def test_convert_tangent_euclidean_to_stereo():
solver = kmap.Solver()
n_steps = 10
s3_map = map_sens.Sensitivity(solver, n_steps)
n_runup = 10
u_init = solver.u_init
s0 = solver.s0
u_init = solver.primal_step(u_init, s0, n_runup)
u_trj = s3_map.solve_primal(solver,\
u_init, n_steps, solver.s0)
u_trj = u_trj.T
q = rand(2, n_steps)
q[0] = 0.
q[1] = 1.
q_euclidean = solver.convert_tangent_stereo_to_euclidean(u_trj, q)
q_euclidean = vstack(q_euclidean)
q_spherical = solver.convert_tangent_euclidean_to_spherical(u_trj,\
q_euclidean)
q_spherical = vstack(q_spherical)
p_euclidean = solver.convert_tangent_spherical_to_euclidean(
u_trj, q_spherical)
p_euclidean = vstack(p_euclidean)
p_stereo = solver.convert_tangent_euclidean_to_stereo(u_trj, p_euclidean)
p_stereo = vstack(p_stereo)
print(p_stereo)
assert (norm(p_stereo - q) < 1.e-10)
return
def test_step_primal():
holes = array([1./sqrt(2.),0,1./sqrt(2.0),\
-1./sqrt(2.),0,1./sqrt(2.0), \
1./sqrt(2.),0,-1./sqrt(2.0), \
-1./sqrt(2.),0,-1./sqrt(2.0)])
holes = holes.reshape(4, 3)
solver_map = kmap.Solver()
u_init = solver_map.u_init
n_map = solver_map.n_poincare
n_steps = 500
s0 = solver_map.s0
sens_object = map_sens.Sensitivity()
solve_primal = sens_object.solve_primal
u = solve_primal(solver_map,\
u_init,n_steps,s0)
flag = 1
epsi = 1.e-1
for i in range(1, n_steps):
if (not (flag)):
break
for j in range(holes.shape[0]):
if (norm(u[i, :3] - holes[j]) < epsi):
flag = 0
if (norm(u[i, :3]) > 1.0 + epsi):
flag = 0
assert (i == n_steps - 1)
def visualize_poincare_primal():
solver_map = kmap.Solver()
u_init = solver_map.u_init
n_steps = 500
s0 = solver_map.s0
sens_object = map_sens.Sensitivity()
solve_primal = sens_object.solve_primal
u_map = solve_primal(solver_map,\
u_init,n_steps,s0)
stereo_real, stereo_imag = \
solver_map.stereographic_projection(u_map.T)
figure()
plot(stereo_real, stereo_imag, '.')
savefig('st_proj_poincare_attractor.png')
def visualize_unstable_direction():
solver_map = kmap.Solver()
u_init = solver_map.u_init
n_map = solver_map.n_poincare
n_steps = 500
s0 = solver_map.s0
sens_object = map_sens.Sensitivity()
solve_primal = sens_object.solve_primal
u_map = solve_primal(solver_map,\
u_init,n_steps,s0)
solve_unstable_direction = sens_object.\
solve_unstable_direction
v = solve_unstable_direction(solver_map,\
u_map, rand(solver_map.state_dim),n_steps,\
s0)
visualize_tangent_stereographic(u_map,\
v)
def test_convert_tangent_euclidean_to_spherical():
solver = kmap.Solver()
n_steps = 10
s3_map = map_sens.Sensitivity(solver, n_steps)
n_runup = 10
u_init = solver.u_init
s0 = solver.s0
u_init = solver.primal_step(u_init, s0, n_runup)
u_trj = s3_map.solve_primal(solver,\
u_init, n_steps, solver.s0)
q = rand(2, n_steps)
q_euclidean = solver.convert_tangent_spherical_to_euclidean(u_trj.T, q)
q_euclidean = vstack(q_euclidean)
q_spherical = solver.convert_tangent_euclidean_to_spherical(u_trj.T,\
q_euclidean)
assert (norm(q - q_spherical) < 1.e-10)
return
def test_convert_tangent_stereo_to_euclidean():
solver = kmap.Solver()
n_steps = 10
s3_map = map_sens.Sensitivity(solver, n_steps)
n_runup = 10
u_init = solver.u_init
s0 = solver.s0
u_init = solver.primal_step(u_init, s0, n_runup)
u_trj = s3_map.solve_primal(solver,\
u_init, n_steps, solver.s0)
p1 = solver.convert_tangent_stereo_to_euclidean(u_trj.T,\
vstack((ones(n_steps),zeros(n_steps))))
p2 = solver.convert_tangent_stereo_to_euclidean(u_trj.T,\
vstack((zeros(n_steps),ones(n_steps))))
p1 = vstack(p1)
p2 = vstack(p2)
dot_product_p1_p2 = sum(p1 * p2, axis=0)
print(norm(dot_product_p1_p2))
assert (norm(dot_product_p1_p2) <= 1.e-10)
return
def test_convert_tangent_spherical_to_euclidean():
solver = kmap.Solver()
n_steps = 10
s3_map = map_sens.Sensitivity(solver, n_steps)
n_runup = 10
u_init = solver.u_init
s0 = solver.s0
u_init = solver.primal_step(u_init, s0, n_runup)
u_trj = s3_map.solve_primal(solver,\
u_init, n_steps, solver.s0)
q1 = solver.convert_tangent_spherical_to_euclidean(u_trj.T,\
vstack((ones(n_steps),zeros(n_steps))))
q2 = solver.convert_tangent_spherical_to_euclidean(u_trj.T,\
vstack((zeros(n_steps),ones(n_steps))))
q1 = array(q1)
q2 = array(q2)
dot_product_q1_q2 = sum(q1 * q2, axis=0)
print(norm(dot_product_q1_q2))
assert (norm(dot_product_q1_q2) <= 1.e-10)
return
u_ode = plykin.solve_primal(solver_ode,solver_ode.u_init,\
n_steps,
solver_ode.s0)
t1 = clock()
print('{:<60s}{:>16.10f}'.format(\
"Time to solve primal: ",\
t1-t0))
t2 = clock()
plykin.precompute_sources(solver_ode, u_ode)
t3 = clock()
print('{:<60s}{:>16.10f}'.format(\
"Time for flow precomputations: ",\
t3-t2))
poincare_plykin = map_sens.Sensitivity(solver_map, n_steps_map)
#u_map = poincare_plykin.solve_primal(solver_map,\
# solver_map.u_init, n_steps_map, solver_map.s0)
u_map = u_ode[::n_map]
t4 = clock()
poincare_plykin.precompute_sources(solver_map, u_map)
t5 = clock()
print('{:<60s}{:>16.10f}'.format(\
"Time for map precomputations: ",\
t5-t4))
# Comparison of pre-computations
v0 = plykin.v0[::n_map]
v0_map = poincare_plykin.v0
gradJ = zeros_like(u)
gradJ[:, index] = 1.0
return gradJ
@jit(nopython=True)
def compute_dJds(gradJ, v):
dJds = diag(dot(gradJ, v.T))
return dJds
#def compute_sensitivity():
if __name__ == "__main__":
solver = kp.Solver()
n_steps = 50
s3_map = map_sens.Sensitivity(solver, n_steps)
n_runup = 500
u_init = solver.u_init
s0 = solver.s0
state_dim = solver.state_dim
param_dim = s0.size
u_init = solver.primal_step(u_init, s0, n_runup)
u_trj = s3_map.solve_primal(solver,\
solver.u_init, n_steps, solver.s0)
dFds = s3_map.compute_source_tangent(solver,\
u_trj, n_steps, s0)[:,0,:]
v0_init = zeros_like(u_init)
v_trj = solve_tangent_equation(solver, u_trj, v0_init, dFds, n_steps, s0)
dJ_trj = gradient_J(u_trj, 0)
dJds = compute_dJds(dJ_trj, v_trj)
fig = figure(figsize=[15, 10])