def test_eval(self):
#
# 1D
#
# Define Kernel function
def k_fn(x,y,c = 2):
return x*y + c
k = Explicit(f=lambda x,y,c: x*y+c, parameters={'c':1},
n_variables=2, dim=1)
# Construct kernel, specifying parameter c
kernel = Kernel(k)
# Evaluation points
x = np.ones((11,1))
y = np.linspace(0,1,11)[:,None]
# Check accuracy
self.assertTrue(np.allclose(kernel.eval((x,y)), x*y+1))
# Define kernel with default parameters
k.set_parameters({'c':2})
kernel = Kernel(k)
# Check accuracy
self.assertTrue(np.allclose(kernel.eval((x,y)), x*y+2))
def test_n_samples(self):
#
# 1D
#
# Define mesh
mesh = Mesh1D(resolution=(10,))
# Define function
f = Explicit([lambda x: x, lambda x: -2+2*x**2], dim=1)
n_samples = f.n_samples()
k = Kernel(f)
n_points = 101
x0, x1 = mesh.bounding_box()
x = np.linspace(x0,x1,n_points)
self.assertEqual(k.eval(x).shape, (n_points, n_samples))
self.assertTrue(np.allclose(k.eval(x)[:,0],x))
self.assertTrue(np.allclose(k.eval(x)[:,1], -2+2*x**2))
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())