance(of=['PolyRVDV'])
std_dev_j = np.sqrt(var_j['PolyRVDV'][0, 0])
dmu_j_complex[i] = mu_j['PolyRVDV'].imag / pert.imag
dvar_j_complex[i] = var_j['PolyRVDV'].imag / pert.imag
dstd_dev_complex[i] = std_dev_j.imag / pert.imag
dv[i] -= pert
err1 = dmu_j['PolyRVDV']['dv'] - dmu_j_complex
self.assertTrue((err1 < 1.e-13).all())
err2 = dvar_j['PolyRVDV']['dv'] - dvar_j_complex
self.assertTrue((err2 < 1.e-10).all())
err3 = dstd_dev['PolyRVDV']['dv'] - dstd_dev_complex
self.assertTrue((err2 < 1.e-13).all())
sample_radius=1.e-2)
dominant_space.getDominantDirections(UQObj.QoI,
UQObj.jdist,
max_eigenmodes=4)
# # Full collocation
# sc_obj = StochasticCollocation2(UQObj.jdist, 3, 'MvNormal', UQObj.QoI_dict,
# include_derivs=False)
# sc_obj.evaluateQoIs(UQObj.jdist, include_derivs=False)
# REduced collocation
dominant_dir = dominant_space.iso_eigenvecs[:, dominant_space.
dominant_indices]
sc_obj = StochasticCollocation2(UQObj.jdist,
3,
'MvNormal',
UQObj.QoI_dict,
include_derivs=False,
reduced_collocation=True,
dominant_dir=dominant_dir)
sc_obj.evaluateQoIs(UQObj.jdist, include_derivs=False)
# print('fvals = ')
# print(sc_obj.QoI_dict['fuelburn']['fvals'])
# Print initial value
init_mu_j = sc_obj.mean(of=['fuelburn'])
init_var_j = sc_obj.variance(of=['fuelburn'])
# print("Mean fuelburn = ", init_mu_j['fuelburn'][0])
# print("Variance fuelburn = ", init_var_j['fuelburn'][0])
# print()
mfmc_sol = {
'twist_cp': np.array([-1.59, 0.34, 4.50]),
def run_hadamard(systemsize, eigen_decayrate, std_dev, n_eigenmodes):
n_collocation_pts = 3
# Create Hadmard Quadratic object
QoI = examples.HadamardQuadratic(systemsize, eigen_decayrate)
# Initialize chaospy distribution
x = np.random.rand(QoI.systemsize)
jdist = cp.MvNormal(x, np.diag(std_dev))
threshold_factor = 1.0
use_exact_Hessian = False
if use_exact_Hessian:
dominant_space = DimensionReduction(threshold_factor=threshold_factor,
exact_Hessian=True)
dominant_space.getDominantDirections(QoI, jdist)
dominant_dir = dominant_space.iso_eigenvecs[:, 0:n_eigenmodes]
else:
dominant_space = DimensionReduction(threshold_factor=threshold_factor,
exact_Hessian=False,
n_arnoldi_sample=71,
min_eigen_accuracy=1.e-2)
dominant_space.getDominantDirections(QoI,
jdist,
max_eigenmodes=n_eigenmodes)
dominant_dir = dominant_space.dominant_dir
# print "dominant_indices = ", dominant_space.dominant_indices
# Create stochastic collocation object
# collocation = StochasticCollocation(n_collocation_pts, "Normal")
QoI_dict = {
'Hadamard': {
'QoI_func': QoI.eval_QoI,
'output_dimensions': 1,
},
}
sc_obj = StochasticCollocation2(jdist,
n_collocation_pts,
'MvNormal',
QoI_dict,
include_derivs=False,
reduced_collocation=True,
dominant_dir=dominant_dir)
sc_obj.evaluateQoIs(jdist)
# Collocate
# mu_j = collocation.normal.reduced_mean(QoI, jdist, dominant_space)
mu_j = sc_obj.mean(of=['Hadamard'])
var_j = sc_obj.variance(of=['Hadamard'])
std_dev_j = np.sqrt(var_j['Hadamard'])
# print "mu_j = ", mu_j
# Evaluate the analytical value of the Hadamard Quadratic
covariance = cp.Cov(jdist)
mu_analytic = QoI.eval_analytical_QoI_mean(x, covariance)
var_analytic = QoI.eval_analytical_QoI_variance(x, covariance)
std_dev_analytic = np.sqrt(var_analytic)
# print "mu_analytic = ", mu_analytic
relative_error_mu = np.linalg.norm(
(mu_j['Hadamard'] - mu_analytic) / mu_analytic)
relative_err_var = np.linalg.norm(
(var_j['Hadamard'] - var_analytic) / var_analytic)
relative_err_std_dev = np.linalg.norm(
(std_dev_j - std_dev_analytic) / std_dev_analytic)
# print "relative_error = ", relative_error
return relative_error_mu, relative_err_var, relative_err_std_dev
n_dominant_dir = int(sys.argv[2]) # 11
dominant_dir = eigenvecs[:, 0:n_dominant_dir]
use_surrogate_qoi_obj = False
if use_surrogate_qoi_obj:
surrogate_QoI_dict = {
'time_duration': {
'QoI_func': surrogate_QoI.eval_QoI,
'output_dimensions': 1,
}
}
# Create the stochastic collocation object
sc_obj = StochasticCollocation2(jdist,
3,
'MvNormal',
surrogate_QoI_dict,
reduced_collocation=True,
dominant_dir=dominant_dir,
include_derivs=False)
sc_obj.evaluateQoIs(jdist)
else:
# Create the stochastic collocation object
sc_obj = StochasticCollocation2(jdist,
2,
'MvNormal',
QoI_dict,
reduced_collocation=True,
dominant_dir=dominant_dir,
include_derivs=False)
sc_obj.evaluateQoIs(jdist)
},
}
# Check if plugging back value also yields the expected results
QoI.p['oas_scaneagle.wing.thickness_cp'] = np.array([0.008, 0.008, 0.008])
QoI.p['oas_scaneagle.wing.twist_cp'] = np.array([2.5, 2.5, 5.])
QoI.p['oas_scaneagle.wing.sweep'] = 20.0
QoI.p['oas_scaneagle.alpha'] = 5.0
sample_radii = [1.e-1, 1.e-2, 1.e-3, 1.e-4, 1.e-5, 1.e-6]
if sys.argv[1] == 'full':
# Create the stochastic collocation object
sc_obj_full = StochasticCollocation2(jdist,
3,
'MvNormal',
QoI_dict,
include_derivs=False,
reduced_collocation=False)
start_time = time.time()
sc_obj_full.evaluateQoIs(jdist, include_derivs=False)
t_sampling = time.time()
mu_j_full = sc_obj_full.mean(of=['fuelburn'])
time_elapsed_mu = time.time() - t_sampling
t_before_var = time.time()
var_j_full = sc_obj_full.variance(of=['fuelburn'])
time_elapsed_var = time.time() - t_before_var
print('mu_j_full = ', mu_j_full['fuelburn'][0])
print('var_j_full = ', var_j_full['fuelburn'][0, 0])
n_thickness_intersects = UQObj.QoI.p[
'oas_scaneagle.AS_point_0.wing_perf.thickness_intersects'].size
n_CM = 3
print("eigenvals = ", UQObj.dominant_space.iso_eigenvals)
print('eigenvecs = \n', UQObj.dominant_space.iso_eigenvecs)
print('dominant_dir = \n', UQObj.dominant_space.dominant_dir)
print('\n#-----------------------------------------------------------#')
# Full collocation
use_stochastic_collocation = False
use_monte_carlo = True
if use_stochastic_collocation:
colloc_obj = StochasticCollocation2(UQObj.jdist,
3,
'MvNormal',
UQObj.QoI_dict,
include_derivs=False)
red_colloc_obj = StochasticCollocation2(
UQObj.jdist,
3,
'MvNormal',
UQObj.QoI_dict,
include_derivs=False,
reduced_collocation=True,
# dominant_dir=custom_eigenvec[:,0:int(sys.argv[1])])
dominant_dir=UQObj.dominant_space.dominant_dir)
elif use_monte_carlo:
nsample = 100
colloc_obj = MonteCarlo(nsample,
UQObj.jdist,
