def get_performance_data(periodic=False):
"""obtain the data used in the performance plot
Args:
periodic (bool): The boundary conditions of the underlying grid
Returns:
dict: The durations of calculating the Laplacian on different grids
using different methods
"""
sizes = 2**np.arange(3, 13)
statistics = {}
for size in display_progress(sizes):
data = {}
grid = UnitGrid([size] * 2, periodic=periodic)
field = ScalarField.random_normal(grid)
field.set_ghost_cells(bc="auto_periodic_neumann")
for backend in ["numba", "scipy"]:
op = grid.make_operator("laplace",
bc="auto_periodic_neumann",
backend=backend)
data[backend] = time_function(op, field.data)
op = grid.make_operator_no_bc("laplace", backend="numba")
data["numba_no_bc"] = time_function(op, field._data_full, use_out=True)
if opencv_laplace:
data["opencv"] = time_function(opencv_laplace, field.data)
statistics[int(size)] = data
return statistics
def main():
""" main routine testing the performance """
print("Reports calls-per-second (larger is better)\n")
# Cartesian grid with different shapes and boundary conditions
for size in [32, 512]:
grid = UnitGrid((size, size), periodic=False)
print(grid)
field = ScalarField.random_normal(grid)
bc_value = np.ones(size)
result = field.laplace(bc={"value": 1}).data
for bc in ["scalar", "array", "linked"]:
if bc == "scalar":
bcs = {"value": 1}
elif bc == "array":
bcs = {"value": bc_value}
elif bc == "linked":
bcs = Boundaries.from_data(grid, {"value": bc_value}, rank=0)
for ax, upper in grid._iter_boundaries():
bcs[ax][upper].link_value(bc_value)
# result = field.laplace(bc=bcs).data
laplace = grid.get_operator("laplace", bc=bcs)
# call once to pre-compile and test result
np.testing.assert_allclose(laplace(field.data), result)
speed = estimate_computation_speed(laplace, field.data)
print(f"{bc:>6s}:{int(speed):>9d}")
print()
def test_pde_bcs_error(bc):
"""test PDE with wrong boundary conditions"""
eq = PDE({"u": "laplace(u)"}, bc=bc)
grid = grids.UnitGrid([8, 8])
field = ScalarField.random_normal(grid)
for backend in ["numpy", "numba"]:
with pytest.raises(BCDataError):
eq.solve(field, t_range=1, dt=0.01, backend=backend, tracker=None)
def test_pde_scalar():
"""test PDE with a single scalar field"""
eq = PDE({"u": "laplace(u) + exp(-t) + sin(t)"})
assert eq.explicit_time_dependence
assert not eq.complex_valued
grid = grids.UnitGrid([8])
field = ScalarField.random_normal(grid)
res_a = eq.solve(field, t_range=1, dt=0.01, backend="numpy", tracker=None)
res_b = eq.solve(field, t_range=1, dt=0.01, backend="numba", tracker=None)
res_a.assert_field_compatible(res_b)
np.testing.assert_allclose(res_a.data, res_b.data)
def test_pde_bcs(bc):
"""test PDE with boundary conditions"""
eq = PDE({"u": "laplace(u)"}, bc=bc)
assert not eq.explicit_time_dependence
assert not eq.complex_valued
grid = grids.UnitGrid([8])
field = ScalarField.random_normal(grid)
res_a = eq.solve(field, t_range=1, dt=0.01, backend="numpy", tracker=None)
res_b = eq.solve(field, t_range=1, dt=0.01, backend="numba", tracker=None)
res_a.assert_field_compatible(res_b)
np.testing.assert_allclose(res_a.data, res_b.data)
def main():
"""main routine testing the performance"""
print("Reports calls-per-second (larger is better)\n")
# Cartesian grid with different shapes and boundary conditions
for size in [32, 512]:
grid = UnitGrid([size, size], periodic=False)
print(grid)
field = ScalarField.random_normal(grid)
bc_value = np.ones(size)
result = field.laplace(bc={"value": 1}).data
for bc in ["scalar", "array", "function", "time-dependent", "linked"]:
if bc == "scalar":
bcs = {"value": 1}
elif bc == "array":
bcs = {"value": bc_value}
elif bc == "function":
bcs = grid.get_boundary_conditions(
{"virtual_point": "2 - value"})
elif bc == "time-dependent":
bcs = grid.get_boundary_conditions({"value_expression": "t"})
elif bc == "linked":
bcs = grid.get_boundary_conditions({"value": bc_value})
for ax, upper in grid._iter_boundaries():
bcs[ax][upper].link_value(bc_value)
else:
raise RuntimeError
# create the operator with these conditions
laplace = grid.make_operator("laplace", bc=bcs)
if bc == "time-dependent":
args = numba_dict({"t": 1})
# call once to pre-compile and test result
np.testing.assert_allclose(laplace(field.data, args=args),
result)
# estimate the speed
speed = estimate_computation_speed(laplace,
field.data,
args=args)
else:
# call once to pre-compile and test result
np.testing.assert_allclose(laplace(field.data), result)
# estimate the speed
speed = estimate_computation_speed(laplace, field.data)
print(f"{bc:>14s}:{int(speed):>9d}")
print()
def test_custom_operators():
""" test using a custom operator """
grid = grids.UnitGrid([32])
field = ScalarField.random_normal(grid)
eq = PDE({"u": "undefined(u)"})
with pytest.raises(NameError):
eq.evolution_rate(field)
def make_op(state):
return lambda state: state
grids.UnitGrid.register_operator("undefined", make_op)
eq._cache = {} # reset cache
res = eq.evolution_rate(field)
np.testing.assert_allclose(field.data, res.data)
del grids.UnitGrid._operators["undefined"] # reset original state
"""
Visualizing a 3d field interactively
====================================
This example demonstrates how to display 3d data interactively using the
`napari <https://napari.org>`_ viewer.
"""
import numpy as np
from pde import CartesianGrid, ScalarField
# create a scalar field with some noise
grid = CartesianGrid([[0, 2 * np.pi]] * 3, 64)
data = ScalarField.from_expression(
grid, "(cos(2 * x) * sin(3 * y) + cos(2 * z))**2")
data += ScalarField.random_normal(grid, std=0.01)
data.label = "3D Field"
data.plot_interactive()
def main():
"""main routine testing the performance"""
print("Reports calls-per-second (larger is better)")
print(" The `CUSTOM` method implemented by hand is the baseline case.")
print(" The `FLEXIBLE` method is a serial implementation using the "
"boundary conditions from the package.")
print(" The other methods use the functions supplied by the package.\n")
# Cartesian grid with different shapes and boundary conditions
for shape in [(32, 32), (512, 512)]:
for periodic in [True, False]:
grid = UnitGrid(shape, periodic=periodic)
print(grid)
field = ScalarField.random_normal(grid)
bcs = grid.get_boundary_conditions("natural", rank=0)
expected = field.laplace("natural")
for method in [
"CUSTOM", "FLEXIBLE", "OPTIMIZED", "numba", "scipy"
]:
if method == "CUSTOM":
laplace = custom_laplace_2d(shape, periodic=periodic)
elif method == "FLEXIBLE":
laplace = flexible_laplace_2d(bcs)
elif method == "OPTIMIZED":
laplace = optimized_laplace_2d(bcs)
elif method in {"numba", "scipy"}:
laplace = grid.make_operator("laplace",
bc=bcs,
backend=method)
else:
raise ValueError(f"Unknown method `{method}`")
# call once to pre-compile and test result
if method == "OPTIMIZED":
result = laplace(field._data_all)
np.testing.assert_allclose(result, expected.data)
speed = estimate_computation_speed(laplace,
field._data_all)
else:
np.testing.assert_allclose(laplace(field.data),
expected.data)
speed = estimate_computation_speed(laplace, field.data)
print(f"{method:>9s}: {int(speed):>9d}")
print()
# Cylindrical grid with different shapes
for shape in [(32, 64), (512, 512)]:
grid = CylindricalSymGrid(shape[0], [0, shape[1]], shape)
print(f"Cylindrical grid, shape={shape}")
field = ScalarField.random_normal(grid)
bcs = Boundaries.from_data(grid, "derivative")
expected = field.laplace(bcs)
for method in ["CUSTOM", "numba"]:
if method == "CUSTOM":
laplace = custom_laplace_cyl_neumann(shape)
elif method == "numba":
laplace = grid.make_operator("laplace", bc=bcs)
else:
raise ValueError(f"Unknown method `{method}`")
# call once to pre-compile and test result
np.testing.assert_allclose(laplace(field.data), expected.data)
speed = estimate_computation_speed(laplace, field.data)
print(f"{method:>8s}: {int(speed):>9d}")
print()
# Spherical grid with different shapes
for shape in [32, 512]:
grid = SphericalSymGrid(shape, shape)
print(grid)
field = ScalarField.random_normal(grid)
bcs = Boundaries.from_data(grid, "derivative")
for conservative in [True, False]:
laplace = grid.make_operator("laplace",
bcs,
conservative=conservative)
laplace(field.data) # call once to pre-compile
speed = estimate_computation_speed(laplace, field.data)
print(
f" numba (conservative={str(conservative):<5}): {int(speed):>9d}"
)
print()
"""
Visualizing a scalar field
==========================
This example displays methods for visualizing scalar fields.
"""
import numpy as np
import matplotlib.pyplot as plt
from pde import CylindricalGrid, ScalarField
# create a scalar field with some noise
grid = CylindricalGrid(7, [0, 4 * np.pi], 64)
data = ScalarField.from_expression(grid, 'sin(z) * exp(-r / 3)')
data += 0.05 * ScalarField.random_normal(grid)
# manipulate the field
smoothed = data.smooth() # Gaussian smoothing to get rid of the noise
projected = data.project('r') # integrate along the radial direction
sliced = smoothed.slice({'z': 1}) # slice the smoothed data
# create four plots of the field and the modifications
fig, axes = plt.subplots(nrows=2, ncols=2)
data.plot(ax=axes[0, 0], title='Original field')
smoothed.plot(ax=axes[1, 0], title='Smoothed field')
projected.plot(ax=axes[0, 1], title='Projection on axial coordinate')
sliced.plot(ax=axes[1, 1], title='Slice of smoothed field at $z=1$')
plt.subplots_adjust(hspace=0.8)
def get_initial_state(self, grid):
""" prepare a useful initial state """
u = ScalarField(grid, self.a, label="Field $u$")
v = self.b / self.a + 0.1 * ScalarField.random_normal(grid, label="Field $v$")
return FieldCollection([u, v])
Here, :math:`D_0` and :math:`D_1` are the respective diffusivity and the
parameters :math:`a` and :math:`b` are related to reaction rates.
Note that the same result can also be achieved with a
:doc:`full implementation of a custom class <pde_brusselator_class>`, which
allows for more flexibility at the cost of code complexity.
"""
from pde import PDE, FieldCollection, PlotTracker, ScalarField, UnitGrid
# define the PDE
a, b = 1, 3
d0, d1 = 1, 0.1
eq = PDE(
{
"u": f"{d0} * laplace(u) + {a} - ({b} + 1) * u + u**2 * v",
"v": f"{d1} * laplace(v) + {b} * u - u**2 * v",
}
)
# initialize state
grid = UnitGrid([64, 64])
u = ScalarField(grid, a, label="Field $u$")
v = b / a + 0.1 * ScalarField.random_normal(grid, label="Field $v$")
state = FieldCollection([u, v])
# simulate the pde
tracker = PlotTracker(interval=1, plot_args={"vmin": 0, "vmax": 5})
sol = eq.solve(state, t_range=20, dt=1e-3, tracker=tracker)