def gradient(x, u_final_data, f, psi, T, R, M):
""" Gradient of the objective function
$$
F = 0.5 * || u_final - u_final_data ||_2^2
$$
Args:
x: physical space grid
f: source in advection equation
u_final_data: observation of the field at time $T$
psi: advection coefficient
T: final time
M: Nb of time intervals
R: Source support is contained in $[- R / 2, R / 2]$
Returns:
F_f: gradient with respect to the source term
F_psi: derivative with respect to advection coefficient
"""
# grid related definitions
dt = T / M
N = len(x)
dx = np.zeros(N)
dx[0] = x[0] - (- R / 2.0)
dx[1 : N] = x[1 : N] - x[0 : N - 1]
# we will accumulate the gradient over time
F_f = np.zeros(len(x))
F_psi = 0.0
u_final = forward(x, np.zeros(len(x)), psi, f, T, R)
for i in range(1, M + 1):
t = i * dt
# solve adjoint problem in w
w = forward(x, u_final_data - u_final, -psi, np.zeros(len(x)), T - t, R)
# solve forward problem for u_x
u_x = forwardGradient(x, np.zeros(len(x)), psi, f, t, R)
# accumulate gradient
F_f = F_f - dt * w
F_psi = F_psi - np.dot(u_x * w, dx) * dt
return F_f, F_psi
def gradientTest():
# problem parameters
T = 0.0001
R = 1.0
N = 100 # Nb of grid points in physical space
M = 10 # Nb of grid points in time space
# synthetic data
x = np.linspace(- R / 2.0 + R / N, R / 2.0, N)
t_f = T
f_optimal = (
(1.0 / t_f) * np.exp( - x * x / (4.0 * t_f)) /
np.sqrt( 4.0 * np.pi * t_f))
psi_optimal = 2000
u_final_data = forward(x, np.zeros(len(x)), psi_optimal, f_optimal, T, R)
# initial coefficients
f = np.zeros(len(x))
psi = 0.0
F_f, F_psi = gradient(x, u_final_data, f, psi, T, R, M)
return F_f, F_psi
def recoverDemo():
# problem parameters
T = 0.0001
R = 1.0
N = 1000 # Nb of grid points in physical space
M = 10 # Nb of grid points in time space
nb_grad_steps = 1000 # Nb of updates with the gradient
alpha_f = 100000000.0 # gradient update size
alpha_psi = 1000.0
sigma = 0.02
# plots
pp = PdfPages('./images/recover_demo.pdf')
# synthetic data
x = np.linspace(- R / 2.0 + R / N, R / 2.0, N)
t_f = T
f_optimal = (
(1.0 / t_f) * np.exp( - (x - 0.1) * (x - 0.1) / (4.0 * t_f)) /
np.sqrt( 4.0 * np.pi * t_f))
f_optimal = f_optimal + (
(1.0 / t_f) * np.exp( - (x + 0.1) * (x + 0.1) / (4.0 * t_f)) /
np.sqrt( 4.0 * np.pi * t_f))
f_optimal = f_optimal / 2.0
psi_optimal = 900
u_final_data = forward(x, np.zeros(len(x)), psi_optimal, f_optimal, T, R)
# add noise to the data
u_final_data = u_final_data + np.random.normal(0, sigma, len(u_final_data))
# initial coefficients
#f = f_optimal
#f = np.zeros(len(x))
f = (
(1.0 / t_f) * np.exp( - (x) * (x) / (4.0 * t_f)) /
np.sqrt( 4.0 * np.pi * t_f))
psi = 910.0
# plot f
plt.figure()
plt.plot(x, f, 'r', label="initial source")
plt.plot(x, f_optimal, 'b', label="true source")
plt.legend(
bbox_to_anchor=(0., 1.02, 1., .102), loc=3,
ncol=2, mode="expand", borderaxespad=0.)
plt.xlabel('$x$')
plt.ylabel('$f$')
pp.savefig()
for i in range(0, nb_grad_steps):
F_f, F_psi = gradient(x, u_final_data, f, psi, T, R, M)
f = f - alpha_f * F_f
psi = psi - alpha_psi * F_psi
print("psi_recovered : ", psi)
print("psi_optimal : ", psi_optimal)
print("F_psi : ", F_psi)
u_final_recovered = forward(x, np.zeros(len(x)), psi, f, T, R)
# plot u_final_data
plt.figure()
plt.plot(x, u_final_data, 'b', label="data")
plt.plot(x, u_final_recovered, 'r--', label="recovered")
plt.legend(
bbox_to_anchor=(0., 1.02, 1., .102), loc=3,
ncol=2, mode="expand", borderaxespad=0.)
plt.xlabel('$x$')
plt.ylabel('$u_{T}$')
pp.savefig()
# plot f
plt.figure()
plt.plot(x, f, 'r', label="source recovered")
plt.plot(x, f_optimal, 'b', label="true source")
plt.legend(
bbox_to_anchor=(0., 1.02, 1., .102), loc=3,
ncol=2, mode="expand", borderaxespad=0.)
plt.xlabel('$x$')
plt.ylabel('$f$')
pp.savefig()
#plot F_f
plt.figure()
plt.plot(x, F_f, label="source sensitivity")
plt.xlabel('$x$')
plt.ylabel('$F_f$')
pp.savefig()
# show and close pdf file
pp.close()
plt.show()
