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 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 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 test_set_data(self):
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]
}
}
parms = {1: {1: [{}, {}], 2: [{}, {}]}, 2: {1: [{}, {}], 2: [{}, {}]}}
for dim in [1, 2]:
# Get mesh and element
mesh = meshes[dim]
element = elements[dim]
# Set dofhandler
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
n_dofs = dofhandler.n_dofs()
# Set basis
basis = Basis(dofhandler)
# Determine the shapes of the data
det_shapes = {1: (n_dofs, 1), 2: (n_dofs, n_dofs, 1)}
smp_shapes = {1: (n_dofs, 2), 2: (n_dofs, n_dofs, 2)}
# Get a vertex
i = np.random.randint(n_dofs)
j = np.random.randint(n_dofs)
dofhandler.set_dof_vertices()
x = dofhandler.get_dof_vertices()
x1 = np.array([x[i, :]])
x2 = np.array([x[j, :]])
for n_variables in [1, 2]:
fn = fns[dim][n_variables]
parm = parms[dim][n_variables]
#
# Deterministic
#
f = Nodal(f=fn, basis=basis, n_variables=n_variables)
# Check shape
self.assertEqual(f.data().shape, det_shapes[n_variables])
# Check value
if n_variables == 1:
val = fn(x1)[0]
self.assertEqual(val, f.data()[i])
else:
val = fn(x1, x2)
self.assertEqual(val[0], f.data()[i, j])
#
# Sampled
#
f = Nodal(f=fn,
parameters=parm,
basis=basis,
n_variables=n_variables)
# Check shape
self.assertEqual(f.data().shape, smp_shapes[n_variables])
# Check that samples are stored in the right place
if n_variables == 1:
self.assertTrue(np.allclose(f.data()[:, 0],
f.data()[:, 1]))
elif n_variables == 2:
self.assertTrue(
np.allclose(f.data()[:, :, 0],
f.data()[:, :, 1]))
#
mesh = Mesh1D(resolution=(20,))
# Mark Dirichlet Vertices
mesh.mark_region('left', lambda x: np.abs(x)<1e-9)
mesh.mark_region('right', lambda x: np.abs(x-1)<1e-9)
#
# Element
#
Q1 = QuadFE(mesh.dim(), 'Q1')
# Dofhandler
dofhandler = DofHandler(mesh, Q1)
dofhandler.distribute_dofs()
x = dofhandler.get_dof_vertices()
n = dofhandler.n_dofs()
# -----------------------------------------------------------------------------
# Sample full dimensional parameter space
# -----------------------------------------------------------------------------
#
# Set up sparse grid on coarse parameter space
#
tasmanian_library="/home/hans-werner/bin/TASMANIAN-6.0/libtasmaniansparsegrid.so"
grid = TasmanianSG.TasmanianSparseGrid(tasmanian_library=tasmanian_library)
dimensions = 4
outputs = 1
max_depth = 8
type = 'level'
rule = 'gauss-legendre'
def test_constructor(self):
# =====================================================================
# Test 1D
# =====================================================================
#
# Kernel consists of a single explicit Function:
#
f1 = lambda x: x+2
f = Explicit(f1, dim=1)
k = Kernel(f)
x = np.linspace(0,1,100)
n_points = len(x)
# Check that it evaluates correctly.
self.assertTrue(np.allclose(f1(x), k.eval(x).ravel()))
# Check shape of kernel
self.assertEqual(k.eval(x).shape, (n_points,1))
#
# Kernel consists of a combination of two explicit functions
#
f1 = Explicit(lambda x: x+2, dim=1)
f2 = Explicit(lambda x: x**2 + 1, dim=1)
F = lambda f1, f2: f1**2 + f2
f_t = lambda x: (x+2)**2 + x**2 + 1
k = Kernel([f1,f2], F=F)
# Check evaluation
self.assertTrue(np.allclose(f_t(x), k.eval(x).ravel()))
# Check shape
self.assertEqual(k.eval(x).shape, (n_points,1))
#
# Same thing as above, but with nodal functions
#
mesh = Mesh1D(resolution=(1,))
Q1 = QuadFE(1,'Q1')
Q2 = QuadFE(1,'Q2')
dQ1 = DofHandler(mesh,Q1)
dQ2 = DofHandler(mesh,Q2)
# Distribute dofs
[dQ.distribute_dofs() for dQ in [dQ1,dQ2]]
# Basis functions
phi1 = Basis(dQ1,'u')
phi2 = Basis(dQ2,'u')
f1 = Nodal(lambda x: x+2, basis=phi1)
f2 = Nodal(lambda x: x**2 + 1, basis=phi2)
k = Kernel([f1,f2], F=F)
# Check evaluation
self.assertTrue(np.allclose(f_t(x), k.eval(x).ravel()))
#
# Replace f2 above with its derivative
#
k = Kernel([f1,f2], derivatives=['f', 'fx'], F=F)
f_t = lambda x: (x+2)**2 + 2*x
# Check derivative evaluation F = F(f1, df2_dx)
self.assertTrue(np.allclose(f_t(x), k.eval(x).ravel()))
#
# Sampling
#
one = Constant(1)
f1 = Explicit(lambda x: x**2 + 1, dim=1)
# Sampled function
a = np.linspace(0,1,11)
n_samples = len(a)
# Define Dofhandler
dh = DofHandler(mesh, Q2)
dh.distribute_dofs()
dh.set_dof_vertices()
xv = dh.get_dof_vertices()
n_dofs = dh.n_dofs()
phi = Basis(dh, 'u')
# Evaluate parameterized function at mesh dof vertices
f2_m = np.empty((n_dofs, n_samples))
for i in range(n_samples):
f2_m[:,i] = xv.ravel() + a[i]*xv.ravel()**2
f2 = Nodal(data=f2_m, basis=phi)
# Define kernel
F = lambda f1, f2, one: f1 + f2 + one
k = Kernel([f1,f2,one], F=F)
# Evaluate on a fine mesh
x = np.linspace(0,1,100)
n_points = len(x)
self.assertEqual(k.eval(x).shape, (n_points, n_samples))
for i in range(n_samples):
# Check evaluation
self.assertTrue(np.allclose(k.eval(x)[:,i], f1.eval(x)[:,i] + x + a[i]*x**2+ 1))
#
# Sample multiple constant functions
#
f1 = Constant(data=a)
f2 = Explicit(lambda x: 1 + x**2, dim=1)
f3 = Nodal(data=f2_m[:,-1], basis=phi)
F = lambda f1, f2, f3: f1 + f2 + f3
k = Kernel([f1,f2,f3], F=F)
x = np.linspace(0,1,100)
for i in range(n_samples):
self.assertTrue(np.allclose(k.eval(x)[:,i], \
a[i] + f2.eval(x)[:,i] + f3.eval(x)[:,i]))
#
# Submeshes
#
mesh = Mesh1D(resolution=(1,))
mesh_labels = Tree(regular=False)
mesh = Mesh1D(resolution=(1,))
Q1 = QuadFE(1,'Q1')
Q2 = QuadFE(1,'Q2')
dQ1 = DofHandler(mesh,Q1)
dQ2 = DofHandler(mesh,Q2)
# Distribute dofs
[dQ.distribute_dofs() for dQ in [dQ1,dQ2]]
# Basis
p1 = Basis(dQ1)
p2 = Basis(dQ2)
f1 = Nodal(lambda x: x, basis=p1)
f2 = Nodal(lambda x: -2+2*x**2, basis=p2)
one = Constant(np.array([1,2]))
F = lambda f1, f2, one: 2*f1**2 + f2 + one
I = mesh.cells.get_child(0)
kernel = Kernel([f1,f2, one], F=F)
rule1D = GaussRule(5,shape='interval')
x = I.reference_map(rule1D.nodes())
def reference_solution():
"""
Use sparse grid method to compute a benchmark solution
"""
mesh = QuadMesh(resolution=(10,10))
mesh.mark_region('boundary', lambda x,y:True,
entity_type='half_edge', on_boundary=True)
element = QuadFE(mesh.dim(), 'Q1')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
n = dofhandler.n_dofs()
phi = Basis(dofhandler, 'u')
phi_x = Basis(dofhandler, 'ux')
phi_y = Basis(dofhandler, 'uy')
yd_fn = Explicit(lambda x: np.sin(2*np.pi*x[:,0])*np.sin(2*np.pi*x[:,1]),dim=2)
g = np.ones((n,1))
#
# Random diffusion coefficient
#
# Sparse grid
tasmanian_library="/home/hans-werner/bin/TASMANIAN-6.0/libtasmaniansparsegrid.so"
grid = TasmanianSG.TasmanianSparseGrid(tasmanian_library=tasmanian_library)
dimensions = 4
outputs = 1
depth = 4
type = 'tensor'
rule = 'gauss-legendre'
grid.makeGlobalGrid(dimensions, outputs, depth, type, rule)
Y = grid.getPoints()
w = grid.getQuadratureWeights()
n_samples = grid.getNumPoints()
x = dofhandler.get_dof_vertices()
a_nodal = 1 + 0.1*(np.outer(np.cos(np.pi*x[:,1]),Y[:,0])+\
np.outer(np.cos(np.pi*x[:,0]),Y[:,1])+\
np.outer(np.sin(2*np.pi*x[:,1]),Y[:,2])+\
np.outer(np.sin(2*np.pi*x[:,0]),Y[:,3]))
a = Nodal(data=a_nodal, dofhandler=dofhandler)
yd_vec = yd_fn.eval(x)
problems = [[Form(a, test=phi_x, trial=phi_x),
Form(a, test=phi_y, trial=phi_y)],
[Form(1, test=phi, trial=phi)]]
assembler = Assembler(problems, mesh)
assembler.assemble()
A = assembler.af[0]['bilinear'].get_matrix()
M = assembler.af[1]['bilinear'].get_matrix()
state = LS(phi)
state.add_dirichlet_constraint('boundary')
adjoint = LS(phi)
adjoint.add_dirichlet_constraint('boundary')
tau = 10
k_max = 20
alpha = 0.1
u = np.zeros((n,1))
norm_dJ_iter = []
J_iter = []
u_iter = []
for k in range(k_max):
print('iteration', k)
#
# Compute average cost and gradient
#
dJ = np.zeros((n,1))
J = 0
print('sampling')
for n in range(n_samples):
print(n)
yn, pn, Jn, dJn = cost_gradient(state,adjoint,A[n],M,
g,u,yd_vec,alpha)
J += w[n]*Jn
dJ += w[n]*dJn
print('')
norm_dJ = np.sqrt(dJ.T.dot(M.dot(dJ)))
#
# Store current iterates
#
norm_dJ_iter.append(norm_dJ)
J_iter.append(J)
u_iter.append(u)
#
# Check for convergence
#
if norm_dJ<1e-8:
break
#
# Update iterate
#
u -= tau*dJ
phi_0 = Basis(dhQ1)
print(phi_0.subforest_flag())
phi_1 = Basis(dhQ1, subforest_flag='coarse')
cell = mesh.cells.get_leaves(subforest_flag='coarse')[0]
cell.info()
print(mesh.cells.is_contained_in('fine', 'coarse'))
# Evaluate phi_1 on cell
sub_dofs = []
for child in cell.get_leaves(flag='fine'):
sub_dofs.extend(dhQ1.get_cell_dofs(child))
sub_dofs = list(set(sub_dofs))
print(sub_dofs)
x = dhQ1.get_dof_vertices(dofs=sub_dofs)
x_ref = cell.reference_map(x, mapsto='reference')
print(x_ref)
data = phi_1.eval(x_ref, cell)
fig = plt.figure()
ax = plt.axes(projection='3d')
print(data.shape, x.shape)
ax.scatter3D(x[:, 0], x[:, 1], data[:, 0])
plt.show()
print(data)
phi_2 = Basis(dhQ1, subforest_flag='fine')
phi_0 = Basis(dhQ0)
flag = tuple(mtags.get_node_address())
mesh.cells.record(flag)
mtags.add_child()
new_flag = tuple(mtags.get_child(0).get_node_address())
mesh.cells.refine(subforest_flag=flag, \
new_label=new_flag)
for leaf in mesh.cells.get_leaves(subforest_flag=flag):
leaf.info()
print('==' * 20)
for leaf in mesh.cells.get_leaves(subforest_flag=new_flag):
leaf.info()
element = QuadFE(1, 'Q2')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
dofhandler.set_dof_vertices()
dofs = dofhandler.get_global_dofs(subforest_flag=flag)
print(dofs)
dofs = dofhandler.get_global_dofs(subforest_flag=new_flag)
print(dofs)
dv = dofhandler.get_dof_vertices(dofs)
print(dv)
n_production = (4, 3) # resolution
x_production = np.linspace(0.5, 1.5, n_production[0])
y_production = np.linspace(0.2, 0.8, n_production[1])
X, Y = np.meshgrid(x_production, y_production)
xy = np.array([X.ravel(), Y.ravel()]).T
cells_production = mesh.bin_points(xy)
# Mark vertices
for cell, dummy in cells_production:
cell.get_vertex(2).mark('production')
# Extract degrees of freedom
production_dofs = dh_Q1.get_region_dofs(entity_type='vertex',
entity_flag='production')
v_production = dh_Q1.get_dof_vertices(dofs=production_dofs)
# Target pressure at production wells
z_fn = Explicit(f=lambda x: 3 - 4 * (x[:, 0] - 1)**2 - 8 * (x[:, 1] - 0.5)**2,
dim=2,
mesh=mesh)
y_target = z_fn.eval(v_production)
#
# Locations of injection wells
#
n_injection = (5, 4) # resolution
x_injection = np.linspace(0.5, 1.5, n_injection[0])
y_injection = np.linspace(0.25, 0.75, n_injection[1])
X, Y = np.meshgrid(x_injection, y_injection)
xy = np.array([X.ravel(), Y.ravel()]).T
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)
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_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, ))
def test_set_hanging_nodes(self):
"""
Check that functions in the finite element space can be interpolated
by linear combinations of shape functions at supporting nodes.
TODO: Move this test to tests for system
"""
#
# Define QuadMesh with hanging node
#
mesh = QuadMesh(resolution=(1,1))
mesh.cells.refine()
mesh.cells.get_child(0).get_child(0).mark(flag=0)
mesh.cells.refine(refinement_flag=0)
c_00 = mesh.cells.get_child(0).get_child(0)
#
# Define test functions to interpolate
#
test_functions = {'Q1': lambda x,y: x + y, \
'Q2': lambda x,y: x**2 + y**2,\
'Q3': lambda x,y: x**3*y + y**2*x**2}
etypes = ['Q1', 'Q2', 'Q3']
plot = Plot()
for etype in etypes:
#print(etype)
# Define new element
element = QuadFE(2,etype)
# Distribute dofs and set vertices
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
dofhandler.set_dof_vertices()
# Determine hanging nodes
dofhandler.set_hanging_nodes()
hanging_nodes = dofhandler.get_hanging_nodes()
for dof, support in hanging_nodes.items():
# Get hanging node vertex
x_hgnd = dofhandler.get_dof_vertices(dof)
# Extract support indices and weights
js, ws = support
# Extract
dofs_glb = dofhandler.get_cell_dofs(c_00)
#print(dof, js)
# Local dof numbers for supporting nodes
dofs_loc_supp = [i for i in range(element.n_dofs()) if dofs_glb[i] in js]
#x_dofs = c_00.reference_map(element.reference_nodes())
#phi_supp = element.shape(x_hgnd, cell=c_00, local_dofs=dofs_loc_supp)
#print(phi_supp, js)
# Evaluate test function at hanging node
#f_hgnd = test_functions[etype](x_hgnd[0],x_hgnd[1])
#print('Weighted sum of support function', np.dot(phi_supp,ws))
#print(f_hgnd - np.dot(phi_supp, ws))
#phi_hgnd = element.shape(x_dofs, cell=c_00, local_dofs=dofs_loc_hgnd)
#print(phi_supp)
#print(phi_hgnd)
#plot.mesh(mesh, dofhandler=dofhandler, dofs=True)
# Evaluate
c_01 = mesh.cells.get_child(0).get_child(1)
c_022 = mesh.cells.get_child(0).get_child(2).get_child(2)
#print(dofhandler.get_global_dofs(c_022))
x_ref = element.reference_nodes()
#print(dofhandler.get_global_dofs(c_01))
#print(dofhandler.get_hanging_nodes())
x = c_01.reference_map(x_ref)
dh_y = DofHandler(mesh, Q1)
dh_y.distribute_dofs()
ny = dh_y.n_dofs()
# Basis functions
phi = Basis(dh_y, 'v')
phi_x = Basis(dh_y, 'vx')
# -----------------------------------------------------------------------------
# Observations
# -----------------------------------------------------------------------------
# Determine vertices corresponding to production wells
n_prod = 4
h = (x_max-x_min)/(n_prod+2)
x_prod = np.array([(i+1)*h for i in range(n_prod)])
v = dh_y.get_dof_vertices()
dofs_prod = []
for x in x_prod:
dofs_prod.append(np.argmin(abs(v-x)))
#
# Target pressure at production wells
#
z_fn = Explicit(f=lambda x: 3-4*(x[:,0]-1)**2, dim=1, mesh=mesh)
z_data = z_fn.eval(v[dofs_prod])
# -----------------------------------------------------------------------------
# Control
# -----------------------------------------------------------------------------
# Determine the vertices corresponding to the injection wells
n_inj = 6
h = (x_max-x_min)/(n_inj+2)
# Mark Dirichlet Vertices
mesh.mark_region('left', lambda x: np.abs(x) < 1e-9)
mesh.mark_region('right', lambda x: np.abs(x - 2) < 1e-9)
#
# Finite element spaces
#
Q1 = QuadFE(mesh.dim(), 'Q1')
# Dofhandler for state
dh = DofHandler(mesh, Q1)
dh.distribute_dofs()
m = dh.n_dofs()
dh.set_dof_vertices()
x = dh.get_dof_vertices()
# Basis functions
phi = Basis(dh, 'v')
phi_x = Basis(dh, 'vx')
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()
def test_eval(self):
#
# Out of the box covariance kernels
#
fig, ax = plt.subplots(6, 4, figsize=(5, 7))
cov_names = [
'constant', 'linear', 'gaussian', 'exponential', 'matern',
'rational'
]
anisotropies = {1: [None, 2], 2: [None, np.diag([2, 1])]}
m_count = 0
for mesh in [Mesh1D(resolution=(10, )), QuadMesh(resolution=(10, 10))]:
# Dimension
dim = mesh.dim()
#
# Construct computational mesh
#
# Piecewise constant elements
element = QuadFE(dim, 'DQ0')
# Define dofhandler -> get vertices
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
dofhandler.set_dof_vertices()
v = dofhandler.get_dof_vertices()
# Define meshgrid for 1 and 2 dimensions
n_dofs = dofhandler.n_dofs()
M1, M2 = np.mgrid[0:n_dofs, 0:n_dofs]
if dim == 1:
X = v[:, 0][M1].ravel()
Y = v[:, 0][M2].ravel()
elif dim == 2:
X = np.array([v[:, 0][M1].ravel(), v[:, 1][M1].ravel()]).T
Y = np.array([v[:, 0][M2].ravel(), v[:, 1][M2].ravel()]).T
x = convert_to_array(X, dim=dim)
y = convert_to_array(Y, dim=dim)
a_count = 0
isotropic_label = ['isotropic', 'anisotropic']
for M in anisotropies[dim]:
# Cycle through anisotropies
# Define covariance parameters
cov_pars = {
'constant': {
'sgm': 1
},
'linear': {
'sgm': 1,
'M': M
},
'gaussian': {
'sgm': 1,
'l': 0.1,
'M': M
},
'exponential': {
'l': 0.1,
'M': M
},
'matern': {
'sgm': 1,
'nu': 2,
'l': 0.5,
'M': M
},
'rational': {
'a': 3,
'M': M
}
}
c_count = 0
for cov_name in cov_names:
C = CovKernel(cov_name, cov_pars[cov_name])
Z = C.eval((x, y)).reshape(M1.shape)
col = int(m_count * 2**1 + a_count * 2**0)
row = c_count
ax[row, col].imshow(Z)
if col == 0:
ax[row, col].set_ylabel(cov_name)
if row == 0:
ax[row, col].set_title('%dD mesh\n %s' %
(dim, isotropic_label[a_count]))
ax[row, col].set_xticks([], [])
ax[row, col].set_yticks([], [])
c_count += 1
a_count += 1
m_count += 1
fig.savefig('test_covkernel_eval.eps')
