def __str__(self):
"""Return ``str(self)``."""
inner_str = '{}'.format(self.domain)
dtype_str = dtype_repr(self.out_dtype)
if self.field == RealNumbers():
if self.out_dtype == np.dtype('float64'):
pass
else:
inner_str += ', out_dtype={}'.format(dtype_str)
elif self.field == ComplexNumbers():
if self.out_dtype == np.dtype('complex128'):
inner_str += ', field={!r}'.format(self.field)
else:
inner_str += ', out_dtype={}'.format(dtype_str)
else: # different field, name explicitly
inner_str += ', field={!r}'.format(self.field)
inner_str += ', out_dtype={}'.format(dtype_str)
return '{}({})'.format(self.__class__.__name__, inner_str)
def __str__(self):
"""Return ``str(self)``."""
inner_str = '{}'.format(self.domain)
dtype_str = dtype_repr(self.out_dtype)
if self.field == RealNumbers():
if self.out_dtype == np.dtype('float64'):
pass
else:
inner_str += ', out_dtype={}'.format(dtype_str)
elif self.field == ComplexNumbers():
if self.out_dtype == np.dtype('complex128'):
inner_str += ', field={!r}'.format(self.field)
else:
inner_str += ', out_dtype={}'.format(dtype_str)
else: # different field, name explicitly
inner_str += ', field={!r}'.format(self.field)
inner_str += ', out_dtype={}'.format(dtype_str)
return '{}({})'.format(self.__class__.__name__, inner_str)
def fn_impl(request):
"""String with an available `FnBase` implementation name."""
return request.param
ntuples_impl_params = odl.NTUPLES_IMPLS.keys()
ntuples_impl_ids = [" impl = '{}' ".format(p) for p in ntuples_impl_params]
@fixture(scope="module", ids=ntuples_impl_ids, params=ntuples_impl_params)
def ntuples_impl(request):
"""String with an available `NtuplesBase` implementation name."""
return request.param
floating_dtype_params = np.sctypes['float'] + np.sctypes['complex']
floating_dtype_ids = [' dtype = {} '.format(dtype_repr(dt))
for dt in floating_dtype_params]
@fixture(scope="module", ids=floating_dtype_ids, params=floating_dtype_params)
def floating_dtype(request):
"""Floating point (real or complex) dtype."""
return request.param
scalar_dtype_params = (floating_dtype_params +
np.sctypes['int'] +
np.sctypes['uint'])
scalar_dtype_ids = [' dtype = {} '.format(dtype_repr(dt))
for dt in scalar_dtype_params]
def _pyfftw_check_args(arr_in, arr_out, axes, halfcomplex, direction):
"""Raise an error if anything is not ok with in and out."""
if len(set(axes)) != len(axes):
raise ValueError('duplicate axes are not allowed')
if direction == 'forward':
out_shape = list(arr_in.shape)
if halfcomplex:
try:
out_shape[axes[-1]] = arr_in.shape[axes[-1]] // 2 + 1
except IndexError:
raise IndexError('axis index {} out of range for array '
'with {} axes'
''.format(axes[-1], arr_in.ndim))
if arr_out.shape != tuple(out_shape):
raise ValueError('expected output shape {}, got {}'
''.format(tuple(out_shape), arr_out.shape))
if is_real_dtype(arr_in.dtype):
out_dtype = complex_dtype(arr_in.dtype)
elif halfcomplex:
raise ValueError('cannot combine halfcomplex forward transform '
'with complex input')
else:
out_dtype = arr_in.dtype
if arr_out.dtype != out_dtype:
raise ValueError('expected output dtype {}, got {}'
''.format(dtype_repr(out_dtype),
dtype_repr(arr_out.dtype)))
elif direction == 'backward':
in_shape = list(arr_out.shape)
if halfcomplex:
try:
in_shape[axes[-1]] = arr_out.shape[axes[-1]] // 2 + 1
except IndexError as err:
raise IndexError('axis index {} out of range for array '
'with {} axes'
''.format(axes[-1], arr_out.ndim))
if arr_in.shape != tuple(in_shape):
raise ValueError('expected input shape {}, got {}'
''.format(tuple(in_shape), arr_in.shape))
if is_real_dtype(arr_out.dtype):
in_dtype = complex_dtype(arr_out.dtype)
elif halfcomplex:
raise ValueError('cannot combine halfcomplex backward transform '
'with complex output')
else:
in_dtype = arr_out.dtype
if arr_in.dtype != in_dtype:
raise ValueError('expected input dtype {}, got {}'
''.format(dtype_repr(in_dtype),
dtype_repr(arr_in.dtype)))
else: # Shouldn't happen
raise RuntimeError
def __repr__(self):
"""Return ``repr(self)``."""
return '{}({}, {})'.format(self.__class__.__name__, self.size,
dtype_repr(self.dtype))
return request.param
ntuples_impl_params = odl.NTUPLES_IMPLS.keys()
ntuples_impl_ids = [" impl = '{}' ".format(p) for p in ntuples_impl_params]
@fixture(scope="module", ids=ntuples_impl_ids, params=ntuples_impl_params)
def ntuples_impl(request):
"""String with an available `NtuplesBase` implementation name."""
return request.param
floating_dtype_params = np.sctypes['float'] + np.sctypes['complex']
floating_dtype_ids = [
' dtype = {} '.format(dtype_repr(dt)) for dt in floating_dtype_params
]
@fixture(scope="module", ids=floating_dtype_ids, params=floating_dtype_params)
def floating_dtype(request):
"""Floating point (real or complex) dtype."""
return request.param
scalar_dtype_params = (floating_dtype_params + np.sctypes['int'] +
np.sctypes['uint'])
scalar_dtype_ids = [
' dtype = {} '.format(dtype_repr(dt)) for dt in scalar_dtype_params
]
def _pyfftw_check_args(arr_in, arr_out, axes, halfcomplex, direction):
"""Raise an error if anything is not ok with in and out."""
if len(set(axes)) != len(axes):
raise ValueError('duplicate axes are not allowed')
if direction == 'forward':
out_shape = list(arr_in.shape)
if halfcomplex:
try:
out_shape[axes[-1]] = arr_in.shape[axes[-1]] // 2 + 1
except IndexError as err:
raise_from(IndexError('axis index {} out of range for array '
'with {} axes'
''.format(axes[-1], arr_in.ndim)),
err)
if arr_out.shape != tuple(out_shape):
raise ValueError('expected output shape {}, got {}'
''.format(tuple(out_shape), arr_out.shape))
if is_real_dtype(arr_in.dtype):
out_dtype = complex_dtype(arr_in.dtype)
elif halfcomplex:
raise ValueError('cannot combine halfcomplex forward transform '
'with complex input')
else:
out_dtype = arr_in.dtype
if arr_out.dtype != out_dtype:
raise ValueError('expected output dtype {}, got {}'
''.format(dtype_repr(out_dtype),
dtype_repr(arr_out.dtype)))
elif direction == 'backward':
in_shape = list(arr_out.shape)
if halfcomplex:
try:
in_shape[axes[-1]] = arr_out.shape[axes[-1]] // 2 + 1
except IndexError as err:
raise_from(IndexError('axis index {} out of range for array '
'with {} axes'
''.format(axes[-1], arr_out.ndim)),
err)
if arr_in.shape != tuple(in_shape):
raise ValueError('expected input shape {}, got {}'
''.format(tuple(in_shape), arr_in.shape))
if is_real_dtype(arr_out.dtype):
in_dtype = complex_dtype(arr_out.dtype)
elif halfcomplex:
raise ValueError('cannot combine halfcomplex backward transform '
'with complex output')
else:
in_dtype = arr_out.dtype
if arr_in.dtype != in_dtype:
raise ValueError('expected input dtype {}, got {}'
''.format(dtype_repr(in_dtype),
dtype_repr(arr_in.dtype)))
else: # Shouldn't happen
raise RuntimeError
def reciprocal_space(space, axes=None, halfcomplex=False, shift=True,
**kwargs):
"""Return the range of the Fourier transform on ``space``.
Parameters
----------
space : `DiscreteLp`
Real space whose reciprocal is calculated. It must be
uniformly discretized.
axes : sequence of ints, optional
Dimensions along which the Fourier transform is taken.
Default: all axes
halfcomplex : bool, optional
If ``True``, take only the negative frequency part along the last
axis for. For ``False``, use the full frequency space.
This option can only be used if ``space`` is a space of
real-valued functions.
shift : bool or sequence of bools, optional
If ``True``, the reciprocal grid is shifted by half a stride in
the negative direction. With a boolean sequence, this option
is applied separately to each axis.
If a sequence is provided, it must have the same length as
``axes`` if supplied. Note that this must be set to ``True``
in the halved axis in half-complex transforms.
Default: ``True``
impl : string, optional
Implementation back-end for the created space.
Default: ``'numpy'``
exponent : float, optional
Create a space with this exponent. By default, the conjugate
exponent ``q = p / (p - 1)`` of the exponent of ``space`` is
used, where ``q = inf`` for ``p = 1`` and vice versa.
dtype : optional
Complex data type of the created space. By default, the
complex counterpart of ``space.dtype`` is used.
Returns
-------
rspace : `DiscreteLp`
Reciprocal of the input ``space``. If ``halfcomplex=True``, the
upper end of the domain (where the half space ends) is chosen to
coincide with the grid node.
"""
if not isinstance(space, DiscreteLp):
raise TypeError('`space` {!r} is not a `DiscreteLp` instance'
''.format(space))
if axes is None:
axes = tuple(range(space.ndim))
axes = normalized_axes_tuple(axes, space.ndim)
if not all(space.is_uniform_byaxis[axis] for axis in axes):
raise ValueError('`space` is not uniformly discretized in the '
'`axes` of the transform')
if halfcomplex and space.field != RealNumbers():
raise ValueError('`halfcomplex` option can only be used with real '
'spaces')
exponent = kwargs.pop('exponent', None)
if exponent is None:
exponent = conj_exponent(space.exponent)
dtype = kwargs.pop('dtype', None)
if dtype is None:
dtype = complex_dtype(space.dtype)
else:
if not is_complex_floating_dtype(dtype):
raise ValueError('{} is not a complex data type'
''.format(dtype_repr(dtype)))
impl = kwargs.pop('impl', 'numpy')
# Calculate range
recip_grid = reciprocal_grid(space.grid, shift=shift,
halfcomplex=halfcomplex, axes=axes)
# Make a partition with nodes on the boundary in the last transform axis
# if `halfcomplex == True`, otherwise a standard partition.
if halfcomplex:
max_pt = {axes[-1]: recip_grid.max_pt[axes[-1]]}
part = uniform_partition_fromgrid(recip_grid, max_pt=max_pt)
else:
part = uniform_partition_fromgrid(recip_grid)
# Use convention of adding a hat to represent fourier transform of variable
axis_labels = list(space.axis_labels)
for i in axes:
# Avoid double math
label = axis_labels[i].replace('$', '')
axis_labels[i] = '$\^{{{}}}$'.format(label)
recip_spc = uniform_discr_frompartition(part, exponent=exponent,
dtype=dtype, impl=impl,
axis_labels=axis_labels)
return recip_spc
def dft_postprocess_data(arr, real_grid, recip_grid, shift, axes,
interp, sign='-', op='multiply', out=None):
"""Post-process the Fourier-space data after DFT.
This function multiplies the given data with the separable
function::
q(xi) = exp(+- 1j * dot(x[0], xi)) * s * phi_hat(xi_bar)
where ``x[0]`` and ``s`` are the minimum point and the stride of
the real-space grid, respectively, and ``phi_hat(xi_bar)`` is the FT
of the interpolation kernel. The sign of the exponent depends on the
choice of ``sign``. Note that for ``op='divide'`` the
multiplication with ``s * phi_hat(xi_bar)`` is replaced by a
division with the same array.
In discretized form on the reciprocal grid, the exponential part
of this function becomes an array::
q[k] = exp(+- 1j * dot(x[0], xi[k]))
and the arguments ``xi_bar`` to the interpolation kernel
are the normalized frequencies::
for 'shift=True' : xi_bar[k] = -pi + pi * (2*k) / N
for 'shift=False' : xi_bar[k] = -pi + pi * (2*k+1) / N
See [Pre+2007], Section 13.9 "Computing Fourier Integrals Using
the FFT" for a similar approach.
Parameters
----------
arr : `array-like`
Array to be pre-processed. An array with real data type is
converted to its complex counterpart.
real_grid : uniform `RectGrid`
Real space grid in the transform.
recip_grid : uniform `RectGrid`
Reciprocal grid in the transform
shift : bool or sequence of bools
If ``True``, the grid is shifted by half a stride in the negative
direction in the corresponding axes. The sequence must have the
same length as ``axes``.
axes : int or sequence of ints
Dimensions along which to take the transform. The sequence must
have the same length as ``shifts``.
interp : string or sequence of strings
Interpolation scheme used in the real-space.
sign : {'-', '+'}, optional
Sign of the complex exponent.
op : {'multiply', 'divide'}, optional
Operation to perform with the stride times the interpolation
kernel FT
out : `numpy.ndarray`, optional
Array in which the result is stored. If ``out is arr``, an
in-place modification is performed.
Returns
-------
out : `numpy.ndarray`
Result of the post-processing. If ``out`` was given, the returned
object is a reference to it.
References
----------
[Pre+2007] Press, W H, Teukolsky, S A, Vetterling, W T, and Flannery, B P.
*Numerical Recipes in C - The Art of Scientific Computing* (Volume 3).
Cambridge University Press, 2007.
"""
arr = np.asarray(arr)
if is_real_floating_dtype(arr.dtype):
arr = arr.astype(complex_dtype(arr.dtype))
elif not is_complex_floating_dtype(arr.dtype):
raise ValueError('array data type {} is not a complex floating point '
'data type'.format(dtype_repr(arr.dtype)))
if out is None:
out = arr.copy()
elif out is not arr:
out[:] = arr
if axes is None:
axes = list(range(arr.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
shift_list = normalized_scalar_param_list(shift, length=len(axes),
param_conv=bool)
if sign == '-':
imag = -1j
elif sign == '+':
imag = 1j
else:
raise ValueError("`sign` '{}' not understood".format(sign))
op, op_in = str(op).lower(), op
if op not in ('multiply', 'divide'):
raise ValueError("kernel `op` '{}' not understood".format(op_in))
# Make a list from interp if that's not the case already
try:
# Duck-typed string check
interp + ''
except TypeError:
pass
else:
interp = [str(interp).lower()] * arr.ndim
onedim_arrs = []
for ax, shift, intp in zip(axes, shift_list, interp):
x = real_grid.min_pt[ax]
xi = recip_grid.coord_vectors[ax]
# First part: exponential array
onedim_arr = np.exp(imag * x * xi)
# Second part: interpolation kernel
len_dft = recip_grid.shape[ax]
len_orig = real_grid.shape[ax]
halfcomplex = (len_dft < len_orig)
odd = len_orig % 2
fmin = -0.5 if shift else -0.5 + 1.0 / (2 * len_orig)
if halfcomplex:
# maximum lies around 0, possibly half a cell left or right of it
if shift and odd:
fmax = - 1.0 / (2 * len_orig)
elif not shift and not odd:
fmax = 1.0 / (2 * len_orig)
else:
fmax = 0.0
else: # not halfcomplex
# maximum lies close to 0.5, half or full cell left of it
if shift:
# -0.5 + (N-1)/N = 0.5 - 1/N
fmax = 0.5 - 1.0 / len_orig
else:
# -0.5 + 1/(2*N) + (N-1)/N = 0.5 - 1/(2*N)
fmax = 0.5 - 1.0 / (2 * len_orig)
freqs = np.linspace(fmin, fmax, num=len_dft)
stride = real_grid.stride[ax]
interp_kernel = _interp_kernel_ft(freqs, intp)
interp_kernel *= stride
if op == 'multiply':
onedim_arr *= interp_kernel
else:
onedim_arr /= interp_kernel
onedim_arrs.append(onedim_arr.astype(out.dtype, copy=False))
fast_1d_tensor_mult(out, onedim_arrs, axes=axes, out=out)
return out
def dft_preprocess_data(arr, shift=True, axes=None, sign='-', out=None):
"""Pre-process the real-space data before DFT.
This function multiplies the given data with the separable
function::
p(x) = exp(+- 1j * dot(x - x[0], xi[0]))
where ``x[0]`` and ``xi[0]`` are the minimum coodinates of
the real-space and reciprocal grids, respectively. The sign of
the exponent depends on the choice of ``sign``. In discretized
form, this function becomes an array::
p[k] = exp(+- 1j * k * s * xi[0])
If the reciprocal grid is not shifted, i.e. symmetric around 0,
it is ``xi[0] = pi/s * (-1 + 1/N)``, hence::
p[k] = exp(-+ 1j * pi * k * (1 - 1/N))
For a shifted grid, we have :math:``xi[0] = -pi/s``, thus the
array is given by::
p[k] = (-1)**k
Parameters
----------
arr : `array-like`
Array to be pre-processed. If its data type is a real
non-floating type, it is converted to 'float64'.
shift : bool or or sequence of bools, optional
If ``True``, the grid is shifted by half a stride in the negative
direction. With a sequence, this option is applied separately on
each axis.
axes : int or sequence of ints, optional
Dimensions in which to calculate the reciprocal. The sequence
must have the same length as ``shift`` if the latter is given
as a sequence.
Default: all axes.
sign : {'-', '+'}, optional
Sign of the complex exponent.
out : `numpy.ndarray`, optional
Array in which the result is stored. If ``out is arr``,
an in-place modification is performed. For real data type,
this is only possible for ``shift=True`` since the factors are
complex otherwise.
Returns
-------
out : `numpy.ndarray`
Result of the pre-processing. If ``out`` was given, the returned
object is a reference to it.
Notes
-----
If ``out`` is not specified, the data type of the returned array
is the same as that of ``arr`` except when ``arr`` has real data
type and ``shift`` is not ``True``. In this case, the return type
is the complex counterpart of ``arr.dtype``.
"""
arr = np.asarray(arr)
if not is_scalar_dtype(arr.dtype):
raise ValueError('array has non-scalar data type {}'
''.format(dtype_repr(arr.dtype)))
elif is_real_dtype(arr.dtype) and not is_real_floating_dtype(arr.dtype):
arr = arr.astype('float64')
if axes is None:
axes = list(range(arr.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
shape = arr.shape
shift_list = normalized_scalar_param_list(shift, length=len(axes),
param_conv=bool)
# Make a copy of arr with correct data type if necessary, or copy values.
if out is None:
if is_real_dtype(arr.dtype) and not all(shift_list):
out = np.array(arr, dtype=complex_dtype(arr.dtype), copy=True)
else:
out = arr.copy()
else:
out[:] = arr
if is_real_dtype(out.dtype) and not shift:
raise ValueError('cannot pre-process real input in-place without '
'shift')
if sign == '-':
imag = -1j
elif sign == '+':
imag = 1j
else:
raise ValueError("`sign` '{}' not understood".format(sign))
def _onedim_arr(length, shift):
if shift:
# (-1)^indices
factor = np.ones(length, dtype=out.dtype)
factor[1::2] = -1
else:
factor = np.arange(length, dtype=out.dtype)
factor *= -imag * np.pi * (1 - 1.0 / length)
np.exp(factor, out=factor)
return factor.astype(out.dtype, copy=False)
onedim_arrs = []
for axis, shift in zip(axes, shift_list):
length = shape[axis]
onedim_arrs.append(_onedim_arr(length, shift))
fast_1d_tensor_mult(out, onedim_arrs, axes=axes, out=out)
return out
def __repr__(self):
"""Return ``repr(self)``."""
return '{}({}, {})'.format(self.__class__.__name__, self.size,
dtype_repr(self.dtype))
def reduce_over_partition(discr_func,
partition,
reduction,
pad_const=0,
out=None):
"""Reduce a discrete function blockwise over a coarser partition.
This helper function is intended as a helper for multi-grid
computations where a finely discretized function needs to undergo
a blockwise reduction operation over a coarser partition of a
containing spatial region. An example is to average the given
function over larger blocks as defined by the partition.
Parameters
----------
discr_func : `DiscreteLpElement`
Element in a uniformly discretized function space that is to be
reduced over blocks defined by ``partition``.
partition : uniform `RectPartition`
Coarser partition than ``discr_func.space.partition`` that defines
the large cells (blocks) over which ``discr_func`` is reduced.
Its ``cell_sides`` must be an integer multiple of
``discr_func.space.cell_sides``.
reduction : callable
Reduction function defining the operation on each block of values
in ``discr_func``. It needs to be callable as
``reduction(array, axes=my_axes)`` or
``reduction(array, axes=my_axes, out=out_array)``, where
``array, out_array`` are `numpy.ndarray`'s, and ``my_axes`` are
sequence of ints specifying over which axes is being reduced.
The typical examples are NumPy reductions like `np.sum` or `np.mean`,
but custom functions are also possible.
pad_const : scalar, optional
This value is filled into the parts that are not covered by the
function.
out : `numpy.ndarray`, optional
Array to which the output is written. It needs to have the same
``shape`` as ``partition`` and a ``dtype`` to which
``discr_func.dtype`` can be cast.
Returns
-------
out : `numpy.ndarray`
Array holding the result of the reduction operation. If ``out``
was given, the returned object is a reference to it.
Examples
--------
Consider a simple 1D example with 4 small cells per large cell,
and summing a constant function over the large cells:
>>> partition = odl.uniform_partition(0, 1, 5)
>>> partition.cell_boundary_vecs
(array([ 0. , 0.2, 0.4, 0.6, 0.8, 1. ]),)
>>> space = odl.uniform_discr(0, 0.5, 10) # 0.5 falls between
>>> func = space.one()
>>> reduce_over_partition(func, partition, reduction=np.sum)
array([ 4., 4., 2., 0., 0.])
>>> # The value 4 is due to summing 4 ones from 4 small cells per
>>> # large cell.
>>> # The "2" in the third cell is expected since it only counts half --
>>> # the overlap of func.domain is only half a cell ([0.4, 0.5]).
In 2D, everything (including partial overlap weighting) works per
axis:
>>> partition = odl.uniform_partition([0, 0], [1, 1], [5, 5])
>>> space = odl.uniform_discr([0, 0], [0.5, 0.7], [10, 14])
>>> func = space.one()
>>> reduce_over_partition(func, partition, reduction=np.sum)
array([[ 16., 16., 16., 8., 0.],
[ 16., 16., 16., 8., 0.],
[ 8., 8., 8., 4., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.]])
>>> # 16 = sum of 16 ones from 4 x 4 small cells per large cell
>>> # 8: cells have half weight due to half overlap
>>> # 4: the corner cell overlaps half in both axes, i.e. 1/4 in
>>> # total
"""
if not isinstance(discr_func, DiscreteLpElement):
raise TypeError('`discr_func` must be a `DiscreteLpElement` instance, '
'got {!r}'.format(discr_func))
if not discr_func.space.is_uniform:
raise ValueError('`discr_func.space` is not uniformly discretized')
if not isinstance(partition, RectPartition):
raise TypeError('`partition` must be a `RectPartition` instance, '
'got {!r}'.format(partition))
if not partition.is_uniform:
raise ValueError('`partition` is not uniform')
# TODO: use different eps in each axis?
dom_eps = 1e-8 * max(discr_func.space.partition.extent)
if not partition.set.contains_set(discr_func.space.domain, atol=dom_eps):
raise ValueError('`partition.set` {} does not contain '
'`discr_func.space.domain` {}'
''.format(partition.set, discr_func.space.domain))
if out is None:
out = np.empty(partition.shape,
dtype=discr_func.dtype,
order=discr_func.dtype)
if not isinstance(out, np.ndarray):
raise TypeError('`out` must be a `numpy.ndarray` instance, got '
'{!r}'.format(out))
if not np.can_cast(discr_func.dtype, out.dtype):
raise ValueError('cannot safely cast from `discr_func.dtype` {} '
'to `out.dtype` {}'
''.format(dtype_repr(discr_func.dtype),
dtype_repr(out.dtype)))
if not np.array_equal(out.shape, partition.shape):
raise ValueError('`out.shape` differs from `partition.shape` '
'({} != {})'.format(out.shape, partition.shape))
if not np.can_cast(pad_const, out.dtype):
raise ValueError('cannot safely cast `pad_const` {} '
'to `out.dtype` {}'
''.format(pad_const, dtype_repr(out.dtype)))
out.fill(pad_const)
# Some abbreviations for easier notation
# All variables starting with "s" refer to properties of
# `discr_func.space`, whereas "p" quantities refer to the (coarse)
# `partition`.
spc = discr_func.space
smin, smax = spc.min_pt, spc.max_pt
scsides = spc.cell_sides
part = partition
pmin = part.min_pt, part.max_pt
func_arr = discr_func.asarray()
ndim = spc.ndim
# Partition cell sides must be an integer multiple of space cell sides
csides_ratio_f = part.cell_sides / spc.cell_sides
csides_ratio = np.around(csides_ratio_f).astype(int)
if not np.allclose(csides_ratio_f, csides_ratio):
raise ValueError('`partition.cell_sides` is a non-integer multiple '
'({}) of `discr_func.space.cell_sides'
''.format(csides_ratio_f))
# Shift must be an integer multiple of space cell sides
rel_shift_f = (smin - pmin) / scsides
if not np.allclose(np.round(rel_shift_f), rel_shift_f):
raise ValueError('shift between `partition` and `discr_func.space` '
'is a non-integer multiple ({}) of '
'`discr_func.space.cell_sides'
''.format(rel_shift_f))
# Calculate relative position of a number of interesting points
# Positions of the space domain min and max vectors relative to the
# partition
cvecs = part.cell_boundary_vecs
smin_idx = np.array(part.index(smin), ndmin=1)
smin_partpt = np.array([cvec[si + 1] for si, cvec in zip(smin_idx, cvecs)])
smax_idx = np.array(part.index(smax), ndmin=1)
smax_partpt = np.array([cvec[si] for si, cvec in zip(smax_idx, cvecs)])
# Inner part of the partition in the space domain, i.e. partition cells
# that are completely contained in the spatial domain and do not touch
# its boundary
p_inner_slc = [slice(li + 1, ri) for li, ri in zip(smin_idx, smax_idx)]
# Positions of the first and last partition points that still lie in
# the spatial domain, relative to the space partition
pl_idx = np.array(np.round(spc.partition.index(smin_partpt,
floating=True)).astype(int),
ndmin=1)
pr_idx = np.array(np.round(spc.partition.index(smax_partpt,
floating=True)).astype(int),
ndmin=1)
s_inner_slc = [slice(li, ri) for li, ri in zip(pl_idx, pr_idx)]
# Slices to constrain to left and right boundary in each axis
pl_slc = [slice(li, li + 1) for li in smin_idx]
pr_slc = [slice(ri, ri + 1) for ri in smax_idx]
# Slices for the overlapping space cells to the left and the right
# (up to left index excl. / from right index incl.)
sl_slc = [slice(None, li) for li in pl_idx]
sr_slc = [slice(ri, None) for ri in pr_idx]
# Shapes for reduction of the inner part by summing over axes.
reduce_inner_shape = []
reduce_axes = tuple(2 * i + 1 for i in range(ndim))
inner_shape = func_arr[s_inner_slc].shape
for n, k in zip(inner_shape, csides_ratio):
reduce_inner_shape.extend([n // k, k])
# Now we loop over boundary parts of all dimensions from 0 to ndim-1.
# They are encoded as follows:
# - We select inner (1) and outer (2) parts per axis by looping over
# `product([1, 2], repeat=ndim)`, using the name `parts`.
# - Wherever there is a 2 in the sequence, 2 slices must be generated,
# one for left and one for right. The total number of slices is the
# product of the numbers in `parts`, i.e. `num_slcs = prod(parts)`.
# - We get the indices of the 2's in the sequence and put them in
# `outer_indcs`.
# - The "p" and "s" slice lists are initialized with the inner parts.
# We need `num_slcs` such lists for this particular sequence `parts`.
# - Now we enumerate `outer_indcs` as `i, oi` and put into the
# (2*i)-th entry of the slice lists the "left" outer slice and into
# the (2*i+1)-th entry the "right" outer slice.
#
# The total number of slices to loop over is equal to
# sum(k=0->ndim, binom(ndim, k) * 2^k) = 3^ndim.
# This should not add too much computational overhead.
for parts in product([1, 2], repeat=ndim):
# Number of slices to consider
num_slcs = np.prod(parts)
# Indices where we need to consider the outer parts
outer_indcs = tuple(np.where(np.equal(parts, 2))[0])
# Initialize the "p" and "s" slice lists with the inner slices.
# Each list contains `num_slcs` of those.
p_slcs = [list(p_inner_slc) for _ in range(num_slcs)]
s_slcs = [list(s_inner_slc) for _ in range(num_slcs)]
# Put the left/right slice in the even/odd sublists at the
# position indexed by the outer_indcs thing.
# We also need to initialize the `reduce_shape`'s for all cases,
# which has the value (n // k, k) for the "inner" axes and
# (1, n) in the "outer" axes.
reduce_shapes = [list(reduce_inner_shape) for _ in range(num_slcs)]
for islc, bdry in enumerate(product('lr', repeat=len(outer_indcs))):
for oi, l_or_r in zip(outer_indcs, bdry):
if l_or_r == 'l':
p_slcs[islc][oi] = pl_slc[oi]
s_slcs[islc][oi] = sl_slc[oi]
else:
p_slcs[islc][oi] = pr_slc[oi]
s_slcs[islc][oi] = sr_slc[oi]
f_view = func_arr[s_slcs[islc]]
for oi in outer_indcs:
reduce_shapes[islc][2 * oi] = 1
reduce_shapes[islc][2 * oi + 1] = f_view.shape[oi]
# Compute the block reduction of all views represented by the current
# `parts`. This is done by reshaping from the original shape to the
# above calculated `reduce_shapes` and reducing over `reduce_axes`.
for p_s, s_s, red_shp in zip(p_slcs, s_slcs, reduce_shapes):
f_view = func_arr[s_s]
out_view = out[p_s]
if 0 not in f_view.shape:
# View not empty, reduction makes sense
_apply_reduction(arr=f_view.reshape(red_shp),
out=out_view,
axes=reduce_axes,
reduction=reduction)
return out
def reciprocal_space(space, axes=None, halfcomplex=False, shift=True,
**kwargs):
"""Return the range of the Fourier transform on ``space``.
Parameters
----------
space : `DiscreteLp`
Real space whose reciprocal is calculated. It must be
uniformly discretized.
axes : sequence of ints, optional
Dimensions along which the Fourier transform is taken.
Default: all axes
halfcomplex : bool, optional
If ``True``, take only the negative frequency part along the last
axis for. For ``False``, use the full frequency space.
This option can only be used if ``space`` is a space of
real-valued functions.
shift : bool or sequence of bools, optional
If ``True``, the reciprocal grid is shifted by half a stride in
the negative direction. With a boolean sequence, this option
is applied separately to each axis.
If a sequence is provided, it must have the same length as
``axes`` if supplied. Note that this must be set to ``True``
in the halved axis in half-complex transforms.
Default: ``True``
impl : string, optional
Implementation back-end for the created space.
Default: ``'numpy'``
exponent : float, optional
Create a space with this exponent. By default, the conjugate
exponent ``q = p / (p - 1)`` of the exponent of ``space`` is
used, where ``q = inf`` for ``p = 1`` and vice versa.
dtype : optional
Complex data type of the created space. By default, the
complex counterpart of ``space.dtype`` is used.
Returns
-------
rspace : `DiscreteLp`
Reciprocal of the input ``space``. If ``halfcomplex=True``, the
upper end of the domain (where the half space ends) is chosen to
coincide with the grid node.
"""
if not isinstance(space, DiscreteLp):
raise TypeError('`space` {!r} is not a `DiscreteLp` instance'
''.format(space))
if not space.is_uniform:
raise ValueError('`space` is not uniformly discretized')
if axes is None:
axes = tuple(range(space.ndim))
axes = normalized_axes_tuple(axes, space.ndim)
if halfcomplex and space.field != RealNumbers():
raise ValueError('`halfcomplex` option can only be used with real '
'spaces')
exponent = kwargs.pop('exponent', None)
if exponent is None:
exponent = conj_exponent(space.exponent)
dtype = kwargs.pop('dtype', None)
if dtype is None:
dtype = complex_dtype(space.dtype)
else:
if not is_complex_floating_dtype(dtype):
raise ValueError('{} is not a complex data type'
''.format(dtype_repr(dtype)))
impl = kwargs.pop('impl', 'numpy')
# Calculate range
recip_grid = reciprocal_grid(space.grid, shift=shift,
halfcomplex=halfcomplex, axes=axes)
# Make a partition with nodes on the boundary in the last transform axis
# if `halfcomplex == True`, otherwise a standard partition.
if halfcomplex:
max_pt = {axes[-1]: recip_grid.max_pt[axes[-1]]}
part = uniform_partition_fromgrid(recip_grid, max_pt=max_pt)
else:
part = uniform_partition_fromgrid(recip_grid)
# Use convention of adding a hat to represent fourier transform of variable
axis_labels = list(space.axis_labels)
for i in axes:
# Avoid double math
label = axis_labels[i].replace('$', '')
axis_labels[i] = '$\^{{{}}}$'.format(label)
recip_spc = uniform_discr_frompartition(part, exponent=exponent,
dtype=dtype, impl=impl,
axis_labels=axis_labels)
return recip_spc
def dft_postprocess_data(arr, real_grid, recip_grid, shift, axes,
interp, sign='-', op='multiply', out=None):
"""Post-process the Fourier-space data after DFT.
This function multiplies the given data with the separable
function::
q(xi) = exp(+- 1j * dot(x[0], xi)) * s * phi_hat(xi_bar)
where ``x[0]`` and ``s`` are the minimum point and the stride of
the real-space grid, respectively, and ``phi_hat(xi_bar)`` is the FT
of the interpolation kernel. The sign of the exponent depends on the
choice of ``sign``. Note that for ``op='divide'`` the
multiplication with ``s * phi_hat(xi_bar)`` is replaced by a
division with the same array.
In discretized form on the reciprocal grid, the exponential part
of this function becomes an array::
q[k] = exp(+- 1j * dot(x[0], xi[k]))
and the arguments ``xi_bar`` to the interpolation kernel
are the normalized frequencies::
for 'shift=True' : xi_bar[k] = -pi + pi * (2*k) / N
for 'shift=False' : xi_bar[k] = -pi + pi * (2*k+1) / N
See [Pre+2007]_, Section 13.9 "Computing Fourier Integrals Using
the FFT" for a similar approach.
Parameters
----------
arr : `array-like`
Array to be pre-processed. An array with real data type is
converted to its complex counterpart.
real_grid : `RegularGrid`
Real space grid in the transform
recip_grid : `RegularGrid`
Reciprocal grid in the transform
shift : bool or sequence of bools
If ``True``, the grid is shifted by half a stride in the negative
direction in the corresponding axes. The sequence must have the
same length as ``axes``.
axes : int or sequence of ints
Dimensions along which to take the transform. The sequence must
have the same length as ``shifts``.
interp : string or sequence of strings
Interpolation scheme used in the real-space.
sign : {'-', '+'}, optional
Sign of the complex exponent.
op : {'multiply', 'divide'}
Operation to perform with the stride times the interpolation
kernel FT
out : `numpy.ndarray`, optional
Array in which the result is stored. If ``out is arr``, an
in-place modification is performed.
Returns
-------
out : `numpy.ndarray`
Result of the post-processing. If ``out`` was given, the returned
object is a reference to it.
"""
arr = np.asarray(arr)
if is_real_floating_dtype(arr.dtype):
arr = arr.astype(complex_dtype(arr.dtype))
elif not is_complex_floating_dtype(arr.dtype):
raise ValueError('array data type {} is not a complex floating point '
'data type'.format(dtype_repr(arr.dtype)))
if out is None:
out = arr.copy()
elif out is not arr:
out[:] = arr
if axes is None:
axes = list(range(arr.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
shift_list = normalized_scalar_param_list(shift, length=len(axes),
param_conv=bool)
if sign == '-':
imag = -1j
elif sign == '+':
imag = 1j
else:
raise ValueError("`sign` '{}' not understood".format(sign))
op, op_in = str(op).lower(), op
if op not in ('multiply', 'divide'):
raise ValueError("kernel `op` '{}' not understood".format(op_in))
# Make a list from interp if that's not the case already
try:
# Duck-typed string check
interp + ''
except TypeError:
pass
else:
interp = [str(interp).lower()] * arr.ndim
onedim_arrs = []
for ax, shift, intp in zip(axes, shift_list, interp):
x = real_grid.min_pt[ax]
xi = recip_grid.coord_vectors[ax]
# First part: exponential array
onedim_arr = np.exp(imag * x * xi)
# Second part: interpolation kernel
len_dft = recip_grid.shape[ax]
len_orig = real_grid.shape[ax]
halfcomplex = (len_dft < len_orig)
odd = len_orig % 2
fmin = -0.5 if shift else -0.5 + 1.0 / (2 * len_orig)
if halfcomplex:
# maximum lies around 0, possibly half a cell left or right of it
if shift and odd:
fmax = - 1.0 / (2 * len_orig)
elif not shift and not odd:
fmax = 1.0 / (2 * len_orig)
else:
fmax = 0.0
else: # not halfcomplex
# maximum lies close to 0.5, half or full cell left of it
if shift:
# -0.5 + (N-1)/N = 0.5 - 1/N
fmax = 0.5 - 1.0 / len_orig
else:
# -0.5 + 1/(2*N) + (N-1)/N = 0.5 - 1/(2*N)
fmax = 0.5 - 1.0 / (2 * len_orig)
freqs = np.linspace(fmin, fmax, num=len_dft)
stride = real_grid.stride[ax]
if op == 'multiply':
onedim_arr *= stride * _interp_kernel_ft(freqs, intp)
else:
onedim_arr /= stride * _interp_kernel_ft(freqs, intp)
onedim_arrs.append(onedim_arr.astype(out.dtype, copy=False))
fast_1d_tensor_mult(out, onedim_arrs, axes=axes, out=out)
return out
def dft_preprocess_data(arr, shift=True, axes=None, sign='-', out=None):
"""Pre-process the real-space data before DFT.
This function multiplies the given data with the separable
function::
p(x) = exp(+- 1j * dot(x - x[0], xi[0]))
where ``x[0]`` and ``xi[0]`` are the minimum coodinates of
the real-space and reciprocal grids, respectively. The sign of
the exponent depends on the choice of ``sign``. In discretized
form, this function becomes an array::
p[k] = exp(+- 1j * k * s * xi[0])
If the reciprocal grid is not shifted, i.e. symmetric around 0,
it is ``xi[0] = pi/s * (-1 + 1/N)``, hence::
p[k] = exp(-+ 1j * pi * k * (1 - 1/N))
For a shifted grid, we have :math:``xi[0] = -pi/s``, thus the
array is given by::
p[k] = (-1)**k
Parameters
----------
arr : `array-like`
Array to be pre-processed. If its data type is a real
non-floating type, it is converted to 'float64'.
shift : bool or or sequence of bools, optional
If ``True``, the grid is shifted by half a stride in the negative
direction. With a sequence, this option is applied separately on
each axis.
axes : int or sequence of ints, optional
Dimensions in which to calculate the reciprocal. The sequence
must have the same length as ``shift`` if the latter is given
as a sequence.
Default: all axes.
sign : {'-', '+'}, optional
Sign of the complex exponent.
out : `numpy.ndarray`, optional
Array in which the result is stored. If ``out is arr``,
an in-place modification is performed. For real data type,
this is only possible for ``shift=True`` since the factors are
complex otherwise.
Returns
-------
out : `numpy.ndarray`
Result of the pre-processing. If ``out`` was given, the returned
object is a reference to it.
Notes
-----
If ``out`` is not specified, the data type of the returned array
is the same as that of ``arr`` except when ``arr`` has real data
type and ``shift`` is not ``True``. In this case, the return type
is the complex counterpart of ``arr.dtype``.
"""
arr = np.asarray(arr)
if not is_scalar_dtype(arr.dtype):
raise ValueError('array has non-scalar data type {}'
''.format(dtype_repr(arr.dtype)))
elif is_real_dtype(arr.dtype) and not is_real_floating_dtype(arr.dtype):
arr = arr.astype('float64')
if axes is None:
axes = list(range(arr.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
shape = arr.shape
shift_list = normalized_scalar_param_list(shift, length=len(axes),
param_conv=bool)
# Make a copy of arr with correct data type if necessary, or copy values.
if out is None:
if is_real_dtype(arr.dtype) and not all(shift_list):
out = np.array(arr, dtype=complex_dtype(arr.dtype), copy=True)
else:
out = arr.copy()
else:
out[:] = arr
if is_real_dtype(out.dtype) and not shift:
raise ValueError('cannot pre-process real input in-place without '
'shift')
if sign == '-':
imag = -1j
elif sign == '+':
imag = 1j
else:
raise ValueError("`sign` '{}' not understood".format(sign))
def _onedim_arr(length, shift):
if shift:
# (-1)^indices
factor = np.ones(length, dtype=out.dtype)
factor[1::2] = -1
else:
factor = np.arange(length, dtype=out.dtype)
factor *= -imag * np.pi * (1 - 1.0 / length)
np.exp(factor, out=factor)
return factor.astype(out.dtype, copy=False)
onedim_arrs = []
for axis, shift in zip(axes, shift_list):
length = shape[axis]
onedim_arrs.append(_onedim_arr(length, shift))
fast_1d_tensor_mult(out, onedim_arrs, axes=axes, out=out)
return out
def __init__(self, map_type, fspace, partition, tspace, linear=False):
"""Initialize a new instance.
Parameters
----------
map_type : {'sampling', 'interpolation'}
The type of operator
fspace : `FunctionSpace`
The non-discretized (abstract) set of functions to be
discretized
partition : `RectPartition`
Partition of (a subset of) ``fspace.domain`` based on a
`RectGrid`.
tspace : `TensorSpace`
Space providing containers for the values/coefficients of a
discretized object. Its `TensorSpace.shape` must be equal
to ``partition.shape``.
linear : bool, optional
Create a linear operator if ``True``, otherwise a non-linear
operator.
"""
map_type, map_type_in = str(map_type).lower(), map_type
if map_type not in ('sampling', 'interpolation'):
raise ValueError('`map_type` {!r} not understood'
''.format(map_type_in))
if not isinstance(fspace, FunctionSpace):
raise TypeError('`fspace` {!r} is not a `FunctionSpace` '
'instance'.format(fspace))
if not isinstance(partition, RectPartition):
raise TypeError('`partition` {!r} is not a `RectPartition` '
'instance'.format(partition))
if not isinstance(tspace, TensorSpace):
raise TypeError('`tspace` {!r} is not a `TensorSpace` instance'
''.format(tspace))
if not fspace.domain.contains_set(partition):
raise ValueError('{} not contained in the domain {} '
'of the function set {}'
''.format(partition, fspace.domain, fspace))
if tspace.shape != partition.shape:
raise ValueError('`tspace.shape` not equal to `partition.shape`: '
'{} != {}'
''.format(tspace.shape, partition.shape))
domain = fspace if map_type == 'sampling' else tspace
range = tspace if map_type == 'sampling' else fspace
super(FunctionSpaceMapping, self).__init__(domain,
range,
linear=linear)
self.__partition = partition
if self.is_linear:
if self.domain.field is None:
raise TypeError('`fspace.field` cannot be `None` for '
'`linear=True`')
if not is_numeric_dtype(tspace.dtype):
raise TypeError('`tspace.dtype` must be a numeric data type '
'for `linear=True`, got {}'
''.format(dtype_repr(tspace)))
if fspace.field != tspace.field:
raise ValueError('`fspace.field` not equal to `tspace.field`: '
'{} != {}'
''.format(fspace.field, tspace.field))