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,
axis=ax,
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_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_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')