def test_chol_sample(self):
"""
Sample field using Cholesky factorization of the covariance and of
the precision.
"""
#
# Initialize Gaussian Random Field
#
n = 201 # size
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n - 1)
Q = linalg.solve(K, I)
# Define mean
mean = np.random.rand(n - 1, 1)
# Define Gaussian field
u_cov = GaussianField(n - 1, mean=mean, K=K, mode='covariance')
u_prc = GaussianField(n - 1, mean=mean, K=Q, mode='precision')
# Define generating white noise
z = u_cov.iid_gauss(n_samples=10)
u_chol_prec = u_prc.sample(z=z, mode='precision', decomposition='chol')
u_chol_cov = u_cov.sample(z=z, mode='covariance', decomposition='eig')
u_chol_can = u_prc.sample(z=z, mode='canonical', decomposition='chol')
fig, ax = plt.subplots(1, 3, figsize=(7, 3))
ax[0].plot(u_chol_prec, linewidth=0.5)
ax[0].set_title('Precision')
ax[0].axis('tight')
ax[1].plot(u_chol_cov, linewidth=0.5)
ax[1].set_title('Covariance')
ax[1].axis('tight')
ax[2].plot(u_chol_can, linewidth=0.5)
ax[2].set_title('Canonical')
ax[2].axis('tight')
fig.suptitle('Samples')
fig.tight_layout()
fig.subplots_adjust(top=0.8)
fig.savefig('gaussian_field_chol_samples.eps')
def test_degenerate_sample(self):
"""
Test support and reduced covariance
"""
oort = 1 / np.sqrt(2)
V = np.array([[0.5, oort, 0, 0.5], [0.5, 0, -oort, -0.5],
[0.5, -oort, 0, 0.5], [0.5, 0, oort, -0.5]])
# Eigenvalues
d = np.array([4, 3, 2, 0], dtype=float)
Lmd = np.diag(d)
# Covariance matrix
K = V.dot(Lmd.dot(V.T))
# Zero mean Gaussian field
u_ex = GaussianField(4, K=K, mode='covariance', support=V[:, 0:3])
u_im = GaussianField(4, K=K, mode='covariance')
u_im.update_support()
# Check reduced covariances
self.assertTrue(
np.allclose(u_ex.covariance().get_matrix(),
u_im.covariance().get_matrix().toarray()))
# Check supports
V_ex = u_ex.support()
V_im = u_im.support()
# Ensure they have the same sign
for i in range(V_ex.shape[1]):
if V_ex[0, i] < 0:
V_ex[:, i] = -V_ex[:, i]
if V_im[0, i] < 0:
V_im[:, i] = -V_im[:, i]
self.assertTrue(np.allclose(V_ex, V_im))
u_ex.set_support(V_ex)
u_im.set_support(V_im)
# Compare samples
z = u_ex.iid_gauss(n_samples=1)
u_ex_smp = u_ex.sample(z=z, decomposition='chol')
u_im_smp = u_im.sample(z=z, decomposition='chol')
self.assertTrue(np.allclose(u_ex_smp, u_im_smp))
def test_gaussian_random_field(self):
"""
Reproduce statistics of Gaussian random field
"""
#
# Define Gaussian Field with degenerate support
#
oort = 1 / np.sqrt(2)
V = np.array([[0.5, oort, 0, 0.5], [0.5, 0, -oort, -0.5],
[0.5, -oort, 0, 0.5], [0.5, 0, oort, -0.5]])
# Eigenvalues
d = np.array([4, 3, 2, 1], dtype=float)
Lmd = np.diag(d)
# Covariance matrix
K = V.dot(Lmd.dot(V.T))
mu = np.array([1, 2, 3, 4])[:, None]
# Zero mean Gaussian field
dim = 4
eta = GaussianField(dim, mean=mu, K=K, mode='covariance')
n_vars = eta.covariance().size()
level = 1
# Define Gauss-Hermite physicist's rule exp(-x**2)
grid = Tasmanian.makeGlobalGrid(n_vars, 4, level, "level",
"gauss-hermite-odd")
# Evaluate the Gaussian random field at the Gauss points
z = grid.getPoints()
y = np.sqrt(2) * z
const_norm = np.sqrt(np.pi)**n_vars
# Evaluate the random field at the Gauss points
w = grid.getQuadratureWeights()
etay = eta.sample(z=y.T)
n = grid.getNumPoints()
I = np.zeros(4)
II = np.zeros((4, 4))
for i in range(n):
II += w[i] * np.outer(etay[:, i] - mu.ravel(),
etay[:, i] - mu.ravel())
I += w[i] * etay[:, i]
I /= const_norm
II /= const_norm
self.assertTrue(np.allclose(II, K))
self.assertTrue(np.allclose(I, mu.ravel()))
def test02_variance():
"""
Compute the variance of J(q) for different mesh refinement levels
and compare with MC estimates.
"""
l_max = 8
for i_res in np.arange(2, l_max):
# Computational mesh
mesh = Mesh1D(resolution=(2**i_res, ))
# Element
element = QuadFE(mesh.dim(), 'DQ0')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Linear Functional
mesh.mark_region('integrate',
lambda x: x >= 0.75,
entity_type='cell',
strict_containment=False)
phi = Basis(dofhandler)
assembler = Assembler(Form(4, test=phi, flag='integrate'))
assembler.assemble()
L = assembler.get_vector()
# Define Gaussian random field
C = Covariance(dofhandler, name='gaussian', parameters={'l': 0.05})
C.compute_eig_decomp()
eta = GaussianField(dofhandler.n_dofs(), K=C)
eta.update_support()
n_samples = 100000
J_paths = L.dot(eta.sample(n_samples=n_samples))
var_mc = np.var(J_paths)
lmd, V = C.get_eig_decomp()
LV = L.dot(V)
var_an = LV.dot(np.diag(lmd).dot(LV.transpose()))
print(var_mc, var_an)
phiy_1 = Basis(dQ1, 'vy')
phi_2 = Basis(dQ2)
phix_2 = Basis(dQ2, 'vx')
phiy_2 = Basis(dQ2, 'vy')
#
# Define Random field
#
cov = Covariance(dQ0, name='gaussian', parameters={'l': 0.01})
cov.compute_eig_decomp()
q = GaussianField(dQ0.n_dofs(), K=cov)
# Sample Random field
n_samples = 100
eq = Nodal(basis=phi_0, data=np.exp(q.sample(n_samples)))
plot.contour(eq, n_sample=25)
#
# Compute state
#
# Define weak form
state = [[
Form(eq, test=phix_1, trial=phix_1),
Form(eq, test=phiy_1, trial=phiy_1),
Form(1, test=phi_1)
], [Form(1, test=phi_1, flag='dmn')]]
# Assemble system
phi = Basis(dQ1)
phi_x = Basis(dQ1, derivative='vx')
phi_y = Basis(dQ1, derivative='vy')
#
# Diffusion Parameter
#
# Covariance Matrix
K = Covariance(dQ1, name='exponential', parameters={'sgm': 1, 'l': 0.1})
# Gaussian random field θ
tht = GaussianField(dQ1.n_dofs(), K=K)
# Sample from field
tht_fn = Nodal(data=tht.sample(n_samples=3), basis=phi)
plot = Plot()
plot.contour(tht_fn)
#
# Advection
#
v = [0.1, -0.1]
plot.mesh(mesh, regions=[('in', 'edge'), ('out', 'edge'), ('reg', 'cell')])
k = Kernel(tht_fn, F=lambda tht: np.exp(tht))
adv_diff = [
Form(k, trial=phi_x, test=phi_x),
Form(k, trial=phi_y, test=phi_y),
Form(0, test=phi),
#
# Regularization parameter
#
#gamma = 0.00001
gamma = 0.1
#
# Random diffusion coefficient
#
n_samples = 200
cov = Covariance(dh_y, name='gaussian', parameters={'l':0.1})
k = GaussianField(ny, K=cov)
k.update_support()
kfn = Nodal(dofhandler=dh_y, data=k.sample(n_samples=n_samples))
# =============================================================================
# Assembly
# =============================================================================
K = Kernel(kfn, F=lambda f:np.exp(f)) # diffusivity
problems = [[Form(K, test=phi_x, trial=phi_x)],
[Form(test=phi, trial=phi)]]
assembler = Assembler(problems, mesh)
assembler.assemble()
# Mass matrix (for control)
M = assembler.af[1]['bilinear'].get_matrix()
entity_flag='injection')
# Initial control
n_Q1 = dh_Q1.n_dofs()
data = np.zeros((n_Q1, 1))
data[production_dofs, :] = 1
u = Nodal(dofhandler=dh_Q1, data=data, dim=2)
#
# Random diffusion coefficient
#
cov = Covariance(dh_Q1, name='gaussian', parameters={'l': 0.1})
n_samples = 1000
k = GaussianField(n_Q1, K=cov)
k.update_support()
kfn = Nodal(dofhandler=dh_Q1, data=k.sample(n_samples=n_samples))
# =============================================================================
# Assembly
# =============================================================================
vb.comment('assembling system')
vb.tic()
K = Kernel(kfn, F=lambda f: np.exp(f)) # diffusivity
problems = [[
Form(K, test=phi_x, trial=phi_x),
Form(K, test=phi_y, trial=phi_y),
Form(0, test=phi)
], [Form(test=phi, trial=phi)]]
assembler = Assembler(problems, mesh)
# Random field
# =============================================================================
n_samples = 10
n_train, n_test = 8, 2
i_train = np.arange(n_train)
i_test = np.arange(n_train, n_samples)
cov = Covariance(dofhandler, name='gaussian', parameters={'l': 0.1})
cov.compute_eig_decomp()
d, V = cov.get_eig_decomp()
plt.semilogy(d, '.')
plt.show()
log_q = GaussianField(n, K=cov)
log_q.update_support()
qfn = Nodal(dofhandler=dofhandler,
data=np.exp(log_q.sample(n_samples=n_samples)))
plot = Plot()
plot.line(qfn, i_sample=np.arange(n_samples))
# =============================================================================
# Generate Snapshot Set
# =============================================================================
phi = Basis(dofhandler, 'u')
phi_x = Basis(dofhandler, 'ux')
problems = [[Form(kernel=qfn, trial=phi_x, test=phi_x),
Form(1, test=phi)], [Form(1, test=phi, trial=phi)]]
assembler = Assembler(problems, mesh)
assembler.assemble()
lmd = np.arange(len(d))
ax.semilogy(lmd, d, '.-', label='level=%d' % i)
plt.legend()
plt.show()
#
# Define random field on the fine mesh
#
C = Covariance(dofhandler, name='gaussian', parameters={'l': 0.05})
C.compute_eig_decomp()
eta = GaussianField(dofhandler.n_dofs(), K=C)
eta.update_support()
#eta_path = Nodal(data=eta.sample(), basis=phi)
eta0 = P.dot(eta.sample())
eg0 = eta.condition(P, eta0, n_samples=100)
eg0_paths = Nodal(data=eg0, basis=Basis(dofhandler))
e0_path = Nodal(data=eta0, basis=Basis(dofhandler, subforest_flag=0))
plot = Plot(quickview=False)
ax = plt.subplot(111)
for i in range(30):
ax = plot.line(eg0_paths,
axis=ax,
mesh=mesh,
i_sample=i,
plot_kwargs={
'color': 'k',
'linewidth': 0.5
})
ax = plot.line(e0_path,
u_data = np.zeros((ny, 1))
u_data[dofs_inj] = 1
u = Nodal(dofhandler=dh_y, data=u_data, dim=1)
#
# Regularization parameter
#
gamma = 0.00001
#gamma = 0.1
#
# Random diffusion coefficient
#
cov = Covariance(dh_y, name='gaussian', parameters={'l': 0.1})
k = GaussianField(ny, K=cov)
k.update_support()
kfn = Nodal(dofhandler=dh_y, data=k.sample(n_samples=1))
# =============================================================================
# Assembly
# =============================================================================
K = Kernel(kfn, F=lambda f: np.exp(f)) # diffusivity
problems = [[Form(K, test=phi_x, trial=phi_x)], [Form(test=phi, trial=phi)]]
assembler = Assembler(problems, mesh)
assembler.assemble()
# Mass matrix (for control)
A = assembler.af[0]['bilinear'].get_matrix()
M = assembler.af[1]['bilinear'].get_matrix()
def test_condition_pointswise(self):
"""
Generate samples and random field by conditioning on pointwise data
"""
#
# Initialize Gaussian Random Field
#
# Resolution
max_res = 10
n = 2**max_res + 1 # size
# Hurst parameter
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n - 1)
Q = linalg.solve(K, I)
# Define mean
mean = np.random.rand(n - 1, 1)
# Define Gaussian field
u_cov = GaussianField(n - 1, mean=mean, K=K, mode='covariance')
u_prc = GaussianField(n - 1, mean=mean, K=Q, mode='precision')
# Define generating white noise
z = u_cov.iid_gauss(n_samples=10)
u_obs = u_cov.sample(z=z)
# Index of measured observations
A = np.arange(0, n - 1, 2)
# observed quantities
e = u_obs[A, 0][:, None]
#print('e shape', e.shape)
# change A into matrix
k = len(A)
rows = np.arange(k)
cols = A
vals = np.ones(k)
AA = sp.coo_matrix((vals, (rows, cols)), shape=(k, n - 1)).toarray()
AKAt = AA.dot(K.dot(AA.T))
KAt = K.dot(AA.T)
U, S, Vt = linalg.svd(AA)
#print(U)
#print(S)
#print(Vt)
#print(AA.dot(u_obs)-e)
k = e.shape[0]
Ko = 0.01 * np.identity(k)
# Debug
K = u_cov.covariance()
#U_spp = u_cov.support()
#A_cmp = A.dot(U_spp)
u_cond = u_cov.condition(A, e, Ko=Ko, n_samples=100)
"""
def test_condition_ptwise(self):
#
# Initialize Gaussian Random Field
#
# Resolution
l_max = 9
n = 2**l_max + 1 # size
# Hurst parameter
H = 0.5 # Hurst parameter in [0.5,1]
# Form covariance and precision matrices
x = np.arange(1, n + 1)
X, Y = np.meshgrid(x, x)
K = fbm_cov(X, Y, H)
# Compute the precision matrix
I = np.identity(n)
Q = linalg.solve(K, I)
# Plot meshes
fig, ax = plt.subplots(1, 1)
n = 2**l_max + 1
for l in range(l_max):
nl = 2**l + 1
i_spp = [i * 2**(l_max - l) for i in range(nl)]
ax.plot(x[i_spp], l * np.ones(nl), '.')
#ax.plot(x,'.', markersize=0.1)
#plt.show()
# Plot conditioned field
fig, ax = plt.subplots(3, 3)
# Define original field
u = []
n = 2**(l_max) + 1
for l in range(l_max):
nl = 2**l + 1
i_spp = [i * 2**(l_max - l) for i in range(nl)]
V_spp = I[:, i_spp]
if l == 0:
u_fne = GaussianField(n, K=K, mode='covariance',\
support=V_spp)
u_obs = u_fne.sample()
i_obs = np.array(i_spp)
else:
u_fne = GaussianField(n, K=K, mode='covariance',\
support=V_spp)
u_cnd = u_fne.condition(i_obs, u_obs[i_obs], output='field')
u_obs = u_cnd.sample()
i_obs = np.array(i_spp)
u.append(u_obs)
# Plot
for ll in range(l, l_max):
i, j = np.unravel_index(ll, (3, 3))
if ll == l:
ax[i, j].plot(x[i_spp],
5 * np.exp(0.01 * u_obs[i_spp]),
linewidth=0.5)
else:
ax[i, j].plot(x[i_spp],
5 * np.exp(0.01 * u_obs[i_spp]),
'g',
linewidth=0.1,
alpha=0.1)
fig.savefig('successive_conditioning.pdf')
def test_condition_with_nullspace(self):
"""
Test conditioning with an existing nullspace
"""
#
# Define Gaussian Field with degenerate support
#
oort = 1 / np.sqrt(2)
V = np.array([[0.5, oort, 0, 0.5], [0.5, 0, -oort, -0.5],
[0.5, -oort, 0, 0.5], [0.5, 0, oort, -0.5]])
# Eigenvalues
d = np.array([4, 3, 2, 0], dtype=float)
Lmd = np.diag(d)
# Covariance matrix
K = V.dot(Lmd.dot(V.T))
mu = np.array([1, 2, 3, 4])[:, None]
# Zero mean Gaussian field
u_ex = GaussianField(4,
mean=mu,
K=K,
mode='covariance',
support=V[:, 0:3])
#
# Conditioned random field (hard constraint)
#
# Condition on Ax=e (A full rank)
A = np.array([[1, 2, 3, 2], [2, 4, 6, 4]])
e = np.array([[1], [5]])
# Matrix A is not full rank -> error
with self.assertRaises(np.linalg.LinAlgError):
u_ex.condition(A, e)
# Full rank matrix
A = np.array([[1, 2, 3, 2], [3, 9, 8, 7]])
# Compute conditioned field
u_cnd = u_ex.condition(A, e, output='field')
X_cnd = u_cnd.sample(n_samples=100)
# Sample by Kriging
X_kriged = u_cnd.sample(n_samples=100)
# Check that both samples satisfy constraint
self.assertTrue(np.allclose(A.dot(X_cnd) - e, 0))
self.assertTrue(np.allclose(A.dot(X_kriged) - e, 0))
# Check that the support of the conditioned field is contained in
# that of the unconditioned one
self.assertTrue(
np.allclose(u_ex.project(u_cnd.support(), 'nullspace'), 0))
plt.close('all')
fig, ax = plt.subplots(1, 3, figsize=(7, 3))
ax[0].plot(u_ex.sample(n_samples=100), 'k', linewidth=0.1)
ax[0].set_title('Unconditioned Field')
ax[1].plot(X_kriged, 'k', linewidth=0.1)
ax[1].plot(u_cnd.mean())
ax[1].set_title('Kriged Sample')
ax[2].plot(X_cnd, 'k', linewidth=0.1)
ax[2].set_title('Sample of conditioned field')
fig.suptitle('Samples')
fig.tight_layout()
fig.subplots_adjust(top=0.8)
fig.savefig('degenerate_gf_conditioned_samples.eps')