def test_est_2dkernel():
"""example from Harsha."""
x, xi, xj = symbols('x xi xj')
y, yi, yj = symbols('y yi yj')
X = Tuple(x, y)
Xi = Tuple(xi, yi)
Xj = Tuple(xj, yj)
u = Unknown('u')
phi = Constant('phi')
theta = Constant('theta')
expr = phi * u + dx(u) + dy(dy(u))
print('> generic_kernel := ', expand(generic_kernel(expr, u, (Xi, Xj))))
print('')
kuu = theta * exp(-0.5 * ((xi - xj)**2 + (yi - yj)**2))
kuf = compute_kernel(expr, kuu, Xi)
kfu = compute_kernel(expr, kuu, Xj)
kff = compute_kernel(expr, kuu, (Xi, Xj))
print('> kuf := ', kuf)
print('> kfu := ', kfu)
print('> kff := ', kff)
def test_2d():
x, xi, xj = symbols('x xi xj')
y, yi, yj = symbols('y yi yj')
X = Tuple(x, y)
Xi = Tuple(xi, yi)
Xj = Tuple(xj, yj)
u = Unknown('u')
alpha = Constant('alpha')
beta = Constant('beta')
mu = Constant('mu')
nu = Constant('nu')
zeta = Constant('zeta')
theta = Constant('theta')
# expr = alpha * u
# expr = alpha * dx(u)
# expr = alpha * dy(u)
# expr = alpha * u + beta * dx(u)
# expr = alpha * u + beta * dy(u)
# expr = mu * u + alpha * dx(u) + beta * dx(dx(u))
# expr = mu * u + alpha * dx(u) + beta * dy(dy(u))
expr = mu * u + alpha * dx(u) + beta * dx(dx(u)) + nu * dy(
dy(u)) + zeta * dx(dy(u))
# print('> generic_kernel := ', expand(generic_kernel(expr, u, Xi)))
# print('> generic_kernel := ', expand(generic_kernel(expr, u, Xj)))
print('> generic_kernel := ', expand(generic_kernel(expr, u, (Xi, Xj))))
def test_generic_kernel_3d():
x, xi, xj = symbols('x xi xj')
y, yi, yj = symbols('y yi yj')
z, zi, zj = symbols('z zi zj')
X = Tuple(x, y, z)
Xi = Tuple(xi, yi, zi)
Xj = Tuple(xj, yj, zj)
u = Unknown('u')
# ... testing u
assert (generic_kernel(u, u, xi) == Function('u')(xi))
assert (generic_kernel(u, u, xj) == Function('u')(xj))
assert (generic_kernel(u, u, (xi, xj)) == Function('u')(xi, xj))
# ...
# ... testing dx(u)
assert (generic_kernel(dx(u), u, Xi) == Derivative(Function('u')(*Xi), xi))
assert (generic_kernel(dx(u), u, Xj) == Derivative(Function('u')(*Xj), xj))
assert (generic_kernel(dx(u), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), xi, xj))
# ...
# ... testing dy(u)
assert (generic_kernel(dy(u), u, Xi) == Derivative(Function('u')(*Xi), yi))
assert (generic_kernel(dy(u), u, Xj) == Derivative(Function('u')(*Xj), yj))
assert (generic_kernel(dy(u), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), yi, yj))
# ...
# ... testing dz(u)
assert (generic_kernel(dz(u), u, Xi) == Derivative(Function('u')(*Xi), zi))
assert (generic_kernel(dz(u), u, Xj) == Derivative(Function('u')(*Xj), zj))
assert (generic_kernel(dz(u), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), zi, zj))
# ...
# ... testing dx(dx(u))
assert (generic_kernel(dx(dx(u)), u,
Xi) == Derivative(Function('u')(*Xi), xi, xi))
assert (generic_kernel(dx(dx(u)), u,
Xj) == Derivative(Function('u')(*Xj), xj, xj))
assert (generic_kernel(dx(dx(u)), u, (Xi, Xj)) == Derivative(
Function('u')(*Xi, *Xj), xi, xi, xj, xj))