def _eval(self, z):
"""
Solves the dynamical system for given parameters x.
"""
z = view_as_column(z)
x = np.exp(z)
# Points where the solution will be evaluated
t = np.array([0., 1./6, 1./3, 1./2, 2./3, 5./6, 1.])
t = view_as_column(t)
# Initial condition
y0 = np.array([1., 0., 0., 0., 0., 0.])
y0 = view_as_column(y0)
assert x.shape[0] == 5
sol = f(x[:,0], y0[:,0], t[:,0])
J = df(x[:,0], y0[:,0], t[:,0])
H = df2(x[:,0], y0[:,0], t[:,0])
y = np.delete(sol.reshape((7,6)), 2, 1).flatten() # The 3rd species is unobservable
dy = np.array([np.delete(J[:,i].reshape((7,6)), 2, 1).reshape(35) for i in range(J.shape[1])]) # Delete the 3rd species
d2y = np.zeros((35, H.shape[1], H.shape[2]))
for i in range(H.shape[1]):
for j in range(H.shape[2]):
d2y[:,i,j]= np.delete(H[:,i,j].reshape((7,6)), 2, 1).reshape(35) # Delete the 3rd species
state = {}
state['f'] = y
state['f_grad'] = dy.T * x.T
xx = np.kron(x, x.T)
state['f_grad_2'] = d2y * xx
return state
def _eval(self, x):
"""
Solves the dynamical system for given parameters x.
"""
x = view_as_column(x)
# Points where the solution will be evaluated
t = np.array([0., 30., 60., 90., 120., 150., 180.])
t = view_as_column(t)
# Initial condition
y0 = np.array([500., 0., 0., 0., 0., 0.])
y0 = view_as_column(y0)
assert x.shape[0] == 5
sol = f(x[:, 0], y0[:, 0], t[:, 0])
J = df(x[:, 0], y0[:, 0], t[:, 0])
H = df2(x[:, 0], y0[:, 0], t[:, 0])
y = np.delete(sol.reshape((7, 6)), 2,
1).flatten() # The 3rd species is unobservable
dy = np.array([
np.delete(J[:, i].reshape((7, 6)), 2, 1).reshape(35)
for i in range(J.shape[1])
]) # Delete the 3rd species
d2y = np.zeros((35, H.shape[1], H.shape[2]))
for i in range(H.shape[1]):
for j in range(H.shape[2]):
d2y[:, i, j] = np.delete(H[:, i, j].reshape(
(7, 6)), 2, 1).reshape(35) # Delete the 3rd species
state = {}
state['f'] = y
state['f_grad'] = dy.T
state['f_grad_2'] = d2y
return state
def _eval(self, xs):
"""
Solves the advection equation for u and its derivatives for a given
source location xs.
"""
xs = view_as_column(xs)
assert xs.shape[0] == 2
nx = 50
ny = nx
dx = 0.1
dy = dx
mesh = fp.Grid2D(dx=dx, dy=dy, nx=nx, ny=ny)
vx, vy = make_V_field(mesh)
u = f(xs[:, 0], mesh, vx, vy)
du1 = df(xs[:, 0], mesh, vx, vy, 1)
du2 = df(xs[:, 0], mesh, vx, vy, 2)
d2u11 = df2(xs[:, 0], mesh, vx, vy, 1, 1)
d2u22 = df2(xs[:, 0], mesh, vx, vy, 1, 1)
d2u12 = df2(xs[:, 0], mesh, vx, vy, 1, 2)
dU = np.hstack([view_as_column(du1), view_as_column(du2)])
d2U = np.hstack([
view_as_column(d2u11),
view_as_column(d2u12),
view_as_column(d2u12),
view_as_column(d2u22)
])
d2U = d2U.reshape((d2U.shape[0], 2, 2))
state = {}
state['f'] = u #view_as_column(u)
state['f_grad'] = dU
state['f_grad_2'] = d2U
return state
def _eval(self, xs):
"""
Solves the diffusion equations for u and its derivatives for a given
source location xs.
"""
xs = view_as_column(xs)
assert xs.shape[0] == 2
nx = 25
ny = nx
dx = 0.04
dy = dx
mesh = fp.Grid2D(dx=dx, dy=dy, nx=nx, ny=ny)
u = f(xs[:,0], mesh)
du1 = df(xs[:,0], mesh, 1)
du2 = df(xs[:,0], mesh, 2)
d2u11 = df2(xs[:,0], mesh, 1, 1)
d2u22 = df2(xs[:,0], mesh, 1, 1)
d2u12 = df2(xs[:,0], mesh, 1, 2)
dU = np.hstack([view_as_column(du1), view_as_column(du2)])
d2U = np.hstack([view_as_column(d2u11), view_as_column(d2u12), view_as_column(d2u12), view_as_column(d2u22)])
d2U = d2U.reshape((d2U.shape[0], 2, 2))
state = {}
state['f'] = u #view_as_column(u)
state['f_grad'] = dU
state['f_grad_2'] = d2U
return state
def _eval(self, x):
"""
Solves the dynamical system for given parameters x.
"""
x = view_as_column(x)
# Points where the solution will be evaluated
t = np.array([0., 1. / 6, 1. / 3, 1. / 2, 2. / 3, 5. / 6, 1.])
t = view_as_column(t)
# Initial condition
y0 = np.array([1., 0., 0., 0., 0., 0.])
y0 = view_as_column(y0)
assert x.shape[0] == 5
sol = f(x[:, 0], y0[:, 0], t[:, 0])
J = df(x[:, 0], y0[:, 0], t[:, 0])
H = df2(x[:, 0], y0[:, 0], t[:, 0])
y = np.delete(sol.reshape((7, 6)), 2,
1).flatten() # The 3rd species is unobservable
sol_y = np.delete(y.reshape((7, 5)), 0, 0).reshape(
30) # Delete the initial conditions from the data
dy = np.array([
np.delete(J[:, i].reshape((7, 6)), 2, 1).reshape(35)
for i in range(J.shape[1])
]) # Delete the 3rd species
sol_dy = np.array([
np.delete(dy[i, :].reshape((7, 5)), 0, 0).flatten()
for i in range(dy.shape[0])
]) # Delete the initial cond.
sol_d2y = np.zeros((30, H.shape[1], H.shape[2]))
for i in range(H.shape[1]):
for j in range(H.shape[2]):
d2y = np.delete(H[:, i, j].reshape((7, 6)), 2,
1).reshape(35) # Delete the 3rd species
sol_d2y[:, i, j] = np.delete(d2y.reshape((7, 5)), 0,
0).flatten()
state = {}
state['f'] = sol_y
state['f_grad'] = sol_dy.T
state['f_grad_2'] = sol_d2y
return state
def _eval_u(self, xs):
"""
Solves only the diffusion equation for u for a given
source location xs.
"""
xs = view_as_column(xs)
assert xs.shape[0] == 2
nx = 25
ny = nx
dx = 0.04
dy = dx
mesh = fp.Grid2D(dx=dx, dy=dy, nx=nx, ny=ny)
u = f(xs[:,0], mesh)
return u
