def proj_noncongested(points,
domain,
center=None,
mass=None,
radial_func=RadialFuncInBall(),
verbose=None):
nb_points = len(points)
assert (nb_points != 0)
if mass is None:
mass = np.ones(nb_points) / nb_points
laguerre = OptimalTransport(positions=points,
weights=None,
masses=mass,
domain=domain,
radial_func=radial_func,
linear_solver="CuPyx")
if not center is None:
initialize_weights(power_diagram=laguerre.pd,
center=center,
verbose=verbose)
laguerre.adjust_weights(relax=1)
if np.linalg.norm(laguerre.pd.integrals() - mass) > 1e-5:
print("The Newton algorithm did not converge!")
laguerre.display_vtk("debug_file/bad_Newton.vtk", points=True)
laguerre.get_domain().display_boundaries_vtk(
"debug_file/bad_Newton_domain.vtk")
np.save("debug_file/bad_positions", laguerre.get_positions())
np.save("debug_file/integrals", laguerre.pd.integrals())
np.save("debug_file/bad_weights", laguerre.get_weights())
assert (False)
return laguerre.pd
def optimal_transport(density, Y, masses, psi0 = None, err=1e-8):
center = (density.min_position() + density.max_position())/2
halfsides = (density.max_position() - density.min_position())/2
ratio = 1/np.max(np.abs((Y-center)/halfsides))
psi = (1-ratio)*np.sum((Y-center)**2, axis=-1)
ot = OptimalTransport(Y, psi, density, radial_func=RadialFuncUnit(), obj_max_dw=err)
ot.set_masses(masses)
ot.adjust_weights()
return ot.pd
def make_case(name, nb_diracs, dim):
# default domain
if name == "random":
positions = np.random.rand(nb_diracs, dim)
elif name == "grid":
positions = make_grid(nb_diracs, dim)
elif name == "grid_with_rand":
positions = make_grid(nb_diracs, dim, rand_val=1)
elif name == "faces":
# voronoi with 100 points
pd = PowerDiagram(np.random.rand(5, dim))
# quantization
lot = OptimalTransport(positions=make_grid(nb_diracs, dim))
lot.obj_max_dw = 1e-5
lot.verbosity = 1
for ratio in [1 - 0.85**n for n in range(50)]:
# density
img_size = 1000
img_points = []
items = [range(img_size) for i in range(dim)]
for i in itertools.product(*items):
img_points.append(i)
img = pd.distances_from_boundaries(
np.array(img_points) / img_size).reshape((img_size, img_size))
img = (1 - ratio) + ratio * np.exp(-(100 * img)**2)
lot.set_domain(ScaledImage([0, 0], [1, 1], img / np.mean(img)))
# opt
for _ in range(10):
lot.adjust_weights()
B = lot.get_centroids()
lot.set_positions(lot.get_positions() + 0.3 *
(B - lot.get_positions()))
positions = lot.get_positions()
plt.plot(positions[:, 0], positions[:, 1], ".")
plt.show()
np.save("/data/{}_n{}_d{}_voro.npy".format(name, nb_diracs, dim),
(positions[:, 0], positions[:, 1]))
# solve
if nb_diracs < 32000000:
ot = OptimalTransport(positions)
# ot.verbosity = 1
# solve
ot.adjust_weights()
# display
# ot.display_vtk( "results/pd.vtk" )
np.save("/data/{}_n{}_d{}.npy".format(name, nb_diracs, dim),
(positions[:, 0], positions[:, 1], ot.get_weights()))
def run(n, base_filename, l=0.5):
# domain
domain = ConvexPolyhedraAssembly()
domain.add_box([0, 0], [1, 1])
# initial positions, weights and masses
positions = []
if n == 1:
radius = 0.3
mass = 3.14159 * radius**2
positions.append([0.5, radius])
else:
radius = l / (2 * (n - 1))
mass = l**2 / n**2
for y in np.linspace(radius, l - radius, n):
for x in np.linspace(0.5 - l / 2 + radius, 0.5 + l / 2 - radius,
n):
nx = x + 0.0 * radius * (np.random.rand() - 0.5)
ny = y + 0.0 * radius * (np.random.rand() - 0.5) + 0.5 * radius
positions.append([nx, ny])
positions = np.array(positions)
nb_diracs = positions.shape[0]
# dim = positions.shape[ 1 ]
# OptimalTransport
ot = OptimalTransport(domain, RadialFuncInBall())
ot.set_weights(np.ones(nb_diracs) * radius**2)
ot.set_masses(np.ones(nb_diracs) * mass)
ot.set_positions(positions)
ot.max_iter = 100
ot.adjust_weights()
ot.display_vtk(base_filename + "0.vtk", points=True, centroids=True)
# history of centroids
ce = ot.get_centroids()
ce[:, 1] += radius / 10
bh = [ce]
dt = 1.0
for num_iter in range(200):
print("num_iter", num_iter)
bh.append(ot.get_centroids())
fit_positions(ot, bh, dt)
# display
n1 = int(num_iter / 1) + 1
ot.display_vtk(base_filename + "{}.vtk".format(n1),
points=True,
centroids=True)
def run(n, base_filename, l=0.5):
# domain
domain = ConvexPolyhedraAssembly()
domain.add_box([0, 0], [1, 1])
# initial positions, weights and masses
positions = []
if n == 1:
radius = 0.3
mass = 3.14159 * radius**2
positions.append([0.5, radius])
else:
radius = l / (2 * (n - 1))
mass = l**2 / n**2
for y in np.linspace(radius, l - radius, n):
for x in np.linspace(radius, l - radius, n):
nx = x # + 0.2 * radius * (np.random.rand() - 0.5)
ny = y # + 0.2 * radius * (np.random.rand() - 0.5)
positions.append([nx, ny])
positions = np.array(positions)
nb_diracs = positions.shape[0]
dim = positions.shape[1]
# OptimalTransport
ot = OptimalTransport(domain, RadialFuncInBall())
ot.set_weights(np.ones(nb_diracs) * radius**2)
ot.set_masses(np.ones(nb_diracs) * mass)
ot.set_positions(positions)
ot.adjust_weights()
ot.display_vtk(base_filename + "0.vtk", points=True)
g = np.zeros((nb_diracs, dim))
g[:, 1] = -0.001
bh = [ot.get_centroids()] # history of centroids
for num_iter in range(50):
bh.append(ot.get_centroids())
print("num_iter", num_iter)
# proposition for centroids
bn = 2 * bh[-1] - bh[-2] + g
# find a new set of diracs parameters (position and weight)
# to be as close to the new centroids as possible
update_positions_to_get_centroids(ot, bn)
# display
n1 = num_iter + 1
ot.display_vtk(base_filename + "{}.vtk".format(n1), points=True)
def test_base_ot(self, nb_diracs=1000):
for _ in range(10):
ot = OptimalTransport(self.domain)
# diracs
ot.set_positions(np.random.rand(nb_diracs, 2))
ot.set_weights(np.ones(nb_diracs))
# optimal weights
ot.adjust_weights()
# integrals
areas = ot.pd.integrals()
self.assertAlmostEqual(np.min(areas), 1.0 / nb_diracs, places=6)
self.assertAlmostEqual(np.max(areas), 1.0 / nb_diracs, places=6)
def run(n, base_filename):
domain = ConvexPolyhedraAssembly()
domain.add_box([0, 0], [1, 1])
domain.add_box([0.2, -0.5], [0.8, 0])
positions = []
radius = 0.5 / (2 * (n - 1))
for y in np.linspace(radius, 0.5 + radius, n):
for x in np.linspace(radius, 0.5 + radius, n):
positions.append([x, y])
nb_diracs = len(positions)
#
ot = OptimalTransport(domain, RadialFuncInBall())
ot.set_masses(np.ones(nb_diracs) * 0.8 * 0.5**2 / nb_diracs)
ot.set_weights(np.ones(nb_diracs) * radius**2)
ot.set_positions(np.array(positions))
b_old = ot.pd.centroids()
ot.adjust_weights()
ot.display_vtk(base_filename + "0.vtk")
nb_timesteps = int(20 / radius)
v = np.zeros((nb_diracs, 2))
dt = 0.003 * radius
for i in range(nb_timesteps):
print(i, "/", nb_timesteps)
# first trial
v[:, 1] -= 1
p_old = ot.get_positions()
p_tst = p_old + dt * v
ot.set_positions(p_tst)
ot.adjust_weights()
# display
d = int(n / 5)
if i % d == 0:
ot.display_vtk(base_filename + "{:03}.vtk".format(1 + int(i / d)))
# corrections
b_new = ot.pd.centroids()
v = (b_new - b_old) / dt
ot.set_positions(b_new)
b_old = b_new
def test_ball_cut(self, nb_diracs=100):
for _ in range(10):
ot = OptimalTransport(self.domain, RadialFuncInBall())
positions = np.random.rand(nb_diracs, 2)
positions[:, 1] *= 0.5
radius = 0.25 / nb_diracs**0.5
mass = np.pi * radius**2
# diracs
ot.set_positions(positions)
ot.set_weights(np.ones(nb_diracs) * radius**2)
ot.set_masses(np.ones(nb_diracs) * mass)
# optimal weights
ot.adjust_weights()
# integrals
areas = ot.pd.integrals()
self.assertAlmostEqual(np.min(areas), mass, places=6)
self.assertAlmostEqual(np.max(areas), mass, places=6)
ot = OptimalTransport(domain, RadialFuncInBall())
ot.set_weights( np.ones( positions.shape[ 0 ] ) * target_radius ** 2 )
ot.set_masses( np.ones( positions.shape[ 0 ] ) * np.pi * target_radius ** 2 )
nb_timesteps = int( 20 / target_radius )
for i in range( nb_timesteps ):
# change positions
weights = ot.get_weights()
for n in range( positions.shape[ 0 ] ):
c = 1 / ( 10 * weights[ n ] / 7e-4 + 1 )
positions[ n, : ] += c * 0.5 * target_radius * g.grad( positions[ n, : ] )
# optimal weights
ot.set_positions( positions )
ot.adjust_weights()
# display
d = 5
if i % d == 0:
ot.display_vtk( directory + "/pd_{:03}.vtk".format( int( i / d ) ) )
# update positions
positions = ot.get_centroids()
if np.min( positions[ :, 0 ] ) > pos_door:
times.append( i )
break
print( times )
def run(n, base_filename, l=0.5):
# domain
domain = ConvexPolyhedraAssembly()
# domain.add_box([0, 0], [1, 1])
domain.add_convex_polyhedron([
# x y Nx Ny (ext)
[0, 0, -1, -0.2],
[0, 0, -0.2, -1],
[1, 1, +1, 0],
[1, 1, 0, +1],
])
# initial positions, weights and masses
positions = []
if n == 1:
radius = 0.2
else:
radius = l / (2 * (n - 1))
for y in np.linspace(radius, l - radius, n):
for x in np.linspace(radius, l - radius, n):
nx = x + 0.2 * radius * (np.random.rand() - 0.5)
ny = y + 0.2 * radius * (np.random.rand() - 0.5)
positions.append([nx, ny])
positions = np.array(positions)
nb_diracs = positions.shape[0]
# OptimalTransport
ot = OptimalTransport(domain, RadialFuncInBall())
ot.set_weights(np.ones(nb_diracs) * radius**2)
ot.set_masses(np.ones(nb_diracs) * l**2 / nb_diracs)
ot.set_positions(positions)
ot.adjust_weights()
ot.display_vtk(base_filename + "0.vtk")
ot.set_positions(ot.get_centroids())
velocity = 0.0 * positions
for num_iter in range(200):
print("num_iter", num_iter)
# barycenters at the beginning
ot.adjust_weights()
b_o = ot.get_centroids()
# trial for the new barycenters
velocity[:, 0] -= 0.05 * radius * 0.707
velocity[:, 1] -= 0.05 * radius * 0.707
b_n = b_o + velocity
# optimisation of positions to go to the target barycenters
ropt = scipy.optimize.minimize(obj,
b_n.flatten(), (ot, b_n),
tol=1e-4,
method='BFGS',
options={'eps': 1e-4 * radius})
positions = ropt.x.reshape((-1, 2))
ot.set_positions(positions)
ot.adjust_weights()
# new barycenters, corrected (minimize have update the weights)
b_n = ot.get_centroids()
velocity = b_n - b_o
print(positions, velocity)
# display
ot.pd.display_vtk_points(base_filename +
"pts_{}.vtk".format(num_iter + 1))
ot.display_vtk(base_filename + "{}.vtk".format(num_iter + 1))
dws = 5
img = imageio.imread("clay.jpg")
img = img[:, :, 1]
beg = int(img.shape[1] / 2 - img.shape[0] / 2)
end = beg + img.shape[0]
img = img[:, beg:end]
img = downscale_local_mean(img, (dws, dws))
img = np.max(img) * 1.05 - img
img /= np.sum(img)
# plt.imshow( img )
# plt.show()
# domain
domain = ConvexPolyhedraAssembly()
domain.add_img([0, 0], [1, 1], img)
# diracs
nd = 100000
ot = OptimalTransport(domain)
ot.set_positions(np.random.rand(nd, 2))
ot.set_weights(np.ones(nd) / nd)
ot.obj_max_dw = 1e-5
ot.verbosity = True
# solve
ot.adjust_weights(relax=1.0)
# display
ot.pd.display_vtk("results/pd.vtk", centroids=True)
class FluidSystem:
def __init__( self, domain, positions, velocities, masses, base_filename ):
self.ot = OptimalTransport(domain, RadialFuncInBall())
self.ot.set_positions(np.array(positions))
self.ot.set_weights(np.array(masses)/np.pi)
self.ot.set_masses(np.array(masses))
self.base_filename = base_filename
self.cpt_display = 0
self.max_iter = 200
self.time = 0
# initial centroid positions and velocities
self.ot.adjust_weights()
self.centroids = self.ot.get_centroids()
self.velocities = np.array(velocities)
self.coeff_centroid_force = 1e-4
def display( self ):
fn = "{}{}.vtk".format( self.base_filename, self.cpt_display )
self.ot.display_vtk( fn, points=True, centroids=True )
self.cpt_display += 1
def make_step( self ):
ratio_dt = 1.0
while self.try_step( ratio_dt ) == False:
ratio_dt *= 0.5
print( " dt ratio:", ratio_dt )
def try_step( self, ratio_dt ):
old_p = self.ot.get_positions()
# find dt
radii_ap = ( np.array( self.ot.get_masses() ) / np.pi ) ** 0.5
vn2 = np.linalg.norm( self.velocities, axis=1, ord=2 )
dt = ratio_dt * 0.2 / np.max( np.abs( vn2 / radii_ap ) )
adv = dt * self.velocities
# target centroid positions + initial guess for the dirac positions
target_centroids = self.centroids + adv
self.ot.set_positions( old_p + adv )
# stuff to extract centroids, masses, ...
d = self.ot.dim()
n = self.ot.nb_diracs()
rd = np.arange( d * n, dtype=np.int )
b0 = ( d + 1 ) * np.floor_divide( rd, d )
l0 = b0 + rd % d # l1 = (d + 1) * np.arange(n, dtype=np.int) + d
# find positions to fit the target centroid positions
ratio = 1.0
for num_iter in range( self.max_iter + 1 ):
if num_iter == self.max_iter:
self.ot.set_positions( old_p )
return False
# search dir
mvs = self.ot.pd.der_centroids_and_integrals_wrt_weight_and_positions()
if mvs.error:
self.ot.set_positions( old_p )
ratio *= 0.5
if ratio < 1e-2:
return False
print( " solve X ratio:", ratio )
continue
M = csr_matrix( ( mvs.m_values, mvs.m_columns, mvs.m_offsets ) )[ l0, : ][ :, l0 ]
V = mvs.v_values[ l0 ] - target_centroids.flatten()
c = self.coeff_centroid_force * np.max( M )
V += c * ( self.ot.get_positions() - target_centroids ).flatten()
M += c * diag( 2 * n )
X = spsolve( M, V ).reshape( ( -1, d ) )
# if np.linalg.norm( X, ord=np.inf ) > self.max_disp_at_each_sub_iter:
# X *= self.max_disp_at_each_sub_iter / np.linalg.norm( X, ord=np.inf )
self.ot.set_positions( self.ot.get_positions() - ratio * X )
e = np.linalg.norm( X )
# print( " e", e )
if e < 1e-6:
break
# projection
# self.ot.verbosity = 1
self.ot.adjust_weights( relax=0.75 )
# update centroid pos and speed
self.time += dt
old_centroids = self.centroids
self.centroids = self.ot.get_centroids()
self.velocities = ( self.centroids - old_centroids ) / dt
return True
def run(n, base_filename, l=0.5):
# domain
domain = ConvexPolyhedraAssembly()
domain.add_box([0, 0], [1, 1])
# initial positions, weights and masses
positions = []
radius = l / (2 * (n - 1))
mass = l**2 / n**2
for y in np.linspace(radius, l - radius, n):
for x in np.linspace(0.5 - l / 2 + radius, 0.5 + l / 2 - radius, n):
nx = x + 0.0 * radius * (np.random.rand() - 0.5)
ny = y + 0.0 * radius * (np.random.rand() - 0.5)
positions.append([nx, ny])
positions = np.array(positions)
nb_diracs = positions.shape[0]
dim = positions.shape[1]
# OptimalTransport
ot = OptimalTransport(domain, RadialFuncInBall())
ot.set_weights(np.ones(nb_diracs) * radius**2)
ot.set_masses(np.ones(nb_diracs) * mass)
ot.set_positions(positions)
ot.max_iter = 100
ot.adjust_weights()
ot.display_vtk(base_filename + "0.vtk", points=True, centroids=True)
# gravity
G = np.zeros((nb_diracs, dim))
G[:, 1] = -9.81
#
eps = 0.5
dt = radius * 0.1
V = np.zeros((nb_diracs, dim))
M = np.stack([ot.get_masses() for d in range(dim)]).transpose()
for num_iter in range(500):
print("num_iter:", num_iter, "dt:", dt)
C = ot.get_centroids()
X = ot.get_positions()
A = G + (C - ot.get_positions()) / (M * eps**2)
while True:
dV = dt * A
dX = dt * (V + dV)
if np.max(np.linalg.norm(dX, axis=1, ord=2)) < 0.2 * radius:
dt *= 1.05
V += dV
X += dX
break
dt *= 0.5
ot.set_positions(X)
ot.adjust_weights()
# display
n1 = int(num_iter / 1) + 1
ot.display_vtk(base_filename + "{}.vtk".format(n1),
points=True,
centroids=True)