def test_kernel_2d_6():
L = dy(dy(u))
# ...
expected = Derivative(k, yi, yi)
assert(evaluate(L, u, Kernel('K'), (Tuple(xi, yi),)) == expected)
# ...
# ...
expected = Derivative(k, yj, yj)
assert(evaluate(L, u, Kernel('K'), (Tuple(xj, yj),)) == expected)
# ...
# ...
expected = Derivative(k, yi, yi, yj, yj)
assert(evaluate(L, u, Kernel('K'), (Tuple(xi,yi), Tuple(xj,yj))) == expected)
def test_kernel_2d_11():
phi = Constant('phi')
L = phi * u + dx(u) + dy(u) + dz(dz(u))
# ...
expected = phi * k(xi, yi, zi) + Derivative(k, xi) + Derivative(
k, yi) + Derivative(k, zi, zi)
assert (evaluate(L, u, Kernel('K'), (Tuple(xi, yi, zi), )) == expected)
# ...
# ...
expected = phi * k(xj, yj, zj) + Derivative(k, xj) + Derivative(
k, yj) + Derivative(k, zj, zj)
assert (evaluate(L, u, Kernel('K'), (Tuple(xj, yj, zj), )) == expected)
# ...
# ...
expected = (phi**2 * k(xi, yi, zi, xj, yj, zj) + phi * Derivative(k, xi) +
phi * Derivative(k, xj) + phi * Derivative(k, yi) +
phi * Derivative(k, yj) + phi * Derivative(k, zi, zi) +
phi * Derivative(k, zj, zj) + Derivative(k, xi, xj) +
Derivative(k, xi, yj) + Derivative(k, yi, xj) +
Derivative(k, yi, yj) + Derivative(k, xi, zj, zj) +
Derivative(k, yi, zj, zj) + Derivative(k, zi, zi, xj) +
Derivative(k, zi, zi, yj) + Derivative(k, zi, zi, zj, zj))
assert (evaluate(L, u, Kernel('K'),
(Tuple(xi, yi, zi), Tuple(xj, yj, zj))) == expected)
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)
def test_kernel_2d_10():
L = dx(dx(u)) + dy(dy(u)) + dz(dz(u))
# ...
expected = Derivative(k, xi, xi) + Derivative(k, yi, yi) + Derivative(
k, zi, zi)
assert (evaluate(L, u, Kernel('K'), (Tuple(xi, yi, zi), )) == expected)
# ...
# ...
expected = Derivative(k, xj, xj) + Derivative(k, yj, yj) + Derivative(
k, zj, zj)
assert (evaluate(L, u, Kernel('K'), (Tuple(xj, yj, zj), )) == expected)
# ...
# ...
expected = (Derivative(k, xi, xi, xj, xj) + Derivative(k, xi, xi, yj, yj) +
Derivative(k, xi, xi, zj, zj) + Derivative(k, yi, yi, xj, xj) +
Derivative(k, yi, yi, yj, yj) + Derivative(k, yi, yi, zj, zj) +
Derivative(k, zi, zi, xj, xj) + Derivative(k, zi, zi, yj, yj) +
Derivative(k, zi, zi, zj, zj))
assert (evaluate(L, u, Kernel('K'),
(Tuple(xi, yi, zi), Tuple(xj, yj, zj))) == 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_est_2d_2():
u = Unknown('u', ldim=2)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
L = lambda u: phi * u + dx(u) + dy(dy(u))
L_expected = lambda u: 2. * u + dx(u) + dy(dy(u))
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x, y
from sympy import sin, cos
u_sym = x**2 + y
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, meshgrid, zeros
from numpy.random import rand
u_num = lambdify((x, y), u_sym, "numpy")
f_num = lambdify((x, y), f_sym, "numpy")
x_u = rand(50, 2)
x_f = x_u
us = u_num(x_u[:, 0], x_u[:, 1])
fs = f_num(x_f[:, 0], x_f[:, 1])
# ...
# eps = 1.e-6
eps = 1.e-5
# eps = 1.e-4
niter = 10
# niter = 1
phis = []
scores = []
for i in range(0, niter):
print('> sample ', i)
# compute the likelihood
# nlml = NLML(L(u), u, 'CSE', debug=False)
nlml = NLML(L(u), u, 'SE', debug=False)
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
# ... using pure python implementation
x_start = rand(len(nlml.args))
_nlml = lambda y: nlml(y, s_u=eps, s_f=eps)
m = nelder_mead(_nlml,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
score = m[1]
phis.append(nlml.map_args(args)['phi'])
scores.append(score)
# ...
phis = asarray(phis)
scores = asarray(scores)
i_max = argmax(scores)
print(phis[i_max])
print(scores[i_max])
print(scores)
print(phis)
def test_est_2d_3():
u = Unknown('u', ldim=2)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
L = lambda u: phi * u + dx(u) + dy(dy(u))
L_expected = lambda u: 2. * u + dx(u) + dy(dy(u))
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x, y
from sympy import sin, cos
u_sym = x**2 + y
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, meshgrid, zeros
from numpy.random import rand
u_num = lambdify((x, y), u_sym, "numpy")
f_num = lambdify((x, y), f_sym, "numpy")
x_u = rand(50, 2)
x_f = x_u
us = u_num(x_u[:, 0], x_u[:, 1])
fs = f_num(x_f[:, 0], x_f[:, 1])
# ...
# compute the likelihood
# nlml = NLML(L(u), u, 'SE')
nlml = NLML(L(u), u, 'RBF')
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
from numpy.random import rand, normal
from numpy import exp, ones, log
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
phi_expected = 2.
phi_var = 0.5
n_samples = 1000
x_starts = rand(len(nlml.args), n_samples)
i_phi = list(nlml.args).index('phi')
x_starts[i_phi, :] = log(normal(phi_expected, phi_var, n_samples))
phis = []
scores = []
phiso = []
scoreso = []
for i in range(0, n_samples):
print('> sample ', i)
x_start = x_starts[:, i]
# ...
def f(params):
params[i_phi] = x_starts[i_phi, i]
return nlml(params)
m = nelder_mead(f,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
score = m[1]
phis.append(nlml.map_args(args)['phi'])
scores.append(score)
# ...
# ...
m = nelder_mead(nlml,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
score = m[1]
phiso.append(nlml.map_args(args)['phi'])
scoreso.append(score)
# ...
from numpy import savetxt
savetxt('est_2d/phis.txt', asarray(phis))
savetxt('est_2d/scores.txt', asarray(scores))
savetxt('est_2d/phiso.txt', asarray(phiso))
savetxt('est_2d/scoreso.txt', asarray(scoreso))
plt.plot(phis, scores, '.b', label=r'fixed $\phi$', alpha=0.4)
plt.plot(phiso, scoreso, '.r', label=r'free $\phi$', alpha=0.4)
plt.axvline(x=phi_expected, color='green', alpha=0.5)
plt.xlabel(r'$\phi$')
plt.ylabel(r'Likelihood $\mathcal{L}$')
title = '$L u := {}$'.format(latex(L(u)))
title += '\n'
title += r'$\phi_{exact}' + ' := {}$'.format(phi_expected)
plt.title(title)
plt.legend()
plt.show()
def test_est_2d_1():
u = Unknown('u', ldim=2)
phi = Constant('phi')
# ... define a partial differential operator as a lambda function
L = lambda u: phi * u + dx(u) + dy(dy(u))
L_expected = lambda u: 2. * u + dx(u) + dy(dy(u))
# ...
# ... symbolic functions for unknown and rhs
from sympy.abc import x, y
from sympy import sin, cos
u_sym = x**2 + y
f_sym = L_expected(u_sym)
# ...
# ... lambdification + evaluation
from numpy import linspace, meshgrid, zeros
from numpy.random import rand
u_num = lambdify((x, y), u_sym, "numpy")
f_num = lambdify((x, y), f_sym, "numpy")
# t = linspace(0, 1, 5)
# x,y = meshgrid(t, t)
# x_u = zeros((x.size, 2))
# x_u[:,0] = x.reshape(x.size)
# x_u[:,1] = y.reshape(y.size)
x_u = rand(100, 2)
x_f = x_u
us = u_num(x_u[:, 0], x_u[:, 1])
fs = f_num(x_f[:, 0], x_f[:, 1])
# ...
# compute the likelihood
debug = True
nlml = NLML(L(u), u, 'SE', debug=debug)
# set values
nlml.set_u(x_u, us)
nlml.set_f(x_f, fs)
from numpy.random import rand
from numpy import exp, ones, log
# ... using pure python implementation
from symgp.nelder_mead import nelder_mead
x_start = rand(len(nlml.args))
print('> x_start = ', x_start)
m = nelder_mead(nlml,
x_start,
step=0.1,
no_improve_thr=10e-4,
no_improv_break=4,
max_iter=0,
alpha=.5,
gamma=1.5,
rho=-0.5,
sigma=0.5,
verbose=False)
args = exp(m[0])
print('> estimated phi = ', nlml.map_args(args)['phi'])
if debug:
plt.xlabel('number of data')
plt.ylabel(r'eigenvalues in log-scale')
plt.legend()
plt.show()