def test_kernel_1d_2():
L = dx(u) + alpha*u
# ...
K = evaluate(L, u, Kernel('K'), xi)
K = update_kernel(K, RBF, (xi, xj))
expected = theta_1*(1.0*alpha - 1.0*xi + 1.0*xj)*exp(-0.5*(xi - xj)**2)
assert(K == expected)
# ...
# ...
K = evaluate(L, u, Kernel('K'), xj)
K = update_kernel(K, RBF, (xi, xj))
expected = theta_1*(1.0*alpha + 1.0*xi - 1.0*xj)*exp(-0.5*(xi - xj)**2)
assert(K == expected)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (xi, xj))
K = update_kernel(K, RBF, (xi, xj))
expected = theta_1*(alpha**2 - 1.0*(xi - xj)**2 + 1.0)*exp(-0.5*(xi - xj)**2)
assert(K == expected)
def test_kernel_1d_1():
L = u
# ...
K = evaluate(L, u, Kernel('K'), xi)
K = update_kernel(K, RBF, (xi, xj))
expected = theta_1*exp(-0.5*(xi - xj)**2)
assert(K == expected)
# ...
# ...
K = evaluate(L, u, Kernel('K'), xj)
K = update_kernel(K, RBF, (xi, xj))
expected = theta_1*exp(-0.5*(xi - xj)**2)
assert(K == expected)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (xi, xj))
K = update_kernel(K, RBF, (xi, xj))
expected = theta_1*exp(-0.5*(xi - xj)**2)
assert(K == expected)
def test_kernel_2d_1():
L = u
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xi, yi)))
K = update_kernel(K, RBF, ((xi, yi), (xj, yj)))
expected = theta_1 * theta_2 * exp(-0.5 * (xi - xj)**2) * exp(-0.5 *
(yi - yj)**2)
assert (K == expected)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xj, yj)))
K = update_kernel(K, RBF, ((xi, yi), (xj, yj)))
expected = theta_1 * theta_2 * exp(-0.5 * (xi - xj)**2) * exp(-0.5 *
(yi - yj)**2)
assert (K == expected)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xi, yi), Tuple(xj, yj)))
K = update_kernel(K, RBF, ((xi, yi), (xj, yj)))
expected = theta_1 * theta_2 * exp(-0.5 * (xi - xj)**2) * exp(-0.5 *
(yi - yj)**2)
assert (K == expected)
def test_kernel_3d_2():
L = phi * u + dx(u) + dy(u) + dz(dz(u))
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xi, yi, zi)))
K = update_kernel(K, RBF, ((xi, yi, zi), (xj, yj, zj)))
expected = theta_1 * theta_2 * theta_3 * (
phi**3 + 1.0 * phi**2 * (-xi + xj) + 1.0 * phi**2 *
(-yi + yj) + 1.0 * phi**2 * (-zi + zj) + 1.0 * phi * (xi - xj) *
(yi - yj) + 1.0 * phi * (xi - xj) * (zi - zj) + 1.0 * phi * (yi - yj) *
(zi - zj) - 1.0 * (xi - xj) * (yi - yj) *
(zi - zj)) * exp(-0.5 *
(xi - xj)**2) * exp(-0.5 *
(yi - yj)**2) * exp(-0.5 *
(zi - zj)**2)
assert (simplify(K - expected) == 0)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xj, yj, zj)))
K = update_kernel(K, RBF, ((xi, yi, zi), (xj, yj, zj)))
expected = theta_1 * theta_2 * theta_3 * (
phi**3 + 1.0 * phi**2 * (xi - xj) + 1.0 * phi**2 *
(yi - yj) + 1.0 * phi**2 * (zi - zj) + 1.0 * phi * (xi - xj) *
(yi - yj) + 1.0 * phi * (xi - xj) * (zi - zj) + 1.0 * phi * (yi - yj) *
(zi - zj) + 1.0 * (xi - xj) * (yi - yj) *
(zi - zj)) * exp(-0.5 *
(xi - xj)**2) * exp(-0.5 *
(yi - yj)**2) * exp(-0.5 *
(zi - zj)**2)
assert (simplify(K - expected) == 0)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xi, yi, zi), Tuple(xj, yj, zj)))
K = update_kernel(K, RBF, ((xi, yi, zi), (xj, yj, zj)))
expected = theta_1 * theta_2 * theta_3 * (
phi**2 + 2.0 * phi * ((zi - zj)**2 - 1) - 1.0 * (xi - xj)**2 - 2.0 *
(xi - xj) * (yi - yj) - 1.0 * (yi - yj)**2 + 1.0 * (zi - zj)**4 - 6.0 *
(zi - zj)**2 + 5.0) * exp(-0.5 * (xi - xj)**2) * exp(
-0.5 * (yi - yj)**2) * exp(-0.5 * (zi - zj)**2)
assert (simplify(K - expected) == 0)
def test_kernel_2d_2():
L = phi * u + dx(u) + dy(dy(u))
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xi, yi)))
K = update_kernel(K, RBF, ((xi, yi), (xj, yj)))
expected = theta_1 * theta_2 * (phi**2 - 1.0 * phi *
(xi - xj) - 1.0 * phi * (yi - yj) + 1.0 *
(xi - xj) * (yi - yj)) * exp(
-0.5 *
(xi - xj)**2) * exp(-0.5 *
(yi - yj)**2)
assert (simplify(K - expected) == 0)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xj, yj)))
K = update_kernel(K, RBF, ((xi, yi), (xj, yj)))
expected = theta_1 * theta_2 * (phi**2 + 1.0 * phi *
(xi - xj) + 1.0 * phi * (yi - yj) + 1.0 *
(xi - xj) * (yi - yj)) * exp(
-0.5 *
(xi - xj)**2) * exp(-0.5 *
(yi - yj)**2)
assert (simplify(K - expected) == 0)
# ...
# ...
K = evaluate(L, u, Kernel('K'), (Tuple(xi, yi), Tuple(xj, yj)))
K = update_kernel(K, RBF, ((xi, yi), (xj, yj)))
expected = theta_1 * theta_2 * (phi**2 + 2.0 * phi *
((yi - yj)**2 - 1) - 1.0 *
(xi - xj)**2 + 1.0 * (yi - yj)**4 - 6.0 *
(yi - yj)**2 + 4.0) * exp(
-0.5 *
(xi - xj)**2) * exp(-0.5 *
(yi - yj)**2)
assert (simplify(K - expected) == 0)