def test_assemble_iiform(self):
mesh = Mesh1D(resolution=(1, ))
Q1 = QuadFE(1, 'DQ1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
phi = Basis(dofhandler, 'u')
k = Explicit(lambda x, y: x * y, n_variables=2, dim=1)
kernel = Kernel(k)
form = IIForm(kernel, test=phi, trial=phi)
assembler = Assembler(form, mesh)
assembler.assemble()
Ku = Nodal(lambda x: 1 / 3 * x, basis=phi)
#af = assembler.af[0]['bilinear']
M = assembler.get_matrix().toarray()
u = Nodal(lambda x: x, basis=phi)
u_vec = u.data()
self.assertTrue(np.allclose(M.dot(u_vec), Ku.data()))
def test_constructor(self):
#
# Define mesh, element, and dofhandler
#
mesh = QuadMesh(box=[0, 20, 0, 20],
resolution=(20, 20),
periodic={0, 1})
dim = mesh.dim()
element = QuadFE(dim, 'Q2')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler, 'u')
alph = 2
kppa = 1
# Symmetric tensor gma T + bta* vv^T
gma = 0.1
bta = 25
p = lambda x: 10/np.pi*(0.75*np.sin(np.pi*x[:,0]/10)+\
0.25*np.sin(np.pi*x[:,1]/10))
f = Nodal(f=p, basis=basis)
fx = f.differentiate((1, 0))
fy = f.differentiate((1, 1))
#plot.contour(f)
x = np.linspace(0, 20, 12)
X, Y = np.meshgrid(x, x)
xy = np.array([X.ravel(), Y.ravel()]).T
U = fx.eval(xy).reshape(X.shape)
V = fy.eval(xy).reshape(X.shape)
v1 = lambda x: -0.25 * np.cos(np.pi * x[:, 1] / 10)
v2 = lambda x: 0.75 * np.cos(np.pi * x[:, 0] / 10)
U = v1(xy).reshape(X.shape)
V = v2(xy).reshape(X.shape)
#plt.quiver(X,Y, U, V)
#plt.show()
h11 = Explicit(lambda x: gma + bta * v1(x) * v1(x), dim=2)
h12 = Explicit(lambda x: bta * v1(x) * v2(x), dim=2)
h22 = Explicit(lambda x: gma + bta * v2(x) * v2(x), dim=2)
tau = (h11, h12, h22)
#tau = (Constant(2), Constant(1), Constant(1))
#
# Define default elliptic field
#
u = EllipticField(dofhandler, kappa=1, tau=tau, gamma=2)
Q = u.precision()
v = Nodal(data=u.sample(mode='precision', decomposition='chol'),
basis=basis)
plot = Plot(20)
plot.contour(v)
def test_assemble_ipform(self):
# =====================================================================
# Test 7: Assemble Kernel
# =====================================================================
mesh = Mesh1D(resolution=(10, ))
Q1 = QuadFE(1, 'DQ1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
phi = Basis(dofhandler, 'u')
k = Explicit(lambda x, y: x * y, n_variables=2, dim=1)
kernel = Kernel(k)
form = IPForm(kernel, test=phi, trial=phi)
assembler = Assembler(form, mesh)
assembler.assemble()
#af = assembler.af[0]['bilinear']
M = assembler.get_matrix().toarray()
u = Nodal(lambda x: x, basis=phi)
v = Nodal(lambda x: 1 - x, basis=phi)
u_vec = u.data()
v_vec = v.data()
I = v_vec.T.dot(M.dot(u_vec))
self.assertAlmostEqual(I[0, 0], 1 / 18)
def test03_dJdq():
"""
Compute dJdq for a simple problem, check that it works
"""
#
# Mesh
#
mesh = Mesh1D(resolution=(20, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-10)
#
# Element
#
Q = QuadFE(mesh.dim(), 'Q3')
dh = DofHandler(mesh, Q)
dh.distribute_dofs()
nx = dh.n_dofs()
x = dh.get_dof_vertices()
#
# Basis
#
phi = Basis(dh, 'v')
phi_x = Basis(dh, 'vx')
#
# Parameters
#
# Reference q
q_ref = Nodal(data=np.zeros(nx), basis=phi)
# Perturbation
dq = Nodal(data=np.ones(nx), basis=phi)
#
# Sample Reference QoI
#
J, u_ref = sample_qoi(q_ref.data(), dh, return_state=True)
u_ref = Nodal(data=u_ref, basis=phi)
#
# Compute dJdq
#
# Perturbation method
Jp_per = dJdq_per(q_ref, dq, dh)
# Sensitivity method
Jp_sen = dJdq_sen(q_ref, u_ref, dq)
# Adjoint method
Jp_adj = dJdq_adj(q_ref, u_ref, dq)
# Check that the answers are close to -1
assert np.allclose(Jp_per, -1)
assert np.allclose(Jp_sen, -1)
assert np.allclose(Jp_adj, -1)
def sensitivity_sample_qoi(exp_q, dofhandler):
"""
Sample QoI by means of Taylor expansion
J(q+dq) ~= J(q) + dJdq(q)dq
"""
# Basis
phi = Basis(dofhandler, 'v')
phi_x = Basis(dofhandler, 'vx')
# Define problem
exp_q_fn = Nodal(data=exp_q, basis=phi)
primal = [Form(exp_q_fn, test=phi_x, trial=phi_x), Form(1, test=phi)]
adjoint = [Form(exp_q_fn, test=phi_x, trial=phi_x), Form(0, test=phi)]
qoi = [Form(exp_q_fn, test=phi_x)]
problems = [primal, adjoint, qoi]
# Define assembler
assembler = Assembler(problems)
#
# Dirichlet conditions for primal problem
#
assembler.add_dirichlet('left', 0, i_problem=0)
assembler.add_dirichlet('right', 1, i_problem=0)
# Dirichlet conditions for adjoint problem
assembler.add_dirichlet('left', 0, i_problem=1)
assembler.add_dirichlet('right', -1, i_problem=1)
# Assemble system
assembler.assemble()
# Compute solution and qoi at q (primal)
u = assembler.solve(i_problem=0)
# Compute solution of the adjoint problem
v = assembler.solve(i_problem=1)
# Evaluate J
J = u.dot(assembler.get_vector(2))
#
# Assemble gradient
#
ux_fn = Nodal(data=u, basis=phi_x)
vx_fn = Nodal(data=v, basis=phi_x)
k_int = Kernel(f=[exp_q_fn, ux_fn, vx_fn],
F=lambda exp_q, ux, vx: exp_q * ux * vx)
problem = [Form(k_int, test=phi)]
assembler = Assembler(problem)
assembler.assemble()
dJ = -assembler.get_vector()
return dJ
def test_n_samples(self):
#
# Sampled Case
#
meshes = {1: Mesh1D(), 2: QuadMesh()}
elements = {1: QuadFE(1, 'Q2'), 2: QuadFE(2, 'Q2')}
# Use function to set data
fns = {
1: {
1: lambda x: 2 * x[:, 0]**2,
2: lambda x, y: 2 * x[:, 0] + 2 * y[:, 0]
},
2: {
1: lambda x: x[:, 0]**2 + x[:, 1],
2: lambda x, y: x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1]
}
}
# n_samples = 2
parms = {1: {1: [{}, {}], 2: [{}, {}]}, 2: {1: [{}, {}], 2: [{}, {}]}}
for dim in [1, 2]:
mesh = meshes[dim]
element = elements[dim]
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler)
for n_variables in [1, 2]:
fn = fns[dim][n_variables]
parm = parms[dim][n_variables]
#
# Deterministic
#
f = Nodal(f=fn,
mesh=mesh,
basis=basis,
element=element,
dim=dim,
n_variables=n_variables)
self.assertEqual(f.n_samples(), 1)
#
# Sampled
#
f = Nodal(f=fn,
parameters=parm,
basis=basis,
mesh=mesh,
element=element,
dim=dim,
n_variables=n_variables)
self.assertEqual(f.n_samples(), 2)
def test02a_sensitivity_gradient():
"""
Test whether the sensitivity and adjoint calculations give the same gradient
"""
# Mesh
mesh = Mesh1D(resolution=(100, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(1 - x) < 1e-10)
# Element
Q = QuadFE(mesh.dim(), 'Q2')
dh = DofHandler(mesh, Q)
dh.distribute_dofs()
n_dofs = dh.n_dofs()
phi = Basis(dh, 'u')
# Covariance
cov = Covariance(dh, name='gaussian', parameters={'l': 0.05})
cov.compute_eig_decomp()
lmd, V = cov.get_eig_decomp()
d = len(lmd)
# Coarse field (single sample)
d0 = 2
z0 = np.random.randn(d0, 1)
q0 = sample_q0(V, lmd, d0, z0)
q0_fn = Nodal(data=q0, basis=phi)
# State
J0, u0 = sample_qoi(q0, dh, return_state=True)
u0_fn = Nodal(data=u0, basis=phi)
# Compute gradient using sensitivity
dJs = np.zeros(n_dofs)
for i in range(n_dofs):
# Define perturbation
dq = np.zeros(n_dofs)
dq[i] = 1
dq_fn = Nodal(data=dq, basis=phi)
# Compute gradient using sensitivity
dJs[i] = dJdq_sen(q0_fn, u0_fn, dq_fn)
dJs_fn = Nodal(data=dJs, basis=phi)
plot = Plot()
plot.line(dJs_fn)
# Compute gradient using adjoint method
dJa = dJdq_adj(q0_fn, u0_fn)
dJa_fn = Nodal(data=dJa, basis=phi)
print(dJa)
plot.line(dJa_fn)
def test02_1d_dirichlet_higher_order(self):
mesh = Mesh1D()
for etype in ['Q2', 'Q3']:
element = QuadFE(1, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
ux = Basis(dofhandler, 'ux')
u = Basis(dofhandler, 'u')
# Exact solution
ue = Nodal(f=lambda x: x * (1 - x), basis=u)
# Define coefficient functions
one = Constant(1)
two = Constant(2)
# Define forms
a = Form(kernel=Kernel(one), trial=ux, test=ux)
L = Form(kernel=Kernel(two), test=u)
problem = [a, L]
# Assemble problem
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector()
# Set up linear system
system = LinearSystem(u, A=A, b=b)
# Boundary functions
bnd_left = lambda x: np.abs(x) < 1e-9
bnd_right = lambda x: np.abs(1 - x) < 1e-9
# Mark mesh
mesh.mark_region('left', bnd_left, entity_type='vertex')
mesh.mark_region('right', bnd_right, entity_type='vertex')
# Add Dirichlet constraints to system
system.add_dirichlet_constraint('left', 0)
system.add_dirichlet_constraint('right', 0)
# Solve system
system.solve_system()
system.resolve_constraints()
# Compare solution with the exact solution
ua = system.get_solution(as_function=True)
self.assertTrue(np.allclose(ua.data(), ue.data()))
def test01_solve_2d(self):
"""
Solve a simple 2D problem with no hanging nodes
"""
mesh = QuadMesh(resolution=(5, 5))
# Mark dirichlet boundaries
mesh.mark_region('left',
lambda x, dummy: np.abs(x) < 1e-9,
entity_type='half_edge')
mesh.mark_region('right',
lambda x, dummy: np.abs(x - 1) < 1e-9,
entity_type='half_edge')
Q1 = QuadFE(mesh.dim(), 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
phi = Basis(dQ1, 'u')
phi_x = Basis(dQ1, 'ux')
phi_y = Basis(dQ1, 'uy')
problem = [
Form(1, test=phi_x, trial=phi_x),
Form(1, test=phi_y, trial=phi_y),
Form(0, test=phi)
]
assembler = Assembler(problem, mesh)
assembler.add_dirichlet('left', dir_fn=0)
assembler.add_dirichlet('right', dir_fn=1)
assembler.assemble()
# Get matrix dirichlet correction and right hand side
A = assembler.get_matrix().toarray()
x0 = assembler.assembled_bnd()
b = assembler.get_vector()
ua = np.zeros((phi.n_dofs(), 1))
int_dofs = assembler.get_dofs('interior')
ua[int_dofs, 0] = np.linalg.solve(A, b - x0)
dir_bc = assembler.get_dirichlet()
dir_vals = np.array([dir_bc[dof] for dof in dir_bc])
dir_dofs = [dof for dof in dir_bc]
ua[dir_dofs] = dir_vals
ue_fn = Nodal(f=lambda x: x[:, 0], basis=phi)
ue = ue_fn.data()
self.assertTrue(np.allclose(ue, ua))
self.assertTrue(np.allclose(x0 + A.dot(ua[int_dofs, 0]), b))
def get_solution(self, as_function=True):
"""
Returns the solution of the linear system
"""
if not as_function:
#
# Return solution vector
#
return self.__u
else:
#
# Return solution as nodal function
#
u = Nodal(data=self.__u, basis=self.get_basis())
return u
def test_edge_integrals(self):
"""
Test computing
"""
mesh = QuadMesh(resolution=(1, 1))
Q = QuadFE(2, 'Q1')
dQ = DofHandler(mesh, Q)
dQ.distribute_dofs()
phi = Basis(dQ, 'u')
f = Nodal(data=np.ones((phi.n_dofs(), 1)), basis=phi)
kernel = Kernel(f)
form = Form(kernel, dmu='ds')
assembler = Assembler(form, mesh)
cell = mesh.cells.get_leaves()[0]
shape_info = assembler.shape_info(cell)
xg, wg, phi, dofs = assembler.shape_eval(cell)
def set_diffusion(dQ, z):
"""
Generate sample of diffusion coefficient
Inputs:
dQ: DofHandler, dofhandler object
z: (n_samples, n_dim) array of sample points in [0,1]^2
"""
x = dQ.get_dof_vertices()
q_vals = 1 + 0.1 *np.outer(np.cos(1*np.pi*x[:,0]),z[:,0])+\
0.05 *np.outer(np.cos(2*np.pi*x[:,0]),z[:,1])+\
0.01 *np.outer(np.cos(3*np.pi*x[:,0]),z[:,2])+\
0.005*np.outer(np.cos(4*np.pi*x[:,0]),z[:,3])
q_fn = Nodal(dofhandler=dQ, data=q_vals)
return q_fn
def sample_state(mesh,dQ,z,mflag,reference=False):
"""
Compute the sample output corresponding to a given input
"""
n_samples = z.shape[0]
q = set_diffusion(dQ,z)
phi = Basis(dQ,'u', mflag)
phi_x = Basis(dQ, 'ux', mflag)
if reference:
problems = [[Form(q,test=phi_x,trial=phi_x), Form(1,test=phi)],
[Form(1,test=phi, trial=phi)],
[Form(1,test=phi_x, trial=phi_x)]]
else:
problems = [[Form(q,test=phi_x,trial=phi_x), Form(1,test=phi)]]
assembler = Assembler(problems, mesh, subforest_flag=mflag)
assembler.assemble()
A = assembler.af[0]['bilinear'].get_matrix()
b = assembler.af[0]['linear'].get_matrix()
if reference:
M = assembler.af[1]['bilinear'].get_matrix()
K = assembler.af[2]['bilinear'].get_matrix()
system = LS(phi)
system.add_dirichlet_constraint('left')
system.add_dirichlet_constraint('right')
n_dofs = dQ.n_dofs(subforest_flag=mflag)
y = np.empty((n_dofs,n_samples))
for n in range(n_samples):
system.set_matrix(A[n])
system.set_rhs(b.copy())
system.solve_system()
y[:,n] = system.get_solution(as_function=False)[:,0]
y_fn = Nodal(dofhandler=dQ,subforest_flag=mflag,data=y)
if reference:
return y_fn, M, K
else:
return y_fn
def test01_solve_1d(self):
"""
Test solving 1D systems
"""
mesh = Mesh1D(resolution=(20, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-9)
Q1 = QuadFE(1, 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
phi = Basis(dQ1, 'u')
phi_x = Basis(dQ1, 'ux')
problem = [Form(1, test=phi_x, trial=phi_x), Form(0, test=phi)]
assembler = Assembler(problem, mesh)
assembler.add_dirichlet('left', dir_fn=0)
assembler.add_dirichlet('right', dir_fn=1)
assembler.assemble()
# Get matrix dirichlet correction and right hand side
A = assembler.get_matrix().toarray()
x0 = assembler.assembled_bnd()
b = assembler.get_vector()
ua = np.zeros((phi.n_dofs(), 1))
int_dofs = assembler.get_dofs('interior')
ua[int_dofs, 0] = np.linalg.solve(A, b - x0)
dir_bc = assembler.get_dirichlet()
dir_vals = np.array([dir_bc[dof] for dof in dir_bc])
dir_dofs = [dof for dof in dir_bc]
ua[dir_dofs] = dir_vals
ue_fn = Nodal(f=lambda x: x[:, 0], basis=phi)
ue = ue_fn.data()
self.assertTrue(np.allclose(ue, ua))
self.assertTrue(np.allclose(x0 + A.dot(ua[int_dofs, 0]), b))
def test_constructor(self):
#
# Errors
#
# Nothing specified
self.assertRaises(Exception, Nodal)
# Nominal case
mesh = QuadMesh()
element = QuadFE(2, 'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler)
data = np.arange(0, 4)
f = Nodal(data=data, basis=basis, mesh=mesh, element=element)
self.assertEqual(f.dim(), 2)
self.assertTrue(np.allclose(f.data().ravel(), data))
# Now change the data -> Error
false_data = np.arange(0, 6)
self.assertRaises(
Exception, Nodal, **{
'data': false_data,
'mesh': mesh,
'element': element
})
# Now omit mesh or element
kwargs = {'data': data, 'mesh': mesh}
self.assertRaises(Exception, Nodal, **kwargs)
kwargs = {'data': data, 'element': element}
self.assertRaises(Exception, Nodal, **kwargs)
def test_constructor(self):
"""
GMRF
"""
mesh = Mesh1D(resolution=(1000, ))
element = QuadFE(1, 'Q1')
dofhandler = DofHandler(mesh, element)
cov_kernel = CovKernel(name='matern', dim=2, \
parameters= {'sgm': 1, 'nu': 2,
'l': 0.1, 'M': None})
print('assembling covariance')
covariance = Covariance(cov_kernel, dofhandler)
print('defining gmrf')
X = GMRF(covariance=covariance)
print('sampling')
Xh = Nodal(data=X.chol_sample(), dofhandler=dofhandler)
print('plotting')
plot = Plot()
plot.line(Xh)
'''
def reference_qoi(f, tht, basis, region, n=1000000, verbose=True):
"""
Parameters
----------
f : lambda function,
Function of θ to be integrated.
tht : GaussianField,
Random field θ defined on mesh in terms of its mean and covariance
basis : Basis,
Basis function defining the nodal interpolant. It incorporates the
mesh, the dofhandler, and the derivative.
region : meshflag,
Flag indicating the region of integration
n : int, default=1000000
Sample size
Returns
-------
Q_ref : double,
Reference quantity of interest
err : double,
Expected RMSE given by var(Q)/n.
"""
#
# Assemble integral
#
batch_size = 100000
n_batches = n // batch_size + (0 if (n % batch_size) == 0 else 1)
if verbose:
print('Computing Reference Quantity of Interest')
print('========================================')
print('Sample size: ', n)
print('Batch size: ', batch_size)
print('Number of batches: ', n_batches)
Q_smpl = np.empty(n)
for k in range(n_batches):
# Determine sample sizes for each batch
if k < n_batches - 1:
n_sample = batch_size
else:
# Last sample may be smaller than batch_size
n_sample = n - k * batch_size
if verbose:
print(' - Batch Number ', k)
print(' - Sample Size: ', n_sample)
print(' - Sampling random field')
# Sample from field
tht_smpl = tht.sample(n_sample)
# Define kernel
tht_n = Nodal(data=tht_smpl, basis=basis)
kf = Kernel(tht_n, F=f)
# Define forms
if k == 0:
problems = [[Form(kernel=kf, flag=region)], [Form(flag=region)]]
else:
problems = [Form(kernel=kf, flag=region)]
if verbose:
print(' - Assembling.')
# Compute the integral
assembler = Assembler(problems, basis.mesh())
assembler.assemble()
#
# Compute statistic
#
# Get samples
if k == 0:
dx = assembler.get_scalar(i_problem=1)
if verbose:
print(' - Updating samples \n')
batch_sample = assembler.get_scalar(i_problem=0, i_sample=None)
Q_smpl[k * batch_size:k * batch_size + n_sample] = batch_sample / dx
# Compute mean and MSE
Q_ref = np.mean(Q_smpl)
err = np.var(Q_smpl) / n
# Return reference
return Q_ref, err
z, w = hermite_rule(k, depth)
n_sg = len(w)
# Generate truncated field at the sparse grid points
eta_trunc_sg = V[:, :k].dot(np.diag(np.sqrt(D[:k])).dot(z.T))
# Generate a Monte Carlo sample on top of sparse grid
n_mc = 20
zz = np.random.randn(n_dofs - k, n_mc)
eta_tail_mc = V[:, k:].dot(np.diag(np.sqrt(D[k:]))).dot(zz)
# -----------------------------------------------------------------------------
# Sample and Integrate
# -----------------------------------------------------------------------------
# Samples of random field
theta_trunc = Nodal(data=eta_trunc_sg[:, [50]] + eta_tail_mc, basis=phi_1)
# Assembler
k = Kernel(theta_trunc, F=lambda tht: tht**2)
problem = Form(flag='integration', kernel=k)
assembler = Assembler(problem, mesh)
assembler.assemble()
v = assembler.get_scalar(i_sample=3)
#plot = Plot(quickview=False)
#fig, ax = plt.subplots()
#plot.mesh(mesh,regions=[('region','cell')])
"""
ax = plot.line(theta_trunc, axis=ax, i_sample=np.arange(n_mc),
plot_kwargs={'linewidth':0.2,
def plot_heuristics(f, tht, basis, region, condition):
"""
Parameters
----------
f : lambda,
Function of θ to be integrated.
n : int,
Sample size for Monte Carlo sample
"""
tht.update_support()
#
# Plot samples of the random field
#
n = 10000
tht_sample = tht.sample(n)
tht_fn = Nodal(data=tht_sample, basis=basis)
#
# Compute the quantity of interest
#
# Define the kernel
kf = Kernel(tht_fn, F=f)
# Assemble over the mesh
problems = [[Form(kernel=kf, flag=region)], [Form(flag=region)]]
assembler = Assembler(problems, basis.mesh())
assembler.assemble()
# Extract sample
dx = assembler.get_scalar(i_problem=1)
q_sample = assembler.get_scalar(i_sample=None) / dx
#
# Compute correlation coefficients of q with spatial data
#
plot = Plot(quickview=False)
fig, ax = plt.subplots()
plt_args = {'linewidth': 0.5, 'color': 'k'}
ax = plot.line(tht_fn,
axis=ax,
i_sample=list(range(100)),
plot_kwargs=plt_args)
fig.savefig('ex01_sample_paths.eps')
fig, ax = plt.subplots()
ftht = Nodal(data=f(tht_sample), basis=basis)
ax = plot.line(ftht,
axis=ax,
i_sample=list(range(100)),
plot_kwargs=plt_args)
fig.savefig('ex01_integrand.eps')
fig, ax = plt.subplots(1, 1)
plt.hist(q_sample, bins=50, density=True)
ax.set_title(r'Histogram $Q(\theta)$')
fig.savefig('ex01_histogram.eps')
plt.close('all')
dh = basis.dofhandler()
n_dofs = dh.n_dofs()
# Extract the region on which we condition
cnd_dofs = dh.get_region_dofs(entity_flag=condition)
I = np.eye(n_dofs)
I = I[cnd_dofs, :]
# Measured tht
tht_msr = tht_sample[cnd_dofs, 0][:, None]
n_cnd = 30
cnd_tht = tht.condition(I, tht_msr, n_samples=n_cnd)
#cnd_tht_data = np.array([tht_sample[:,0] for dummy in range(n_cnd)])
#cnd_tht_data[cnd_dofs,:] = cnd_tht
cnd_tht_fn = Nodal(data=f(cnd_tht), basis=basis)
fig, ax = plt.subplots()
ax = plot.line(cnd_tht_fn,
axis=ax,
i_sample=np.arange(n_cnd),
plot_kwargs=plt_args)
fig.tight_layout()
plt.show()
def test08_1d_sampled_rhs(self):
#
# Mesh
#
mesh = Mesh1D(resolution=(1, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9, on_boundary=True)
mesh.mark_region('right',
lambda x: np.abs(1 - x) < 1e-9,
on_boundary=True)
#
# Elements
#
Q3 = QuadFE(1, 'Q3')
dofhandler = DofHandler(mesh, Q3)
dofhandler.distribute_dofs()
#
# Basis
#
v = Basis(dofhandler, 'u')
vx = Basis(dofhandler, 'ux')
#
# Define sampled right hand side and exact solution
#
xv = dofhandler.get_dof_vertices()
n_points = dofhandler.n_dofs()
n_samples = 6
a = np.arange(n_samples)
f = lambda x, a: a * x
u = lambda x, a: a / 6 * (x - x**3) + x
fdata = np.zeros((n_points, n_samples))
udata = np.zeros((n_points, n_samples))
for i in range(n_samples):
fdata[:, i] = f(xv, a[i]).ravel()
udata[:, i] = u(xv, a[i]).ravel()
# Define sampled function
fn = Nodal(data=fdata, basis=v)
ue = Nodal(data=udata, basis=v)
#
# Forms
#
one = Constant(1)
a = Form(Kernel(one), test=vx, trial=vx)
L = Form(Kernel(fn), test=v)
problem = [a, L]
#
# Assembler
#
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector()
#
# Linear System
#
system = LinearSystem(v, A=A, b=b)
# Set constraints
system.add_dirichlet_constraint('left', 0)
system.add_dirichlet_constraint('right', 1)
#system.set_constraint_relation()
#system.incorporate_constraints()
# Solve and resolve constraints
system.solve_system()
#system.resolve_constraints()
# Extract finite element solution
ua = system.get_solution(as_function=True)
# Check that the solution is close
print(ue.data()[:, [0]])
print(ua.data())
self.assertTrue(np.allclose(ue.data()[:, [0]], ua.data()))
def test09_1d_inverse(self):
"""
Compute the inverse of a matrix and apply it to a vector/matrix.
"""
#
# Mesh
#
mesh = Mesh1D(resolution=(1, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9, on_boundary=True)
mesh.mark_region('right',
lambda x: np.abs(1 - x) < 1e-9,
on_boundary=True)
#
# Elements
#
Q3 = QuadFE(1, 'Q3')
dofhandler = DofHandler(mesh, Q3)
dofhandler.distribute_dofs()
#
# Basis
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
#
# Define sampled right hand side and exact solution
#
xv = dofhandler.get_dof_vertices()
n_points = dofhandler.n_dofs()
n_samples = 6
a = np.arange(n_samples)
ffn = lambda x, a: a * x
ufn = lambda x, a: a / 6 * (x - x**3) + x
fdata = np.zeros((n_points, n_samples))
udata = np.zeros((n_points, n_samples))
for i in range(n_samples):
fdata[:, i] = ffn(xv, a[i]).ravel()
udata[:, i] = ufn(xv, a[i]).ravel()
# Define sampled function
fn = Nodal(data=fdata, basis=u)
ue = Nodal(data=udata, basis=u)
#
# Forms
#
one = Constant(1)
a = Form(Kernel(one), test=ux, trial=ux)
L = Form(Kernel(fn), test=u)
problem = [[a], [L]]
#
# Assembler
#
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector(i_problem=1)
#
# Linear System
#
system = LinearSystem(u, A=A)
# Set constraints
system.add_dirichlet_constraint('left', 0)
system.add_dirichlet_constraint('right', 1)
system.solve_system(b)
# Extract finite element solution
ua = system.get_solution(as_function=True)
system2 = LinearSystem(u, A=A, b=b)
# Set constraints
system2.add_dirichlet_constraint('left', 0)
system2.add_dirichlet_constraint('right', 1)
system2.solve_system()
u2 = system2.get_solution(as_function=True)
# Check that the solution is close
self.assertTrue(np.allclose(ue.data()[:, 0], ua.data()[:, 0]))
self.assertTrue(np.allclose(ue.data()[:, [0]], u2.data()))
def test01_1d_dirichlet_linear(self):
"""
Solve one dimensional boundary value problem with dirichlet
conditions on left and right
"""
#
# Define mesh
#
mesh = Mesh1D(resolution=(10, ))
for etype in ['Q1', 'Q2', 'Q3']:
element = QuadFE(1, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
phi = Basis(dofhandler)
#
# Exact solution
#
ue = Nodal(f=lambda x: x, basis=phi)
#
# Define Basis functions
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
#
# Define bilinear form
#
one = Constant(1)
zero = Constant(0)
a = Form(kernel=Kernel(one), trial=ux, test=ux)
L = Form(kernel=Kernel(zero), test=u)
problem = [a, L]
#
# Assemble
#
assembler = Assembler(problem, mesh)
assembler.assemble()
#
# Form linear system
#
A = assembler.get_matrix()
b = assembler.get_vector()
system = LinearSystem(u, A=A, b=b)
#
# Dirichlet conditions
#
# Boundary functions
bm_left = lambda x: np.abs(x) < 1e-9
bm_rght = lambda x: np.abs(x - 1) < 1e-9
# Mark boundary regions
mesh.mark_region('left', bm_left, on_boundary=True)
mesh.mark_region('right', bm_rght, on_boundary=True)
# Add Dirichlet constraints
system.add_dirichlet_constraint('left', ue)
system.add_dirichlet_constraint('right', ue)
#
# Solve system
#
#system.solve_system()
system.solve_system()
#
# Get solution
#
#ua = system.get_solution(as_function=True)
uaa = system.get_solution(as_function=True)
#uaa = uaa.data().ravel()
# Compare with exact solution
#self.assertTrue(np.allclose(ua.data(), ue.data()))
self.assertTrue(np.allclose(uaa.data(), ue.data()))
def test05_2d_dirichlet(self):
"""
Two dimensional Dirichlet problem with hanging nodes
"""
#
# Define mesh
#
mesh = QuadMesh(resolution=(1, 2))
mesh.cells.get_child(1).mark(1)
mesh.cells.refine(refinement_flag=1)
mesh.cells.refine()
#
# Mark left and right boundaries
#
bm_left = lambda x, dummy: np.abs(x) < 1e-9
bm_right = lambda x, dummy: np.abs(1 - x) < 1e-9
mesh.mark_region('left', bm_left, entity_type='half_edge')
mesh.mark_region('right', bm_right, entity_type='half_edge')
for etype in ['Q1', 'Q2', 'Q3']:
#
# Element
#
element = QuadFE(2, etype)
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
#
# Basis
#
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
uy = Basis(dofhandler, 'uy')
#
# Construct forms
#
ue = Nodal(f=lambda x: x[:, 0], basis=u)
ax = Form(kernel=Kernel(Constant(1)), trial=ux, test=ux)
ay = Form(kernel=Kernel(Constant(1)), trial=uy, test=uy)
L = Form(kernel=Kernel(Constant(0)), test=u)
problem = [ax, ay, L]
#
# Assemble
#
assembler = Assembler(problem, mesh)
assembler.assemble()
#
# Get system matrices
#
A = assembler.get_matrix()
b = assembler.get_vector()
#
# Linear System
#
system = LinearSystem(u, A=A, b=b)
#
# Constraints
#
# Add dirichlet conditions
system.add_dirichlet_constraint('left', ue)
system.add_dirichlet_constraint('right', ue)
#
# Solve
#
system.solve_system()
#system.resolve_constraints()
#
# Check solution
#
ua = system.get_solution(as_function=True)
self.assertTrue(np.allclose(ua.data(), ue.data()))
def test04_1d_periodic(self):
#
# Dirichlet Problem on a Periodic Mesh
#
# Define mesh, element
mesh = Mesh1D(resolution=(100, ), periodic=True)
element = QuadFE(1, 'Q3')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
u = Basis(dofhandler, 'u')
ux = Basis(dofhandler, 'ux')
# Exact solution
ue = Nodal(f=lambda x: np.sin(2 * np.pi * x), basis=u)
#
# Mark dirichlet regions
#
bnd_left = lambda x: np.abs(x) < 1e-9
mesh.mark_region('left', bnd_left, entity_type='vertex')
#
# Set up forms
#
# Bilinear form
a = Form(kernel=Kernel(Constant(1)), trial=ux, test=ux)
# Linear form
f = Explicit(lambda x: 4 * np.pi**2 * np.sin(2 * np.pi * x), dim=1)
L = Form(kernel=Kernel(f), test=u)
#
# Assemble
#
problem = [a, L]
assembler = Assembler(problem, mesh)
assembler.assemble()
A = assembler.get_matrix()
b = assembler.get_vector()
#
# Linear System
#
system = LinearSystem(u, A=A, b=b)
# Add dirichlet constraint
system.add_dirichlet_constraint('left', 0, on_boundary=False)
# Assemble constraints
#system.set_constraint_relation()
#system.incorporate_constraints()
system.solve_system()
#system.resolve_constraints()
# Compare with interpolant of exact solution
ua = system.get_solution(as_function=True)
#plot = Plot(2)
#plot.line(ua)
#plot.line(ue)
self.assertTrue(np.allclose(ua.data(), ue.data()))
# Assembler system
#
assembler = Assembler([problem], mesh)
assembler.add_dirichlet(None, ue)
assembler.assemble()
#
# Get solution
#
ua = assembler.solve()
#
# Compute the error
#
e_vec = ua[:, None] - ue.interpolant(dofhandler).data()
efn = Nodal(data=e_vec, basis=u, dim=2)
#
# Record error
#
errors[resolution][eps][etype] = max(np.abs(e_vec))
headers = ('Resolution', 'Q1:1', 'Q1:1e-3', 'Q1:1e-6', 'Q2:1', 'Q2:1e-3',
'Q2:1e-6')
print('{:<12} {:<10} {:<10} {:<10} {:<10} {:<10} {:<10}'.format(*headers))
for resolution in errors.keys():
er = errors[resolution]
row = [resolution[0],er[1]['Q1'][0], er[1e-3]['Q1'][0], er[1e-6]['Q1'][0], \
er[1]['Q2'][0], er[1e-3]['Q2'][0], er[1e-6]['Q2'][0]]
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
def test_eval_projection(self):
"""
Test validity of the local projection-based kernel.
Choose
u, v in V
k(x,y) in VxV (symmetric/non-symm)
cell01, cell02
Compare
v^T*Kloc*u ?= Icell01 Icell02 k(x,y)u(y)dy dx
"""
#
# 1D
#
# Mesh
mesh = Mesh1D(resolution=(3, ))
# Finite element space
etype = 'Q1'
element = QuadFE(1, etype)
# Dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
phi = Basis(dofhandler, 'u')
# Symmetric kernel function
kfns = {
'symmetric': lambda x, y: x * y,
'non_symmetric': lambda x, y: x - y
}
vals = {
'symmetric':
(1 / 2 * 1 / 9 - 1 / 3 * 1 / 27) * (1 / 3 - 1 / 3 * 8 / 27),
'non_symmetric': (1 / 18 - 1 / 3 * 1 / 27) * (1 / 2 - 2 / 9) +
(1 / 18 - 1 / 3) * (1 / 3 - 1 / 3 * 8 / 27)
}
for ktype in ['symmetric', 'non_symmetric']:
# Get kernel function
kfn = kfns[ktype]
# Define integral kernel
kernel = Kernel(Explicit(kfn, dim=1, n_variables=2))
# Define Bilinear Form
form = IPForm(kernel, trial=phi, test=phi)
#
# Compute inputs required for evaluating form_loc
#
# Assembler
assembler = Assembler(form, mesh)
# Cells
ci = mesh.cells.get_child(0)
cj = mesh.cells.get_child(2)
# Shape function info on cells
ci_sinfo = assembler.shape_info(ci)
cj_sinfo = assembler.shape_info(cj)
# Gauss nodes and weights on cell
xi_g, wi_g, phii, dofsi = assembler.shape_eval(ci)
xj_g, wj_g, phij, dofsj = assembler.shape_eval(cj)
#
# Evaluate form
#
form_loc = form.eval((ci,cj), (xi_g,xj_g), \
(wi_g,wj_g), (phii,phij), (dofsi,dofsj))
#
# Define functions
#
u = Nodal(f=lambda x: x, basis=phi)
v = Nodal(f=lambda x: 1 - x, basis=phi)
#
# Get local node values
#
# Degrees of freedom
ci_dofs = phi.dofs(ci)
cj_dofs = phi.dofs(cj)
uj = u.data()[np.array(cj_dofs)]
vi = v.data()[np.array(ci_dofs)]
if ktype == 'symmetric':
# Local form by hand
c10 = 1 / 54
c11 = 1 / 27
c20 = 3 / 2 - 1 - 3 / 2 * 4 / 9 + 8 / 27
c21 = 4 / 27
fl = np.array([[c10 * c20, c10 * c21], [c11 * c20, c11 * c21]])
# Compare computed and explicit local forms
self.assertTrue(np.allclose(fl, form_loc))
# Evaluate Ici Icj k(x,y) y dy (1-x)dx
fa = np.dot(vi.T, form_loc.dot(uj))
fe = vals[ktype]
self.assertAlmostEqual(fa, fe)
def test_eval_interpolation(self):
#
# 1D
#
# Mesh
mesh = Mesh1D(resolution=(3, ))
# Finite element space
etype = 'Q1'
element = QuadFE(1, etype)
# Dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
# Basis functions
phi = Basis(dofhandler, 'u')
# Symmetric kernel function
kfns = {
'symmetric': lambda x, y: x * y,
'non_symmetric': lambda x, y: x - y
}
vals = {
'symmetric':
np.array([0, 1 / 9 * (1 - 8 / 27)]),
'non_symmetric':
np.array([1 / 3 * (-1 + 8 / 27), 1 / 6 * 5 / 9 - 1 / 3 * 19 / 27])
}
for ktype in ['symmetric', 'non_symmetric']:
# Get kernel function
kfn = kfns[ktype]
# Define integral kernel
kernel = Kernel(Explicit(kfn, dim=1, n_variables=2))
# Define Bilinear Form
form = IIForm(kernel, trial=phi, test=phi)
#
# Compute inputs required for evaluating form_loc
#
# Assembler
assembler = Assembler(form, mesh)
# Cells
cj = mesh.cells.get_child(2)
ci = mesh.cells.get_child(0)
# Gauss nodes and weights on cell
xj_g, wj_g, phij, dofsj = assembler.shape_eval(cj)
#
# Evaluate form
#
form_loc = form.eval(cj, xj_g, wj_g, phij, dofsj)
#
# Define functions
#
u = Nodal(lambda x: x, basis=phi)
#
# Get local node values
#
# Degrees of freedom
cj_dofs = phi.dofs(cj)
ci_dofs = phi.dofs(ci)
uj = u.data()[np.array(cj_dofs)]
# Evaluate Ici Icj k(x,y) y dy (1-x)dx
fa = form_loc[ci_dofs].dot(uj)
fe = vals[ktype][:, None]
self.assertTrue(np.allclose(fa, fe))
state = LS(phi)
state.add_dirichlet_constraint('left', 1)
state.add_dirichlet_constraint('right', 0)
state.set_constraint_relation()
adjoint = LS(phi)
adjoint.add_dirichlet_constraint('left', 0)
adjoint.add_dirichlet_constraint('right', 0)
adjoint.set_constraint_relation()
# =============================================================================
# System Parameters
# =============================================================================
# Target
y_target = Nodal(f=lambda x: 3 - 4 * (x[:, 0] - 1)**2, dim=1, dofhandler=dh)
y_data = y_target.data()
# Regularization parameter
#gamma = 0.00001
gamma = 1e-5
# Inital guess
u = np.zeros((m, 1))
#
# Diffusion Parameter
#
# Initialize diffusion
q = Nodal(data=np.empty((m, 1)), dofhandler=dh)
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),