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tensorproductspace.py
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tensorproductspace.py
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"""
Module for implementation of the :class:`.TensorProductSpace` class and
related methods.
"""
from numbers import Number
import functools
import sympy as sp
import numpy as np
from shenfun.fourier.bases import R2CBasis, C2CBasis
from shenfun.utilities import apply_mask
from shenfun.forms.arguments import Function, Array
from shenfun.optimization.cython import evaluate
from shenfun.spectralbase import slicedict, islicedict, SpectralBase
from shenfun.coordinates import Coordinates
from mpi4py_fft.mpifft import Transform, PFFT
from mpi4py_fft.pencil import Subcomm, Pencil
__all__ = ('TensorProductSpace', 'VectorTensorProductSpace',
'MixedTensorProductSpace', 'Convolve')
# Default sympy symbols. Note that order is important
x, y, z, r, s = psi = sp.symbols('x,y,z,r,s', real=True)
class CurvilinearTransform(Transform):
"""Class for performing forward parallel transform using curvilinear
coordinates
Parameters
----------
xfftn : list of serial transform objects
transfer : list of global redistribution objects
pencil : list of two pencil objects
The two pencils represent the input and final output configuration of
the distributed global arrays
T : TensorProductSpace
"""
def __init__(self, xfftn, transfer, pencil, T=None):
assert len(xfftn) == len(transfer) + 1 and len(pencil) == 2
assert T is not None
self._xfftn = tuple(xfftn)
self._transfer = tuple(transfer)
self._pencil = tuple(pencil)
self._T = T
@property
def hi(self):
"""Return scaling factors of Transform"""
return self._T.hi
@property
def local_mesh(self):
"""Return local mesh"""
return self._T.local_mesh(True)
def get_measured_input_array(self):
"""Weigh input array with integral measure
"""
dx = self.hi.prod()
mesh = self.local_mesh
if dx == 1:
return
sym0 = dx.free_symbols
m = []
for sym in sym0:
j = 'xyzrs'.index(str(sym))
m.append(mesh[j])
xj = sp.lambdify(sym0, dx)(*m)
self.input_array[...] = self.input_array*xj
return
def __call__(self, input_array=None, output_array=None, **kw):
"""Compute transform
Parameters
----------
input_array : array, optional
output_array : array, optional
kw : dict
parameters to serial transforms
Note in particular that the keyword 'normalize'=True/False can be
used to turn normalization on or off. Default is to enable
normalization for forward transforms and disable it for backward.
Note
----
If input_array/output_array are not given, then use predefined arrays
as planned with serial transform object _xfftn.
"""
if input_array is not None:
self.input_array[...] = input_array
self.get_measured_input_array()
for i in range(len(self._transfer)):
self._xfftn[i](**kw)
arrayA = self._xfftn[i].output_array
arrayB = self._xfftn[i+1].input_array
self._transfer[i](arrayA, arrayB)
self._xfftn[-1](**kw)
if output_array is not None:
output_array[...] = self.output_array
return output_array
else:
return self.output_array
class TensorProductSpace(PFFT):
"""Class for multidimensional tensorproductspaces.
The tensorproductspaces are created as Cartesian products from a set of 1D
bases. The 1D bases are subclassed instances of the :class:`.SpectralBase`
class.
Parameters
----------
comm : MPI communicator
bases : list
List of 1D bases
axes : tuple of ints, optional
A tuple containing the order of which to perform transforms.
Last item is transformed first. Defaults to range(len(bases))
dtype : data-type, optional
Type of input data in real physical space. If not provided it
will be inferred from the bases.
slab : bool, optional
Use 1D slab decomposition.
collapse_fourier : bool, optional
Collapse axes for Fourier bases if possible
backward_from_pencil : False or Pencil
In case of Pencil configure the transform by starting from the
Pencil distribution in spectral space. This is primarily intended
for a padded space, where the spectral distribution must be
equal to the non-padded space.
coordinates: two tuples, optional
Map for curvilinear coordinatesystem.
First tuple are the coordinate variables in the new coordinate system
Second tuple are the Cartesian coordinates as functions of the variables
in the first tuple. Example::
psi = (theta, r) = sp.symbols('x,y', real=True, positive=True)
rv = (r*sp.cos(theta), r*sp.sin(theta))
where psi and rv are the first and second tuples, respectively.
kw : dict, optional
Dictionary that can be used to plan transforms. Input to method
`plan` for the bases.
"""
def __init__(self, comm, bases, axes=None, dtype=None, slab=False,
collapse_fourier=False, backward_from_pencil=False,
coordinates=None, **kw):
# Note do not call __init__ of super
self.comm = comm
self.bases = bases
coors = coordinates if coordinates is not None else (psi[:len(bases)],)*2
self.coors = Coordinates(*coors)
self.hi = self.coors.get_scaling_factors()
shape = list(self.global_shape())
self.axes = axes
assert shape
assert min(shape) > 0
if axes is not None:
axes = list(axes) if np.ndim(axes) else [axes]
else:
axes = list(range(len(shape)))
for i, ax in enumerate(axes):
if isinstance(ax, (int, np.integer)):
if ax < 0:
ax += len(shape)
axes[i] = (ax,)
else:
assert isinstance(ax, (tuple, list))
ax = list(ax)
for j, a in enumerate(ax):
assert isinstance(a, int)
if a < 0:
a += len(shape)
ax[j] = a
axes[i] = ax
assert min(axes[i]) >= 0
assert max(axes[i]) < len(shape)
assert 0 < len(axes[i]) <= len(shape)
assert sorted(axes[i]) == sorted(set(axes[i]))
if dtype is None:
dtype = np.complex if isinstance(bases[axes[-1][-1]], C2CBasis) else np.float
dtype = np.dtype(dtype)
assert dtype.char in 'fdgFDG'
self.xfftn = []
self.transfer = []
self.pencil = [None, None]
for axis, base in enumerate(self.bases):
base.tensorproductspace = self
base.axis = axis
base.sl = slicedict(axis=axis, dimensions=len(self.bases))
base.si = islicedict(axis=axis, dimensions=len(self.bases))
self.axes = axes
if not backward_from_pencil:
if isinstance(bases[axes[-1][-1]], C2CBasis):
assert np.dtype(dtype).char in 'FDG'
elif isinstance(bases[axes[-1][-1]], R2CBasis):
assert np.dtype(dtype).char in 'fdg'
if isinstance(comm, Subcomm):
assert slab is False
assert len(comm) == len(shape)
assert comm[axes[-1][-1]].Get_size() == 1
self.subcomm = comm
else:
if slab:
axis = (axes[-1][-1] + 1) % len(shape)
dims = [1] * len(shape)
dims[axis] = comm.Get_size()
else:
dims = [0] * len(shape)
for ax in axes[-1]:
dims[ax] = 1
self.subcomm = Subcomm(comm, dims)
# At this points axes is a tuple of tuples of length one.
# Try to collapse some Fourier transforms into one.
if np.any([abs(base.padding_factor - 1.0) > 1e-6 for base in bases]):
collapse_fourier = False
if collapse_fourier:
F = lambda ax: bases[ax].family() == 'fourier' and self.subcomm[ax].Get_size() == 1
axis = self.axes[-1][-1]
groups = [list(self.axes[-1])]
F0 = F(axis)
for ax in reversed(self.axes[:-1]):
axis = ax[-1]
if F0 and F(axis):
groups[0].insert(0, axis)
else:
groups.insert(0, list(ax))
F0 = F(axis)
self.axes = groups
self.axes = tuple(map(tuple, self.axes))
# Configure all transforms
axes = self.axes[-1]
pencil = Pencil(self.subcomm, shape, axes[-1])
self.xfftn.append(self.bases[axes[-1]])
self.xfftn[-1].plan(pencil.subshape, axes, dtype, kw)
self.pencil[0] = pencilA = pencil
if not (shape[axes[-1]] == self.xfftn[-1].forward.output_array.shape[axes[-1]] and
self.xfftn[-1].forward.input_array.dtype == self.xfftn[-1].forward.output_array.dtype):
dtype = self.xfftn[-1].forward.output_array.dtype
shape[axes[-1]] = self.xfftn[-1].forward.output_array.shape[axes[-1]]
pencilA = Pencil(self.subcomm, shape, axes[-1])
for i, axes in enumerate(reversed(self.axes[:-1])):
pencilB = pencilA.pencil(axes[-1])
transAB = pencilA.transfer(pencilB, dtype)
xfftn = self.bases[axes[-1]]
xfftn.plan(pencilB.subshape, axes, dtype, kw)
self.xfftn.append(xfftn)
self.transfer.append(transAB)
pencilA = pencilB
if not (shape[axes[-1]] == xfftn.forward.output_array.shape[axes[-1]] and
xfftn.forward.input_array.dtype == xfftn.forward.output_array.dtype):
dtype = xfftn.forward.output_array.dtype
shape[axes[-1]] = xfftn.forward.output_array.shape[axes[-1]]
pencilA = Pencil(pencilB.subcomm, shape, axes[-1])
self.pencil[1] = pencilA
FT = functools.partial(CurvilinearTransform, T=self) if not self.hi.prod() == 1 else Transform
self.forward = FT(
[o.forward for o in self.xfftn],
[o.forward for o in self.transfer],
self.pencil)
self.backward = Transform(
[o.backward for o in self.xfftn[::-1]],
[o.backward for o in self.transfer[::-1]],
self.pencil[::-1])
self.scalar_product = FT(
[o.scalar_product for o in self.xfftn],
[o.forward for o in self.transfer],
self.pencil)
self.backward_uniform = Transform(
[o.backward_uniform for o in self.xfftn[::-1]],
[o.backward for o in self.transfer[::-1]],
self.pencil[::-1])
else:
self.configure_backwards(backward_from_pencil, dtype, kw)
for i, base in enumerate(bases):
base.axis = i
if base.has_nonhomogeneous_bcs:
base.bc.set_tensor_bcs(base, self)
def configure_backwards(self, pencil, dtype, kw):
"""Configure transforms starting from spectral space
Parameters
----------
pencil : Pencil
The distribution in spectral space
dtype : Numpy dtype
The type of data in spectral space
kw : dict
Any parameters for planning transforms
Note
----
To ensure the same distribution in spectral space, the padded
space must be configured by moving from the spectral space towards the
physical space. The distribution does not have to agree in physical
space, because the padding is done only in the spectral.
"""
shape = list(self.global_shape(True))
axes = self.axes[0]
xfftn = self.bases[axes[-1]]
self.xfftn.append(xfftn)
self.subcomm = pencil.subcomm
subshape = list(pencil.subshape)
if isinstance(xfftn, R2CBasis):
subshape[axes[-1]] = int(np.floor(xfftn.N*xfftn.padding_factor))
dtype = np.float
else:
subshape[axes[-1]] = int(np.floor(subshape[axes[-1]]*xfftn.padding_factor))
self.xfftn[-1].plan(subshape, axes, dtype, kw)
if not (shape[axes[-1]] == self.xfftn[-1].forward.input_array.shape[axes[-1]] and
self.xfftn[-1].forward.input_array.dtype == self.xfftn[-1].forward.output_array.dtype):
dtype = self.xfftn[-1].forward.input_array.dtype
shape[axes[-1]] = self.xfftn[-1].forward.input_array.shape[axes[-1]]
pencilA = Pencil(pencil.subcomm, shape, axes[0])
self.pencil[0] = pencilA
for axes in self.axes[1:]:
pencilB = pencilA.pencil(axes[-1])
transBA = pencilA.transfer(pencilB, dtype)
xfftn = self.bases[axes[-1]]
subshape = list(pencilB.subshape)
if isinstance(xfftn, R2CBasis):
subshape[axes[-1]] = int(np.floor(xfftn.N*xfftn.padding_factor))
dtype = np.float
else:
subshape[axes[-1]] = int(np.floor(subshape[axes[-1]]*xfftn.padding_factor))
xfftn.plan(subshape, axes, dtype, kw)
self.xfftn.append(xfftn)
self.transfer.append(transBA)
pencilA = pencilB
if not (shape[axes[-1]] == xfftn.forward.input_array.shape[axes[-1]] and
xfftn.forward.input_array.dtype == xfftn.forward.output_array.dtype):
dtype = xfftn.forward.input_array.dtype
shape[axes[-1]] = xfftn.forward.input_array.shape[axes[-1]]
pencilA = Pencil(pencilB.subcomm, shape, axes[-1])
self.pencil[1] = pencilA
FT = functools.partial(CurvilinearTransform, T=self) if not self.hi.prod() == 1 else Transform
self.backward = Transform(
[o.backward for o in self.xfftn],
[o.forward for o in self.transfer],
self.pencil)
self.forward = FT(
[o.forward for o in self.xfftn[::-1]],
[o.backward for o in self.transfer[::-1]],
self.pencil[::-1])
self.scalar_product = FT(
[o.scalar_product for o in self.xfftn[::-1]],
[o.forward for o in self.transfer[::-1]],
self.pencil[::-1])
self.backward_uniform = Transform(
[o.backward_uniform for o in self.xfftn],
[o.forward for o in self.transfer],
self.pencil)
def get_dealiased(self, padding_factor=1.5, dealias_direct=False):
if isinstance(padding_factor, Number):
padding_factor = (padding_factor,)*len(self)
elif isinstance(padding_factor, (tuple, list, np.ndarray)):
assert len(padding_factor) == len(self)
padded_bases = [base.get_dealiased(padding_factor=padding_factor[axis],
dealias_direct=dealias_direct)
for axis, base in enumerate(self.bases)]
# Need the correct order of the transforms in case reversed somehow
axes = []
for ax in self.axes:
for ai in ax:
axes.append(ai)
return TensorProductSpace(self.comm, padded_bases, axes=tuple(axes),
dtype=self.forward.output_array.dtype,
backward_from_pencil=self.forward.output_pencil,
coordinates=self.coors.coordinates)
def get_refined(self, N):
if isinstance(N, Number):
N = (N,)*len(self)
elif isinstance(N, (tuple, list, np.ndarray)):
assert len(N) == len(self)
refined_bases = [base.get_refined(N[axis])
for axis, base in enumerate(self.bases)]
return TensorProductSpace(self.subcomm, refined_bases, axes=self.axes,
dtype=self.forward.input_array.dtype,
coordinates=self.coors.coordinates)
def convolve(self, a_hat, b_hat, ab_hat):
"""Convolution of a_hat and b_hat
Parameters
----------
a_hat : array
Input array of shape and type as output array from
self.forward, or instance of :class:`.Function`
b_hat : array
Input array of shape and type as output array from
self.forward, or instance of :class:`.Function`
ab_hat : array
Return array of same type and shape as a_hat and b_hat
Note
----
The return array ab_hat is truncated to the shape of a_hat and b_hat.
Also note that self should have bases with padding for this method to give
a convolution without aliasing. The padding is specified when creating
instances of bases for the :class:`.TensorProductSpace`.
FIXME Efficiency due to allocation
"""
a = self.backward.output_array.copy()
b = self.backward.output_array.copy()
a = self.backward(a_hat, a)
b = self.backward(b_hat, b)
ab_hat = self.forward(a*b, ab_hat)
return ab_hat
def eval(self, points, coefficients, output_array=None, method=2):
"""Evaluate Function at points, given expansion coefficients
Parameters
----------
points : float or array of floats
Array must be of shape (D, N), for N points in D dimensions
coefficients : array
Expansion coefficients, or instance of :class:`.Function`
output_array : array, optional
Return array, function values at points
method : int, optional
Chooses implementation. The default 0 is a low-memory cython
version. Using method = 1 leads to a faster cython
implementation that, on the downside, uses more memory.
The final, method = 2, is a python implementation.
"""
if output_array is None:
output_array = np.zeros(points.shape[1], dtype=self.forward.input_array.dtype)
else:
output_array[:] = 0
if len(self.get_nonperiodic_axes()) > 1:
method = 2
if method == 0:
return self._eval_lm_cython(points, coefficients, output_array)
elif method == 1:
return self._eval_cython(points, coefficients, output_array)
else:
return self._eval_python(points, coefficients, output_array)
def _eval_python(self, points, coefficients, output_array):
"""Evaluate Function at points, given expansion coefficients
Parameters
----------
points : float or array of floats
coefficients : array
Expansion coefficients
output_array : array
Return array, function values at points
"""
P = []
last_conj_index = -1
sl = -1
out = []
previous_axes = []
flataxes = []
for ax in self.axes:
for a in ax:
flataxes.append(a)
for axis in flataxes:
base = self.bases[axis]
x = base.map_reference_domain(points[axis])
D = base.evaluate_basis_all(x=x, argument=1)
P = D[..., self.local_slice()[axis]]
if isinstance(base, R2CBasis):
M = base.N//2+1
if base.N % 2 == 0:
last_conj_index = M-1
else:
last_conj_index = M
offset = self.local_slice()[axis].start
sl = offset
st = self.local_slice()[axis].stop
if sl == 0:
sl = 1
st = min(last_conj_index, st) - offset
sl -= offset
sp = [slice(None), slice(sl, st)]
if len(out) == 0:
out = np.tensordot(P, coefficients, (1, axis))
if isinstance(base, R2CBasis):
ss = [slice(None)]*len(self)
ss[axis] = slice(sl, st)
out += np.conj(np.tensordot(P[tuple(sp)], coefficients[tuple(ss)], (1, axis)))
out = out.real
else:
k = np.count_nonzero([m < axis for m in previous_axes])
if len(self) == 2:
if not isinstance(base, R2CBasis):
out2 = np.sum(P*out, axis=-1)
else:
sp = tuple(sp)
out2 = np.sum(P.real*out.real - P.imag*out.imag, axis=-1)
out2 += np.sum(np.conj(P[sp].real*out[sp].real - P[sp].imag*out[sp].imag), axis=-1)
elif len(self) == 3:
sx = [slice(None)]*out.ndim
sd = [slice(None)]*out.ndim
if len(out.shape) == 3:
kk = 1 if axis-k == 1 else 2
sx[kk] = np.newaxis
if not isinstance(base, R2CBasis):
out2 = np.sum(P[tuple(sx)]*out, axis=1+axis-k)
else:
out2 = np.sum(P[tuple(sx)].real*out.real - P[tuple(sx)].imag*out.imag, axis=1+axis-k)
sx[1+axis-k] = slice(sl, st)
sd[1+axis-k] = slice(sl, st)
sx = tuple(sx)
sd = tuple(sd)
out2 += np.sum(np.conj(P[sx].real*out[sd].real - P[sx].imag*out[sd].imag), axis=1+axis-k)
out = out2
previous_axes.append(axis)
output_array[:] = out
output_array = self.comm.allreduce(output_array)
return output_array
def _eval_lm_cython(self, points, coefficients, output_array):
"""Evaluate Function at points, given expansion coefficients
Parameters
----------
points : float or array of floats
coefficients : array
Expansion coefficients
output_array : array
Return array, function values at points
"""
r2c = -1
last_conj_index = -1
sl = -1
x = []
w = []
for base in self:
axis = base.axis
if isinstance(base, R2CBasis):
r2c = axis
M = base.N//2+1
if base.N % 2 == 0:
last_conj_index = M-1
else:
last_conj_index = M
sl = self.local_slice()[axis].start
x.append(base.map_reference_domain(points[axis]))
w.append(base.wavenumbers(bcast=False)[self.local_slice()[axis]].astype(np.float))
if len(self) == 2:
output_array = evaluate.evaluate_lm_2D(list(self.bases), output_array, coefficients, x[0], x[1], w[0], w[1], r2c, last_conj_index, sl)
elif len(self) == 3:
output_array = evaluate.evaluate_lm_3D(list(self.bases), output_array, coefficients, x[0], x[1], x[2], w[0], w[1], w[2], r2c, last_conj_index, sl)
output_array = np.atleast_1d(output_array)
output_array = self.comm.allreduce(output_array)
return output_array
def _eval_cython(self, points, coefficients, output_array):
"""Evaluate Function at points, given expansion coefficients
Parameters
----------
points : float or array of floats
coefficients : array
Expansion coefficients
output_array : array
Return array, function values at points
"""
P = []
r2c = -1
last_conj_index = -1
sl = -1
for base in self:
axis = base.axis
x = base.map_reference_domain(points[axis])
D = base.evaluate_basis_all(x=x)
P.append(D[..., self.local_slice()[axis]])
if isinstance(base, R2CBasis):
r2c = axis
M = base.N//2+1
if base.N % 2 == 0:
last_conj_index = M-1
else:
last_conj_index = M
sl = self.local_slice()[axis].start
if len(self) == 2:
output_array = evaluate.evaluate_2D(output_array, coefficients, P, r2c, last_conj_index, sl)
elif len(self) == 3:
output_array = evaluate.evaluate_3D(output_array, coefficients, P, r2c, last_conj_index, sl)
output_array = np.atleast_1d(output_array)
output_array = self.comm.allreduce(output_array)
return output_array
def wavenumbers(self, scaled=False, eliminate_highest_freq=False):
"""Return list of wavenumbers of TensorProductSpace
Parameters
----------
scaled : bool, optional
Scale wavenumbers with size of box
eliminate_highest_freq : bool, optional
Set Nyquist frequency to zero for evenly shaped axes of Fourier
bases.
"""
K = []
for base in self:
K.append(base.wavenumbers(scaled=scaled,
eliminate_highest_freq=eliminate_highest_freq))
return K
def local_wavenumbers(self, broadcast=False, scaled=False,
eliminate_highest_freq=False):
"""Return list of local wavenumbers of TensorProductSpace
Parameters
----------
broadcast : bool, optional
Broadcast returned wavenumber arrays to actual
dimensions of TensorProductSpace
scaled : bool, optional
Scale wavenumbers with size of box
eliminate_highest_freq : bool, optional
Set Nyquist frequency to zero for evenly shaped axes of Fourier
bases
"""
k = self.wavenumbers(scaled=scaled, eliminate_highest_freq=eliminate_highest_freq)
lk = []
for axis, (n, s) in enumerate(zip(k, self.local_slice(True))):
ss = [slice(None)]*len(k)
ss[axis] = s
lk.append(n[tuple(ss)])
if broadcast is True:
return [np.broadcast_to(m, self.shape()) for m in lk]
return lk
def mesh(self, uniform=False):
"""Return list of 1D physical meshes for each dimension of
TensorProductSpace
Parameters
----------
uniform : bool, optional
Use uniform mesh for non-periodic bases
"""
X = []
for base in self:
X.append(base.mesh(uniform=uniform))
return X
def local_mesh(self, broadcast=False, uniform=False):
"""Return list of local 1D physical meshes for each dimension of
TensorProductSpace
Parameters
----------
broadcast : bool, optional
Broadcast each 1D mesh to real shape of
:class:`.TensorProductSpace`
uniform : bool, optional
Use uniform mesh for non-periodic bases
"""
mesh = self.mesh(uniform=uniform)
lm = []
for axis, (n, s) in enumerate(zip(mesh, self.local_slice(False))):
ss = [slice(None)]*len(mesh)
ss[axis] = s
lm.append(n[tuple(ss)])
if broadcast is True:
return [np.broadcast_to(m, self.shape(False)) for m in lm]
return lm
def local_curvilinear_mesh(self, uniform=False):
"""Return curvilinear mesh
Parameters
----------
uniform : bool, optional
Use uniform mesh for non-periodic bases
"""
X = self.local_mesh(broadcast=True, uniform=uniform)
xx = []
psi = self.coors.coordinates[0]
for rv in self.coors.coordinates[1]:
xx.append(sp.lambdify(psi, rv)(*X))
return xx
def curvilinear_mesh(self, uniform=False):
"""Return curvilinear mesh
Parameters
----------
uniform : bool, optional
Use uniform mesh for non-periodic bases
"""
X = self.mesh(uniform=uniform)
xx = []
psi = self.coors.coordinates[0]
for rv in self.coors.coordinates[1]:
xx.append(sp.lambdify(psi, rv)(*X))
return xx
def dim(self):
"""Return dimension of ``self`` (degrees of freedom)"""
return np.prod(self.dims())
def dims(self):
"""Return dimensions (degrees of freedom) of all bases in ``self``"""
return tuple([base.dim() for base in self])
def size(self, forward_output=False):
"""Return number of elements in :class:`.TensorProductSpace`"""
return np.prod(self.shape(forward_output))
def global_shape(self, forward_output=False):
"""Return global shape of arrays for TensorProductSpace
Parameters
----------
forward_output : bool, optional
If True then return shape of an array that is the result of a
forward transform. If False then return shape of physical
space, i.e., the input to a forward transform.
"""
#if not forward_output:
# return tuple([int(np.round(base.shape(forward_output)*base.padding_factor)) for base in self])
return tuple([base.shape(forward_output) for base in self])
#return self.shape(forward_output)
def mask_nyquist(self, u_hat, mask=None):
"""Return Function `u_hat` with zero Nyquist coefficients
Parameters
----------
u_hat : array
Function to be masked
mask : array or None, optional
mask array, if not provided then get the mask by calling
:func:`get_mask_nyquist`
"""
if mask is None:
mask = self.get_mask_nyquist()
u_hat = apply_mask(u_hat, mask)
return u_hat
def get_mask_nyquist(self):
"""Return an array with zeros for Nyquist coefficients and one otherwise"""
k = []
do_mask = False
for base in self:
if base.family() == 'fourier':
k.append(base.get_mask_nyquist(bcast=True))
if base.N % 2 == 0:
do_mask = True
else:
k.append(None)
if not do_mask:
return None
lk = []
for axis, (n, s) in enumerate(zip(k, self.local_slice(True))):
if n is None:
lk.append(1)
else:
ss = [slice(None)]*len(k)
ss[axis] = s
lk.append(n[tuple(ss)])
mask = 1
for ll in lk:
mask = mask * ll
return mask
def get_measured_array(self, u):
"""Weigh Array `u` with integral measure
Parameters
----------
u : Array
"""
dx = self.hi.prod()
mesh = self.local_mesh(True)
if dx == 1:
return
sym0 = dx.free_symbols
m = []
for sym in sym0:
j = 'xyzrs'.index(str(sym))
m.append(mesh[j])
xj = sp.lambdify(sym0, dx)(*m)
u *= xj
return u
def __iter__(self):
return iter(self.bases)
@property
def rank(self):
"""Return tensor rank of TensorProductSpace"""
return 0
def __len__(self):
"""Return dimension of TensorProductSpace"""
return len(self.bases)
def num_components(self):
"""Return number of scalar spaces in TensorProductSpace"""
return 1
@property
def dimensions(self):
"""Return dimension of TensorProductSpace"""
return self.__len__()
def get_nonperiodic_axes(self):
"""Return list of axes that are not periodic"""
axes = []
for axis, base in enumerate(self):
if not base.family() == 'fourier':
axes.append(axis)
return axes
@property
def is_orthogonal(self):
ortho = True
for base in self.bases:
ortho *= base.is_orthogonal
return ortho
def get_orthogonal(self):
ortho = []
for base in self.bases:
ortho.append(base.get_orthogonal())
return TensorProductSpace(self.subcomm, ortho, axes=self.axes,
dtype=self.forward.input_array.dtype,
coordinates=self.coors.coordinates)
@property
def is_composite_space(self):
return 0
def __getitem__(self, i):
"""Return instance of base i
Parameters
----------
i : int
"""
return self.bases[i]
def compatible_base(self, space):
"""Return whether space is compatible with self.
Parameters
----------
space : TensorProductSpace
The space compared to
Note
----
Two spaces are deemed compatible if the underlying bases, along each
direction, belong to the same family, has the same number of quadrature
points, and the same quadrature scheme.
"""
compatible = True
for base0, base1 in zip(self.bases, space.bases):
if not hash(base0) == hash(base1):
compatible = False
break
return compatible
class MixedTensorProductSpace(object):
"""Class for composite tensorproductspaces.
The mixed spaces are Cartesian products of TensorproductSpaces or
other MixedTensorProductSpaces.
Parameters
----------
spaces : list
List of spaces
"""
def __init__(self, spaces):
self.spaces = spaces
self.forward = VectorTransform([space.forward for space in spaces])
self.backward = VectorTransform([space.backward for space in spaces])
self.backward_uniform = VectorTransform([space.backward_uniform for space in spaces])
self.scalar_product = VectorTransform([space.scalar_product for space in spaces])
@property
def is_composite_space(self):
return 1
def eval(self, points, coefficients, output_array=None, method=0):
"""Evaluate Function at points, given expansion coefficients
Parameters
----------
points : float or array of floats
coefficients : array
Expansion coefficients
output_array : array, optional
Return array, function values at points
method : int, optional
Chooses implementation. The default 0 is a low-memory cython
version. Using method = 1 leads to a faster cython
implementation that, on the downside, uses more memory.
The final, method = 2, is a python implementation used only
for verification.
"""
if output_array is None:
output_array = np.zeros((len(self.flatten()), points.shape[-1]), dtype=self.forward.input_array.dtype)
for i, space in enumerate(self.flatten()):
output_array.__array__()[i] = space.eval(points, coefficients.__array__()[i], output_array.__array__()[i], method)
return output_array
def convolve(self, a_hat, b_hat, ab_hat):
"""Convolution of a_hat and b_hat
Parameters
----------
a_hat : array
Input array of shape and type as output array from
self.forward, or instance of :class:`.Function`
b_hat : array
Input array of shape and type as output array from
self.forward, or instance of :class:`.Function`
ab_hat : array
Return array of same type and shape as a_hat and b_hat
Note
----
Note that self should have bases with padding for this method to give
a convolution without aliasing. The padding is specified when creating
instances of bases for the TensorProductSpace.
FIXME Efficiency due to allocation
"""
N = list(self.backward.output_array.shape)
a = np.zeros([self.dimensions]+N, dtype=self.backward.output_array.dtype)
b = np.zeros([self.dimensions]+N, dtype=self.backward.output_array.dtype)
a = self.backward(a_hat, a)
b = self.backward(b_hat, b)
ab_hat = self.forward(a*b, ab_hat)
return ab_hat
@property
def dimensions(self):
"""Return dimension of scalar space"""
return self.flatten()[0].dimensions
@property
def rank(self):
"""Return rank of space"""
return None
def dim(self):
"""Return dimension of ``self`` (degrees of freedom)"""
s = 0
for space in self.flatten():