def test_set_parameters(self):
# Define explicit function
rule = lambda x, a: a * x[:, 0] - x[:, 1]
parms = [{'a': 1}, {'a': 2}]
f = Explicit(rule, parameters=parms, dim=2)
x = (1, 2)
self.assertTrue(np.allclose(f.eval(x), np.array([-1, 0])))
# Modify parameter
f.set_parameters({'a': 2}, pos=0)
self.assertTrue(np.allclose(f.eval(x), np.array([0, 0])))
def test_constructor(self):
"""
Test initialization
"""
f = Explicit(lambda x, a: a * x**2,
parameters={'a': 1},
dim=1,
n_variables=1)
f.eval(1)
self.assertAlmostEqual(f.eval(1), 1)
#
# Errors
#
self.assertRaises(Exception, Explicit)
def test_add_rules(self):
# Define explicit function
rule = lambda x, a: a * x**2
parameters = [{'a': 1}, {'a': 2}]
f = Explicit(rule, parameters=parameters, dim=1)
# Raises an error attempt to add multiple functions
self.assertRaises(Exception, f.add_rule, *(lambda x: x, ),
**{'parameters': [{}, {}]})
# Add a single rule and evaluate at a point
f.add_rule(lambda x: x)
x = 2
vals = [4, 8, 2]
for i in range(3):
#(*(x,),**{'a':1}))
self.assertEqual(f.eval(x)[0, i], vals[i])
def test_subsample_deterministic(self):
"""
When evaluating a deterministic function while specifying a subsample,
n_subsample copies of the function output should be returned.
"""
#
# Deterministic functions
#
# Functions
fns = {
1: {
1: lambda x: x[:, 0]**2,
2: lambda x, y: x[:, 0] + y[:, 0]
},
2: {
1: lambda x: x[:, 0]**2 + x[:, 1]**2,
2: lambda x, y: x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1]
}
}
# Singletons
x = {1: {1: 2, 2: (3, 4)}, 2: {1: (1, 2), 2: ((1, 2), (3, 4))}}
xv = {
1: {
1: [(2, ), (2, )],
2: ([(3, ), (3, )], [(4, ), (4, )])
},
2: {
1: [(1, 2), (1, 2)],
2: ([(1, 2), (1, 2)], [(3, 4), (3, 4)])
}
}
vals = {1: {1: 4, 2: 7}, 2: {1: 5, 2: 11}}
subsample = np.array([2, 3], dtype=np.int)
for dim in [1, 2]:
#
# Iterate over dimension
#
# DofHandler
if dim == 1:
mesh = Mesh1D(box=[0, 5], resolution=(1, ))
elif dim == 2:
mesh = QuadMesh(box=[0, 5, 0, 5])
element = QuadFE(dim, 'Q2')
dofhandler = DofHandler(mesh, element)
dofhandler.distribute_dofs()
basis = Basis(dofhandler)
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
#
# Explicit
#
f = fns[dim][n_variables]
# Explicit
fe = Explicit(f, n_variables=n_variables, dim=dim, \
subsample=subsample)
# Nodal
fn = Nodal(f, n_variables=n_variables, basis=basis, dim=dim, \
dofhandler=dofhandler, subsample=subsample)
# Constant
fc = Constant(1, n_variables=n_variables, \
subsample=subsample)
# Singleton input
xn = x[dim][n_variables]
# Explicit
self.assertEqual(fe.eval(xn).shape[1], len(subsample))
self.assertEqual(fe.eval(xn)[0, 0], vals[dim][n_variables])
self.assertEqual(fe.eval(xn)[0, 1], vals[dim][n_variables])
# Nodal
self.assertEqual(fn.eval(xn).shape[1], len(subsample))
self.assertAlmostEqual(
fn.eval(xn)[0, 0], vals[dim][n_variables])
self.assertAlmostEqual(
fn.eval(xn)[0, 1], vals[dim][n_variables])
# Constant
self.assertEqual(fc.eval(xn).shape[1], len(subsample))
self.assertAlmostEqual(fc.eval(xn)[0, 0], 1)
self.assertAlmostEqual(fc.eval(xn)[0, 1], 1)
# Vector input
xn = xv[dim][n_variables]
n_points = 2
# Explicit
self.assertEqual(fe.eval(xn).shape, (2, 2))
for i in range(fe.n_subsample()):
for j in range(n_points):
self.assertEqual(
fe.eval(xn)[i][j], vals[dim][n_variables])
# Nodal
self.assertEqual(fn.eval(xn).shape, (2, 2))
for i in range(fe.n_subsample()):
for j in range(n_points):
self.assertAlmostEqual(
fn.eval(xn)[i][j], vals[dim][n_variables])
# Constant
self.assertEqual(fc.eval(xn).shape, (2, 2))
for i in range(fe.n_subsample()):
for j in range(n_points):
self.assertEqual(fc.eval(xn)[i][j], 1)
def test_eval(self):
#
# Evaluate single univariate/bivariate functions in 1 or 2 dimensions
#
fns = {
1: {
1: lambda x: x**2,
2: lambda x, y: x + y
},
2: {
1: lambda x: x[:, 0]**2 + x[:, 1]**2,
2: lambda x, y: x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1]
}
}
# Singletons
x = {1: {1: 2, 2: (3, 4)}, 2: {1: (1, 2), 2: ((1, 2), (3, 4))}}
vals = {1: {1: 4, 2: 7}, 2: {1: 5, 2: 11}}
for dim in [1, 2]:
#
# Iterate over dimension
#
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
fn = fns[dim][n_variables]
f = Explicit(fn, n_variables=n_variables, dim=dim)
xn = x[dim][n_variables]
self.assertEqual(f.eval(xn), vals[dim][n_variables])
#
# Evaluate sampled functions
#
fns = {
1: {
1: lambda x, a: a * x**2,
2: lambda x, y, a: a * (x + y)
},
2: {
1: lambda x, a: a * (x[:, 0]**2 + x[:, 1]**2),
2: lambda x, y, a: a * (x[:, 0] * y[:, 0] + x[:, 1] * y[:, 1])
}
}
pars = [{'a': 1}, {'a': 2}]
#
# Singletons
#
x = {1: {1: 2, 2: (3, 4)}, 2: {1: (1, 2), 2: ((1, 2), (3, 4))}}
vals = {1: {1: 4, 2: 7}, 2: {1: 5, 2: 11}}
for dim in [1, 2]:
#
# Iterate over dimension
#
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
fn = fns[dim][n_variables]
f = Explicit(fn,
parameters=pars,
n_variables=n_variables,
dim=dim)
xn = x[dim][n_variables]
self.assertEqual(f.eval(xn)[0][0], vals[dim][n_variables])
self.assertEqual(f.eval(xn)[0][1], 2 * vals[dim][n_variables])
#
# 2 points
#
n_points = 2
x = {
1: {
1: [(2, ), (2, )],
2: ([(3, ), (3, )], [(4, ), (4, )])
},
2: {
1: [(1, 2), (1, 2)],
2: ([(1, 2), (1, 2)], [(3, 4), (3, 4)])
}
}
for dim in [1, 2]:
#
# Iterate over dimension
#
for n_variables in [1, 2]:
#
# Iterate over number of variables
#
fn = fns[dim][n_variables]
f = Explicit(fn,
parameters=pars,
n_variables=n_variables,
dim=dim)
xn = x[dim][n_variables]
self.assertEqual(f.eval(xn).shape[0], n_points)
self.assertEqual(f.eval(xn).shape[1], f.n_samples())
for i in range(f.n_samples()):
for j in range(2):
val = pars[i]['a'] * vals[dim][n_variables]
self.assertEqual(f.eval(xn)[j, i], val)
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
# -----------------------------------------------------------------------------
# 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)
x_inj = np.array([(i+1)*h for i in range(n_inj)])
dofs_inj = []
for x in x_inj:
dofs_inj.append(np.argmin(abs(v-x)))
u_data = np.zeros((ny,1))
u_data[dofs_inj] = 1
u = Nodal(dofhandler=dh_y, data=u_data, dim=1)
# 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
cells_injection = mesh.bin_points(xy)
# Mark vertices
for cell, dummy in cells_injection:
cell.get_vertex(0).mark('injection')