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 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
def test_ft():
plot = Plot()
vb = Verbose()
# =============================================================================
# Parameters
# =============================================================================
#
# Flow
#
# permeability field
phi = Constant(1) # porosity
D = Constant(0.0252) # dispersivity
K = Constant(1) # permeability
# =============================================================================
# Mesh and Elements
# =============================================================================
# Mesh
mesh = QuadMesh(resolution=(30, 30))
# Mark left and right regions
mesh.mark_region('left',
lambda x, y: np.abs(x) < 1e-9,
entity_type='half_edge')
mesh.mark_region('right',
lambda x, y: np.abs(x - 1) < 1e-9,
entity_type='half_edge')
# Elements
p_element = QuadFE(2, 'Q1') # element for pressure
c_element = QuadFE(2, 'Q1') # element for concentration
# Dofhandlers
p_dofhandler = DofHandler(mesh, p_element)
c_dofhandler = DofHandler(mesh, c_element)
p_dofhandler.distribute_dofs()
c_dofhandler.distribute_dofs()
# Basis functions
p_ux = Basis(p_dofhandler, 'ux')
p_uy = Basis(p_dofhandler, 'uy')
p_u = Basis(p_dofhandler, 'u')
p_inflow = lambda x, y: np.ones(shape=x.shape)
p_outflow = lambda x, y: np.zeros(shape=x.shape)
c_inflow = lambda x, y: np.zeros(shape=x.shape)
# =============================================================================
# Solve the steady state flow equations
# =============================================================================
vb.comment('Solving flow equations')
# Define problem
flow_problem = [
Form(1, test=p_ux, trial=p_ux),
Form(1, test=p_uy, trial=p_uy),
Form(0, test=p_u)
]
# Assemble
vb.tic('assembly')
assembler = Assembler(flow_problem)
assembler.add_dirichlet('left', 1)
assembler.add_dirichlet('right', 0)
assembler.assemble()
vb.toc()
# Solve linear system
vb.tic('solve')
A = assembler.get_matrix().tocsr()
b = assembler.get_vector()
x0 = assembler.assembled_bnd()
# Interior nodes
pa = np.zeros((p_u.n_dofs(), 1))
int_dofs = assembler.get_dofs('interior')
pa[int_dofs, 0] = spla.spsolve(A, b - x0)
# Resolve Dirichlet conditions
dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False)
pa[dir_dofs] = dir_vals
vb.toc()
# Pressure function
pfn = Nodal(data=pa, basis=p_u)
px = pfn.differentiate((1, 0))
py = pfn.differentiate((1, 1))
#plot.contour(px)
#plt.show()
# =============================================================================
# Transport Equations
# =============================================================================
# Specify initial condition
c0 = Constant(1)
dt = 1e-1
T = 6
N = int(np.ceil(T / dt))
c = Basis(c_dofhandler, 'c')
cx = Basis(c_dofhandler, 'cx')
cy = Basis(c_dofhandler, 'cy')
print('assembling transport equations')
k_phi = Kernel(f=phi)
k_advx = Kernel(f=[K, px], F=lambda K, px: -K * px)
k_advy = Kernel(f=[K, py], F=lambda K, py: -K * py)
tht = 1
m = [Form(kernel=k_phi, test=c, trial=c)]
s = [
Form(kernel=k_advx, test=c, trial=cx),
Form(kernel=k_advy, test=c, trial=cy),
Form(kernel=Kernel(D), test=cx, trial=cx),
Form(kernel=Kernel(D), test=cy, trial=cy)
]
problems = [m, s]
assembler = Assembler(problems)
assembler.add_dirichlet('left', 0, i_problem=0)
assembler.add_dirichlet('left', 0, i_problem=1)
assembler.assemble()
x0 = assembler.assembled_bnd()
# Interior nodes
int_dofs = assembler.get_dofs('interior')
# Dirichlet conditions
dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False)
# System matrices
M = assembler.get_matrix(i_problem=0)
S = assembler.get_matrix(i_problem=1)
# Initialize c0 and cp
c0 = np.ones((c.n_dofs(), 1))
cp = np.zeros((c.n_dofs(), 1))
c_fn = Nodal(data=c0, basis=c)
#
# Compute solution
#
print('time stepping')
for i in range(N):
# Build system
A = M + tht * dt * S
b = M.dot(c0[int_dofs]) - (1 - tht) * dt * S.dot(c0[int_dofs])
# Solve linear system
cp[int_dofs, 0] = spla.spsolve(A, b)
# Add Dirichlet conditions
cp[dir_dofs] = dir_vals
# Record current iterate
c_fn.add_samples(data=cp)
# Update c0
c0 = cp.copy()
#plot.contour(c_fn, n_sample=i)
#
# Quantity of interest
#
def F(c, px, py, entity=None):
"""
Compute c(x,y,t)*(grad p * n)
"""
n = entity.unit_normal()
return c * (px * n[0] + py * n[1])
px.set_subsample(i=np.arange(41))
py.set_subsample(i=np.arange(41))
#kernel = Kernel(f=[c_fn,px,py], F=F)
kernel = Kernel(c_fn)
#print(kernel.n_subsample())
form = Form(kernel, flag='right', dmu='ds')
assembler = Assembler(form, mesh=mesh)
assembler.assemble()
QQ = assembler.assembled_forms()[0].aggregate_data()['array']
Q = np.array([assembler.get_scalar(i_sample=i) for i in np.arange(N + 1)])
t = np.linspace(0, T, N + 1)
plt.plot(t, Q)
plt.show()
print(Q)
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,
'color':'k'})
"""
#plt.show()
def test02_sensitivity():
"""
Check that the sensitivity calculation works. Compare
J(q+eps*dq) - J(q) ~= eps*dJ^T dq
"""
#
# 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')
dQ = DofHandler(mesh, Q)
dQ.distribute_dofs()
nx = dQ.n_dofs()
x = dQ.get_dof_vertices()
#
# Basis
#
phi = Basis(dQ, 'v')
phi_x = Basis(dQ, 'vx')
#
# Parameters
#
# Reference q
q_ref = np.zeros(nx)
# Perturbation
dq = np.ones(nx)
# Perturbed q
n_eps = 10 # Number of refinements
epsilons = [10**(-l) for l in range(n_eps)]
q_per = np.empty((nx, n_eps))
for i in range(n_eps):
q_per[:, i] = q_ref + epsilons[i] * dq
# Define finite element function
exp_qref = Nodal(data=np.exp(q_ref), basis=phi)
exp_qper = Nodal(data=np.exp(q_per), basis=phi)
#
# PDEs
#
# 1. State Equation
state_eqn = [Form(exp_qref, test=phi_x, trial=phi_x), Form(1, test=phi)]
state_dbc = {'left': 0, 'right': 1}
# 2. Perturbed Equation
perturbed_eqn = [
Form(exp_qper, test=phi_x, trial=phi_x),
Form(1, test=phi)
]
perturbed_dbc = {'left': 0, 'right': 1}
# 3. Adjoint Equation
adjoint_eqn = [Form(exp_qref, test=phi_x, trial=phi_x), Form(0, test=phi)]
adjoint_dbc = {'left': 0, 'right': -1}
# Combine
eqns = [state_eqn, perturbed_eqn, adjoint_eqn]
bcs = [state_dbc, perturbed_dbc, adjoint_dbc]
#
# Assembly
#
assembler = Assembler(eqns)
# Boundary conditions
for i, bc in zip(range(3), bcs):
for loc, val in bc.items():
assembler.add_dirichlet(loc, val, i_problem=i)
# Assemble
assembler.assemble()
#
# Solve
#
# Solve state
ur = assembler.solve(i_problem=0)
u_ref = Nodal(data=ur, basis=phi)
ux_ref = Nodal(data=ur, basis=phi_x)
# Solve perturbed state
u_per = Nodal(basis=phi)
ue_per = Nodal(basis=phi)
for i in range(n_eps):
# FEM solution
up = assembler.solve(i_problem=1, i_matrix=i)
u_per.add_samples(up)
# Exact perturbed solution
eps = epsilons[i]
ue_per.add_samples(0.5 * np.exp(-eps) * (x - x**2) + x)
ux_per = Nodal(data=u_per.data(), basis=phi_x)
# Solve adjoint equation
v = assembler.solve(i_problem=2)
v_adj = Nodal(data=v, basis=phi)
vx_adj = Nodal(data=v, basis=phi_x)
#
# Check against exact solution
#
ue = -0.5 * x**2 + 1.5 * x
ve = -x
assert np.allclose(ue, u_ref.data())
assert np.allclose(ve, v_adj.data())
assert np.allclose(ue_per.data(), u_per.data())
#
# Quantities of Interest
#
# Reference
k_ref = Kernel(f=[exp_qref, ux_ref], F=lambda eq, ux: eq * ux)
ref_qoi = [Form(k_ref, dmu='dv', flag='right')]
# Perturbed
k_per = Kernel(f=[exp_qper, ux_per], F=lambda eq, ux: eq * ux)
per_qoi = [Form(k_per, dmu='dv', flag='right')]
# Adjoint
k_adj = Kernel(f=[exp_qref, ux_ref, vx_adj],
F=lambda eq, ux, vx: -eq * ux * vx)
adj_qoi = [Form(k_adj, test=phi)]
qois = [ref_qoi, per_qoi, adj_qoi]
# Assemble
assembler = Assembler(qois)
assembler.assemble()
# Evaluate
J_ref = assembler.get_scalar(0)
J_per = []
for i in range(n_eps):
J_per.append(assembler.get_scalar(1, i))
# Finite difference approximation
dJ = []
for eps, J_p in zip(epsilons, J_per):
dJ.append((J_p - J_ref) / eps)
# Adjoint differential
dJ_adj = assembler.get_vector(2).dot(dq)
#
# Check against exact qois
#
# Check reference sol
Je_ref = 0.5
assert np.allclose(Je_ref, J_ref)
# Check perturbed cost
Je_per = -0.5 + np.exp(np.array(epsilons))
assert np.allclose(Je_per, J_per)
# Check derivative by the adjoint equation
dJdq = 1
assert np.allclose(dJ_adj, dJdq)
# Solve the steady state flow equations
# =============================================================================
# Define problem
flow_problem = [
Form(fNodal),
Form(1, test=p_uy, trial=p_uy),
Form(1, test=p_u)
]
# Assembler
tic = time.time()
assembler = Assembler(flow_problem, mesh)
toc = time.time() - tic
print('initializing assembler', toc)
tic = time.time()
assembler.assemble()
toc = time.time() - tic
print('assembly time', toc)
tic = time.time()
A = assembler.get_matrix()
print('forming bilinear array', time.time() - tic)
b = assembler.get_vector()
c = assembler.get_scalar()
def dJdq_sen(q, u, dq):
"""
Compute the directional derivative dJ(q,dq) by means of the sensitivity
equation.
-(exp(q)*s')' = (exp(q)*dq*u')'
s(0) = s(1) = 0
and computing
dJ(q,dq) = -exp(q(1))*dq(1)*u'(1) - exp(q(1))*s'(1)
"""
#
# Finite Element Specification
#
# Reference state
u_dh = u.basis().dofhandler()
mesh = u_dh.mesh
phi = Basis(u_dh, 'u')
phi_x = Basis(u_dh, 'ux')
ux_fn = Nodal(data=u.data(), basis=phi_x)
# Reference diffusivitity
q_dh = q.basis().dofhandler()
psi = q.basis()
exp_q = Nodal(data=np.exp(q.data()), basis=phi)
# Define sensitivity equation
ker_sen = Kernel(f=[exp_q, dq, ux_fn], F=lambda eq, dq, ux: -eq * dq * ux)
sensitivity_eqn = [
Form(exp_q, test=phi_x, trial=phi_x),
Form(ker_sen, test=phi_x)
]
# Assembler
assembler = Assembler(sensitivity_eqn, u_dh.mesh, n_gauss=(6, 36))
# Apply Dirichlet Boundary conditions
assembler.add_dirichlet('left', 0)
assembler.add_dirichlet('right', 0)
# Assemble system
assembler.assemble()
# Solve for sensitivity
s_fn = Nodal(basis=phi)
for i in range(dq.n_samples()):
# Solve for ith sensitivity
s = assembler.solve(i_vector=i)
s_fn.add_samples(s)
# Derivative of sensitivity
sx_fn = Nodal(data=s_fn.data(), basis=phi_x)
# Sensitivity
k_sens = Kernel(f=[exp_q, dq, ux_fn, sx_fn],
F=lambda eq, dq, ux, sx: -eq * dq * ux - eq * sx)
sens_qoi = Form(k_sens, dmu='dv', flag='right')
# Assemble
assembler = Assembler(sens_qoi, mesh=mesh)
assembler.assemble()
# Differential
dJ = np.array([assembler.get_scalar(i_sample=i) \
for i in range(dq.n_samples())])
return dJ
def dJdq_adj(q, u, dq=None):
"""
Compute the directional derivative dJ(q,dq) by solving the adjoint equation
-(exp(q)v')' = 0,
v(0)=0, v(1)=1
and computing
( v, (exp(q)*dq(1)*u')' ) + exp(q(1))*dq(1)*u'(1) = -(exp(q)*dq u', v')
Inputs:
q: Nodal, single reference log diffusitivity
u: Nodal, single reference system response
dq: Nodal/None, sampled perturbation vector (or None)
Output:
dJdq: Derivative of J wrt q
If dq = Nodal, return directional derivative in direction dq
If dq = None, return gradient
"""
#
# Finite Element Specification
#
# Reference solution
u_dh = u.basis().dofhandler()
phi = Basis(u_dh, 'u')
phi_x = Basis(u_dh, 'ux')
ux = Nodal(data=u.data(), basis=phi_x)
# Reference diffusivity
q_dh = q.basis().dofhandler()
psi = Basis(q_dh, 'q')
exp_q = Nodal(data=np.exp(q.data()), basis=psi)
# Define adjoint equations
adjoint_eqn = [Form(exp_q, test=phi_x, trial=phi_x), Form(0, test=phi)]
# Assembler
assembler = Assembler(adjoint_eqn)
# Apply Dirichlet BC
assembler.add_dirichlet('left', 0)
assembler.add_dirichlet('right', 1)
# Assemble
assembler.assemble()
# Solve for adjoint
v = assembler.solve()
# Get derivative
vx = Nodal(data=v, basis=phi_x)
# Assemble
if dq is None:
#
# Compute the gradient
#
# Kernel
k_adj = Kernel(f=[exp_q, ux, vx], F=lambda eq, ux, vx: -eq * ux * vx)
# Linear form
adj_qoi = [Form(k_adj, test=psi)]
# Assemble
assembler = Assembler(adj_qoi)
assembler.assemble()
# Get gradient vector
dJdq = assembler.get_vector()
else:
#
# Compute the directional derivative
#
# Kernel
k_adj = Kernel(f=[exp_q, dq, ux, vx],
F=lambda eq, dq, ux, vx: -eq * dq * ux * vx)
# Constant form
adj_qoi = [Form(k_adj)]
# Assemble
assembler = Assembler(adj_qoi, mesh=u_dh.mesh)
assembler.assemble()
# Get directional derivatives for each direction
dJdq = np.array(
[assembler.get_scalar(i_sample=i) for i in range(dq.n_samples())])
# Return
return dJdq
def test06_linearization():
"""
Compute samples on fine grid via the linearization
"""
plot = Plot()
#
# Computational mesh
#
mesh = Mesh1D(resolution=(100, ))
mesh.mark_region('left', lambda x: np.abs(x) < 1e-10)
mesh.mark_region('right', lambda x: np.abs(x - 1) < 1e-10)
#
# Element
#
Q1 = QuadFE(mesh.dim(), 'Q1')
dQ1 = DofHandler(mesh, Q1)
dQ1.distribute_dofs()
nx = dQ1.n_dofs()
x = dQ1.get_dof_vertices()
Q3 = QuadFE(mesh.dim(), 'Q3')
dQ3 = DofHandler(mesh, Q3)
dQ3.distribute_dofs()
#
# Basis
#
phi = Basis(dQ1, 'u')
phi_x = Basis(dQ1, 'ux')
psi = Basis(dQ3, 'u')
psi_x = Basis(dQ3, 'ux')
#
# Covariance
#
cov = Covariance(dQ1, name='gaussian', parameters={'l': 0.05})
cov.compute_eig_decomp()
lmd, V = cov.get_eig_decomp()
d = len(lmd)
# Fix coarse truncation level
d0 = 10
#
# Build Sparse Grid
#
grid = TasmanianSG.TasmanianSparseGrid()
dimensions = d0
outputs = 1
depth = 2
type = 'level'
rule = 'gauss-hermite'
grid.makeGlobalGrid(dimensions, outputs, depth, type, rule)
# Sample Points
zzSG = grid.getPoints()
zSG = np.sqrt(2) * zzSG # transform to N(0,1)
# Quadrature Weights
wSG = grid.getQuadratureWeights()
wSG /= np.sqrt(np.pi)**d0 # normalize weights
# Number of grid points
n0 = grid.getNumPoints()
#
# Sample low dimensional input parameter
#
q0 = sample_q0(V, lmd, d0, zSG.T)
J0 = sample_qoi(q0, dQ1)
#
# Sample conditional expectation
#
# Pick a single coarse sample to check
i0 = np.random.randint(0, high=n0)
# Sample fine, conditional on coarse
n_samples = 1
z1 = np.random.randn(d - d0, n_samples)
q = sample_q_given_q0(q0[:, i0], V, lmd, d0, z1)
# Perturbation
log_qref = np.log(q0[:, i0])
dlog_q = np.log(q.ravel()) - log_qref
dlog_qfn = Nodal(data=dlog_q, basis=phi)
# Perturbed q
n_eps = 12 # Number of refinements
epsilons = [10**(-l) for l in range(n_eps)]
log_qper = np.empty((nx, n_eps))
for i in range(n_eps):
log_qper[:, i] = log_qref + epsilons[i] * dlog_q
"""
plt.plot(x, log_qref, label='ref')
for i in range(n_eps):
plt.plot(x, log_qper[:,i],label='%d'%(i))
"""
assert np.allclose(log_qper[:, 0], np.log(q.ravel()))
plt.legend()
plt.show()
# Define finite element function
exp_qref = Nodal(data=q0[:, i0], basis=phi)
exp_qper = Nodal(data=np.exp(log_qper), basis=phi)
#
# PDEs
#
# 1. State Equation
state_eqn = [Form(exp_qref, test=phi_x, trial=phi_x), Form(1, test=phi)]
state_dbc = {'left': 0, 'right': 1}
# 2. Perturbed Equation
perturbed_eqn = [
Form(exp_qper, test=phi_x, trial=phi_x),
Form(1, test=phi)
]
perturbed_dbc = {'left': 0, 'right': 1}
# 3. Adjoint Equation
adjoint_eqn = [Form(exp_qref, test=psi_x, trial=psi_x), Form(0, test=psi)]
adjoint_dbc = {'left': 0, 'right': -1}
# Combine
eqns = [state_eqn, perturbed_eqn, adjoint_eqn]
bcs = [state_dbc, perturbed_dbc, adjoint_dbc]
#
# Assembly
#
assembler = Assembler(eqns, n_gauss=(6, 36))
# Boundary conditions
for i, bc in zip(range(3), bcs):
for loc, val in bc.items():
assembler.add_dirichlet(loc, val, i_problem=i)
# Assemble
assembler.assemble()
#
# Solve
#
# Solve state
ur = assembler.solve(i_problem=0)
u_ref = Nodal(data=ur, basis=phi)
ux_ref = Nodal(data=ur, basis=phi_x)
# Solve perturbed state
u_per = Nodal(basis=phi)
for i in range(n_eps):
# FEM solution
up = assembler.solve(i_problem=1, i_matrix=i)
u_per.add_samples(up)
plt.plot(x, up - ur)
plt.show()
ux_per = Nodal(data=u_per.data(), basis=phi_x)
# Solve adjoint equation
v = assembler.solve(i_problem=2)
v_adj = Nodal(data=v, basis=psi)
vx_adj = Nodal(data=v, basis=psi_x)
#
# Sensitivity
#
# Sensitivity Equation
ker_sen = Kernel(f=[exp_qref, dlog_qfn, ux_ref],
F=lambda eq, dq, ux: -eq * dq * ux)
sensitivity_eqn = [
Form(exp_qref, test=phi_x, trial=phi_x),
Form(ker_sen, test=phi_x)
]
sensitivity_dbc = {'left': 0, 'right': 0}
assembler = Assembler(sensitivity_eqn, n_gauss=(6, 36))
for loc in sensitivity_dbc:
assembler.add_dirichlet(loc, sensitivity_dbc[loc])
assembler.assemble()
s = assembler.solve()
sx = Nodal(data=s, basis=phi_x)
plt.plot(x, s)
plt.show()
#
# Quantities of Interest
#
# Reference
k_ref = Kernel(f=[exp_qref, ux_ref], F=lambda eq, ux: eq * ux)
ref_qoi = [Form(k_ref, dmu='dv', flag='right')]
# Perturbed
k_per = Kernel(f=[exp_qper, ux_per], F=lambda eq, ux: eq * ux)
per_qoi = [Form(k_per, dmu='dv', flag='right')]
# Adjoint
k_adj = Kernel(f=[exp_qref, dlog_qfn, ux_ref, vx_adj],
F=lambda eq, dq, ux, vx: -eq * dq * ux * vx)
adj_qoi = [Form(k_adj)]
# Sensitivity
k_sens = Kernel(f=[exp_qref, dlog_qfn, ux_ref, sx],
F=lambda eq, dq, ux, sx: eq * dq * ux + eq * sx)
sens_qoi = Form(k_sens, dmu='dv', flag='right')
qois = [ref_qoi, per_qoi, adj_qoi, sens_qoi]
# Assemble
assembler = Assembler(qois, mesh=mesh)
assembler.assemble()
# Evaluate
J_ref = assembler.get_scalar(0)
J_per = []
for i in range(n_eps):
J_per.append(assembler.get_scalar(1, i))
# Finite difference approximation
dJ = []
for eps, J_p in zip(epsilons, J_per):
dJ.append((J_p - J_ref) / eps)
# Adjoint differential
dJ_adj = assembler.get_scalar(2)
# Sensitivity differential
dJ_sen = assembler.get_scalar(3)
print(dJ_adj)
print(dJ_sen)
print(dJ)
"""
def test_integrals_2d(self):
"""
Test Assembly of some 2D systems
"""
mesh = QuadMesh(box=[1, 2, 1, 2], resolution=(2, 2))
mesh.cells.get_leaves()[0].mark(0)
mesh.cells.refine(refinement_flag=0)
# Kernel
kernel = Kernel(Explicit(f=lambda x: x[:, 0] * x[:, 1], dim=2))
problem = Form(kernel)
assembler = Assembler(problem, mesh=mesh)
assembler.assemble()
self.assertAlmostEqual(assembler.get_scalar(), 9 / 4)
#
# Linear forms (x,x) and (x,x') over [1,2]^2 = 7/3, 3/2
#
# Elements
Q1 = QuadFE(mesh.dim(), 'Q1')
Q2 = QuadFE(mesh.dim(), 'Q2')
Q3 = QuadFE(mesh.dim(), 'Q3')
# Dofhandlers
dQ1 = DofHandler(mesh, Q1)
dQ2 = DofHandler(mesh, Q2)
dQ3 = DofHandler(mesh, Q3)
# Distribute dofs
[d.distribute_dofs() for d in [dQ1, dQ2, dQ3]]
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
# Kernel function
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
yfn = Nodal(f=lambda x: x[:, 1], basis=phi)
# Kernel
kernel = Kernel(xfn)
# Form
problem = [[Form(kernel, test=phi)], [Form(kernel, test=phi_x)]]
# Assembly
assembler = Assembler(problem, mesh)
assembler.assemble()
# Check b^Tx = (x,y)
b0 = assembler.get_vector(0)
self.assertAlmostEqual(np.sum(b0 * yfn.data()[:, 0]), 9 / 4)
b1 = assembler.get_vector(1)
self.assertAlmostEqual(np.sum(b1 * xfn.data()[:, 0]), 3 / 2)
self.assertAlmostEqual(np.sum(b1 * yfn.data()[:, 0]), 0)
#
# Bilinear forms
#
# Compute (1,x,y) = 9/4, or (xy, 1, 1) = 9/4
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
phi_y = Basis(dQ, 'uy')
# Kernel function
xyfn = Explicit(f=lambda x: x[:, 0] * x[:, 1], dim=2)
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
yfn = Nodal(f=lambda x: x[:, 1], basis=phi)
# Form
problems = [[Form(1, test=phi, trial=phi)],
[Form(Kernel(xfn), test=phi, trial=phi_x)],
[Form(Kernel(xyfn), test=phi_y, trial=phi_x)]]
# Assemble
assembler = Assembler(problems, mesh)
assembler.assemble()
x = xfn.data()[:, 0]
y = yfn.data()[:, 0]
for i_problem in range(3):
A = assembler.get_matrix(i_problem)
self.assertAlmostEqual(y.T.dot(A.dot(x)), 9 / 4)
def test_integrals_1d(self):
"""
Test system assembly
"""
#
# Constant form
#
# Mesh
mesh = Mesh1D(box=[1, 2], resolution=(1, ))
# Kernel
kernel = Kernel(Explicit(f=lambda x: x[:, 0], dim=1))
problem = Form(kernel)
assembler = Assembler(problem, mesh=mesh)
assembler.assemble()
self.assertAlmostEqual(assembler.get_scalar(), 3 / 2)
#
# Linear forms (x,x) and (x,x') over [1,2] = 7/3, 3/2
#
# Elements
Q1 = QuadFE(mesh.dim(), 'Q1')
Q2 = QuadFE(mesh.dim(), 'Q2')
Q3 = QuadFE(mesh.dim(), 'Q3')
# Dofhandlers
dQ1 = DofHandler(mesh, Q1)
dQ2 = DofHandler(mesh, Q2)
dQ3 = DofHandler(mesh, Q3)
# Distribute dofs
[d.distribute_dofs() for d in [dQ1, dQ2, dQ3]]
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
# Kernel function
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
# Kernel
kernel = Kernel(xfn)
# Form
problem = [[Form(kernel, test=phi)], [Form(kernel, test=phi_x)]]
# Assembly
assembler = Assembler(problem, mesh)
assembler.assemble()
# Check b^Tx = (x,x)
b0 = assembler.get_vector(0)
self.assertAlmostEqual(np.sum(b0 * xfn.data()[:, 0]), 7 / 3)
b1 = assembler.get_vector(1)
self.assertAlmostEqual(np.sum(b1 * xfn.data()[:, 0]), 3 / 2)
#
# Bilinear forms
#
# Compute (1,x,x) = 7/3, or (x^2, 1, 1) = 7/3
for dQ in [dQ1, dQ2, dQ3]:
# Basis
phi = Basis(dQ, 'u')
phi_x = Basis(dQ, 'ux')
# Kernel function
x2fn = Explicit(f=lambda x: x[:, 0]**2, dim=1)
xfn = Nodal(f=lambda x: x[:, 0], basis=phi)
# Form
problems = [[Form(1, test=phi, trial=phi)],
[Form(Kernel(xfn), test=phi, trial=phi_x)],
[Form(Kernel(x2fn), test=phi_x, trial=phi_x)]]
# Assemble
assembler = Assembler(problems, mesh)
assembler.assemble()
x = xfn.data()[:, 0]
for i_problem in range(3):
A = assembler.get_matrix(i_problem)
self.assertAlmostEqual(x.T.dot(A.dot(x)), 7 / 3)
# ======================================================================
# Test 1: Assemble simple bilinear form (u,v) on Mesh1D
# ======================================================================
# Mesh
mesh = Mesh1D(resolution=(1, ))
Q1 = QuadFE(mesh.dim(), 'Q1')
Q2 = QuadFE(mesh.dim(), 'Q2')
Q3 = QuadFE(mesh.dim(), 'Q3')
# Test and trial functions
dhQ1 = DofHandler(mesh, QuadFE(1, 'Q1'))
dhQ1.distribute_dofs()
u = Basis(dhQ1, 'u')
v = Basis(dhQ1, 'v')
# Form
form = Form(trial=u, test=v)
# Define system
system = Assembler(form, mesh)
# Get local information
cell = mesh.cells.get_child(0)
si = system.shape_info(cell)
# Compute local Gauss nodes
xg, wg, phi, dofs = system.shape_eval(cell)
self.assertTrue(cell in xg)
self.assertTrue(cell in wg)
# Compute local shape functions
self.assertTrue(cell in phi)
self.assertTrue(u in phi[cell])
self.assertTrue(v in phi[cell])
self.assertTrue(u in dofs[cell])
# Assemble system
system.assemble()
# Extract system bilinear form
A = system.get_matrix()
# Use bilinear form to integrate x^2 over [0,1]
f = Nodal(lambda x: x, basis=u)
fv = f.data()[:, 0]
self.assertAlmostEqual(np.sum(fv * A.dot(fv)), 1 / 3)
# ======================================================================
# Test 3: Constant form (x^2,.,.) over 1D mesh
# ======================================================================
# Mesh
mesh = Mesh1D(resolution=(10, ))
# Nodal kernel function
Q2 = QuadFE(1, 'Q2')
dhQ2 = DofHandler(mesh, Q2)
dhQ2.distribute_dofs()
phiQ2 = Basis(dhQ2)
f = Nodal(lambda x: x**2, basis=phiQ2)
kernel = Kernel(f=f)
# Form
form = Form(kernel=kernel)
# Generate and assemble the system
system = Assembler(form, mesh)
system.assemble()
# Check
self.assertAlmostEqual(system.get_scalar(), 1 / 3)
# =====================================================================
# Test 4: Periodic Mesh
# =====================================================================
#
# TODO: NO checks yet
#
mesh = Mesh1D(resolution=(2, ), periodic=True)
#
Q1 = QuadFE(1, 'Q1')
dhQ1 = DofHandler(mesh, Q1)
dhQ1.distribute_dofs()
u = Basis(dhQ1, 'u')
form = Form(trial=u, test=u)
system = Assembler(form, mesh)
system.assemble()
# =====================================================================
# Test 5: Assemble simple sampled form
# ======================================================================
mesh = Mesh1D(resolution=(3, ))
Q1 = QuadFE(1, 'Q1')
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
phi = Basis(dofhandler)
xv = dofhandler.get_dof_vertices()
n_points = dofhandler.n_dofs()
n_samples = 6
a = np.arange(n_samples)
f = lambda x, a: a * x
fdata = np.zeros((n_points, n_samples))
for i in range(n_samples):
fdata[:, i] = f(xv, a[i]).ravel()
# Define sampled function
fn = Nodal(data=fdata, basis=phi)
kernel = Kernel(fn)
#
# Integrate I[0,1] ax^2 dx by using the linear form (ax,x)
#
v = Basis(dofhandler, 'v')
form = Form(kernel=kernel, test=v)
system = Assembler(form, mesh)
system.assemble()
one = np.ones(n_points)
for i in range(n_samples):
b = system.get_vector(i_sample=i)
self.assertAlmostEqual(one.dot(b), 0.5 * a[i])
#
# Integrate I[0,1] ax^4 dx using bilinear form (ax, x^2, x)
#
Q2 = QuadFE(1, 'Q2')
dhQ2 = DofHandler(mesh, Q2)
dhQ2.distribute_dofs()
u = Basis(dhQ2, 'u')
# Define form
form = Form(kernel=kernel, test=v, trial=u)
# Define and assemble system
system = Assembler(form, mesh)
system.assemble()
# Express x^2 in terms of trial function basis
dhQ2.distribute_dofs()
xvQ2 = dhQ2.get_dof_vertices()
xv_squared = xvQ2**2
for i in range(n_samples):
#
# Iterate over samples
#
# Form sparse matrix
A = system.get_matrix(i_sample=i)
# Evaluate the integral
I = np.sum(xv * A.dot(xv_squared))
# Compare with expected result
self.assertAlmostEqual(I, 0.2 * a[i])
# =====================================================================
# Test 6: Working with submeshes
# =====================================================================
mesh = Mesh1D(resolution=(2, ))