def to_complex_field(self):
"""
Return the ComplexField stored on disk.
.. note::
The mesh stored on disk must be stored with ``mode=complex``
Returns
-------
real : pmesh.pm.ComplexField
an array-like object holding the mesh loaded from disk in Fourier
space
"""
if not self.isfourier:
return NotImplemented
pmread = self.pm
if self.comm.rank == 0:
self.logger.info("reading complex field from %s" % self.path)
with FileMPI(comm=self.comm, filename=self.path)[self.dataset] as ds:
complex2 = ComplexField(pmread)
assert self.comm.allreduce(complex2.size) == ds.size
start = numpy.sum(self.comm.allgather(complex2.size)[:self.comm.rank], dtype='intp')
end = start + complex2.size
complex2.unravel(ds[start:end])
return complex2
def read(self, real):
import bigfile
if self.comm.rank == 0:
self.logger.info("Reading from Nmesh = %d to Nmesh = %d" %(self.Nmesh, real.Nmesh[0]))
if any(real.Nmesh != self.Nmesh):
pmread = ParticleMesh(BoxSize=real.BoxSize, Nmesh=(self.Nmesh, self.Nmesh, self.Nmesh),
dtype='f4', comm=self.comm)
else:
pmread = real.pm
f = bigfile.BigFileMPI(self.comm, self.path)
with f[self.dataset] as ds:
if self.isfourier:
if self.comm.rank == 0:
self.logger.info("reading complex field")
complex2 = ComplexField(pmread)
assert self.comm.allreduce(complex2.size) == ds.size
start = sum(self.comm.allgather(complex2.size)[:self.comm.rank])
end = start + complex2.size
complex2.unsort(ds[start:end])
complex2.resample(real)
else:
if self.comm.rank == 0:
self.logger.info("reading real field")
real2 = RealField(pmread)
start = sum(self.comm.allgather(real2.size)[:self.comm.rank])
end = start + real2.size
real2.unsort(ds[start:end])
real2.resample(real)
def to_complex_field(self):
"""
Return the ComplexField stored on disk.
.. note::
The mesh stored on disk must be stored with ``mode=complex``
Returns
-------
real : pmesh.pm.ComplexField
an array-like object holding the mesh loaded from disk in Fourier
space
"""
if not self.isfourier:
return NotImplemented
pmread = self.pm
if self.comm.rank == 0:
self.logger.info("reading complex field from %s" % self.path)
with BigFileMPI(comm=self.comm,
filename=self.path)[self.dataset] as ds:
complex2 = ComplexField(pmread)
assert self.comm.allreduce(complex2.size) == ds.size
start = sum(self.comm.allgather(complex2.size)[:self.comm.rank])
end = start + complex2.size
complex2.unsort(ds[start:end])
return complex2
def test_c2r_vjp(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
real = pm.generate_whitenoise(1234, type='real', mean=1.0)
comp = real.r2c()
def objective(comp):
real = comp.c2r()
obj = (real.value ** 2).sum()
return comm.allreduce(obj)
grad_real = RealField(pm)
grad_real[...] = real[...] * 2
grad_comp = ComplexField(pm)
grad_comp = grad_real.c2r_vjp(grad_real)
grad_comp.decompress_vjp(grad_comp)
ng = []
ag = []
ind = []
dx = 1e-7
for ind1 in numpy.ndindex(*(list(grad_comp.cshape) + [2])):
dx1, c1 = perturb(comp, ind1, dx)
ng1 = (objective(c1) - objective(comp)) / dx
ag1 = grad_comp.cgetitem(ind1) * dx1 / dx
comm.barrier()
ng.append(ng1)
ag.append(ag1)
ind.append(ind1)
assert_allclose(ng, ag, rtol=1e-5)
def run(self):
"""
Run the algorithm, which computes and returns the grid in C_CONTIGUOUS order partitioned by ranks.
"""
from nbodykit import measurestats
if self.comm.rank == 0:
self.logger.info("importing done")
self.logger.info("Resolution Nmesh : %d" % self.paintNmesh)
self.logger.info("paintbrush : %s" % self.painter.paintbrush)
# setup the particle mesh object, taking BoxSize from the painters
pmpaint = ParticleMesh(BoxSize=self.datasource.BoxSize, Nmesh=[self.paintNmesh] * 3, dtype="f4", comm=self.comm)
pm = ParticleMesh(BoxSize=self.datasource.BoxSize, Nmesh=[self.Nmesh] * 3, dtype="f4", comm=self.comm)
real, stats = self.painter.paint(pmpaint, self.datasource)
if self.writeFourier:
result = ComplexField(pm)
else:
result = RealField(pm)
real.resample(result)
# reuses the memory
result.sort(out=result)
result = result.ravel()
# return all the necessary results
return result, stats
def test_c2r_vjp(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
real = pm.generate_whitenoise(1234, mode='real')
comp = real.r2c()
def objective(comp):
real = comp.c2r()
obj = (real.value ** 2).sum()
return comm.allreduce(obj)
grad_real = RealField(pm)
grad_real[...] = real[...] * 2
grad_comp = ComplexField(pm)
grad_comp = grad_real.c2r_vjp(grad_real)
grad_comp.decompress_vjp(grad_comp)
ng = []
ag = []
ind = []
dx = 1e-7
for ind1 in numpy.ndindex(*(list(grad_comp.cshape) + [2])):
dx1, c1 = perturb(comp, ind1, dx)
ng1 = (objective(c1) - objective(comp)) / dx
ag1 = grad_comp.cgetitem(ind1) * dx1 / dx
comm.barrier()
ng.append(ng1)
ag.append(ag1)
ind.append(ind1)
assert_allclose(ng, ag, rtol=1e-5)
def test_complex_apply(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
complex = ComplexField(pm)
def filter(k, v):
return k[0] + k[1] * 1j
complex.apply(filter, out=Ellipsis)
for i, x, slab in zip(complex.slabs.i, complex.slabs.x, complex.slabs):
assert_array_equal(slab, x[0] + x[1] * 1j)
def test_complex_apply(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
complex = ComplexField(pm)
def filter(k, v):
knormp = k.normp()
assert_allclose(knormp, sum(ki ** 2 for ki in k))
return k[0] + k[1] * 1j
complex.apply(filter, out=Ellipsis)
for i, x, slab in zip(complex.slabs.i, complex.slabs.x, complex.slabs):
assert_array_equal(slab, x[0] + x[1] * 1j)
def test_cmean(comm):
# this tests cmean (collective mean) along with resampling preseves it.
pm1 = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
pm2 = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
complex1 = ComplexField(pm1)
complex2 = ComplexField(pm2)
real2 = RealField(pm2)
real1 = RealField(pm1)
for i, kk, slab in zip(complex1.slabs.i, complex1.slabs.x, complex1.slabs):
slab[...] = sum([k**2 for k in kk]) **0.5
complex1.c2r(real1)
real1.resample(real2)
assert_almost_equal(real1.cmean(), real2.cmean())
def test_hermitian_weights(comm):
numpy.random.seed(42)
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8, 8], comm=comm, dtype='f8')
cfield = ComplexField(pm)
data = numpy.random.random(size=cfield.shape)
data = data[:] + 1j * data[:]
cfield[...] = data[:]
x = cfield.x
# iterate over symmetry axis
for i, slab in enumerate(SlabIterator(x, axis=2, symmetry_axis=2)):
# nonsingular weights give indices of positive frequencies
nonsig = slab.nonsingular
weights = slab.hermitian_weights
# weights == 2 when iterating frequency is positive
if numpy.float(slab.coords(2)) > 0.:
assert weights > 1
assert numpy.all(nonsig == True)
else:
assert weights == 1.0
assert numpy.all(nonsig == False)
def test_cmean(comm):
# this tests cmean (collective mean) along with resampling preseves it.
pm1 = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
pm2 = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
complex1 = ComplexField(pm1)
complex2 = ComplexField(pm2)
real2 = RealField(pm2)
real1 = RealField(pm1)
for i, kk, slab in zip(complex1.slabs.i, complex1.slabs.x, complex1.slabs):
slab[...] = sum([k**2 for k in kk])**0.5
complex1.c2r(real1)
real1.resample(real2)
assert_almost_equal(real1.cmean(), real2.cmean())
def test_whitenoise(comm):
# the whitenoise shall preserve the large scale.
pm0 = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8, 8], comm=comm, dtype='f8')
pm1 = ParticleMesh(BoxSize=8.0, Nmesh=[16, 16, 16], comm=comm, dtype='f8')
pm2 = ParticleMesh(BoxSize=8.0, Nmesh=[32, 32, 32], comm=comm, dtype='f8')
complex1_down = ComplexField(pm0)
complex2_down = ComplexField(pm0)
complex1 = pm1.generate_whitenoise(seed=8, unitary=True)
complex2 = pm2.generate_whitenoise(seed=8, unitary=True)
complex1.resample(complex1_down)
complex2.resample(complex2_down)
mask1 = complex1_down.value != complex2_down.value
assert_array_equal(complex1_down.value, complex2_down.value)
def test_complex_iter(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
complex = ComplexField(pm)
for x, slab in zip(complex.slabs.x, complex.slabs):
assert_array_equal(slab.shape,
sum(x[d]**2 for d in range(len(pm.Nmesh))).shape)
for a, b in zip(slab.x, x):
assert_almost_equal(a, b)
def test_sort(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 6], comm=comm, dtype='f8')
real = RealField(pm)
truth = numpy.arange(8 * 6)
real[...] = truth.reshape(8, 6)[real.slices]
unsorted = real.copy()
real.sort(out=Ellipsis)
conjecture = numpy.concatenate(comm.allgather(real.value.ravel()))
assert_array_equal(conjecture, truth)
real.unravel(real)
assert_array_equal(real, unsorted)
complex = ComplexField(pm)
truth = numpy.arange(8 * 4)
complex[...] = truth.reshape(8, 4)[complex.slices]
complex.ravel(out=Ellipsis)
conjecture = numpy.concatenate(comm.allgather(complex.value.ravel()))
assert_array_equal(conjecture, truth)
def test_sort(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 6], comm=comm, dtype='f8')
real = RealField(pm)
truth = numpy.arange(8 * 6)
real[...] = truth.reshape(8, 6)[real.slices]
unsorted = real.copy()
real.sort(out=Ellipsis)
conjecture = numpy.concatenate(comm.allgather(real.value.ravel()))
assert_array_equal(conjecture, truth)
real.unravel(real)
assert_array_equal(real, unsorted)
complex = ComplexField(pm)
truth = numpy.arange(8 * 4)
complex[...] = truth.reshape(8, 4)[complex.slices]
complex.ravel(out=Ellipsis)
conjecture = numpy.concatenate(comm.allgather(complex.value.ravel()))
assert_array_equal(conjecture, truth)
def run(self):
"""
Run the algorithm, which computes and returns the grid in C_CONTIGUOUS order partitioned by ranks.
"""
from nbodykit import measurestats
if self.comm.rank == 0:
self.logger.info('importing done')
self.logger.info('Resolution Nmesh : %d' % self.paintNmesh)
self.logger.info('paintbrush : %s' % self.painter.paintbrush)
# setup the particle mesh object, taking BoxSize from the painters
pmpaint = ParticleMesh(BoxSize=self.datasource.BoxSize, Nmesh=[self.paintNmesh] * 3, dtype='f4', comm=self.comm)
pm = ParticleMesh(BoxSize=self.datasource.BoxSize, Nmesh=[self.Nmesh] * 3, dtype='f4', comm=self.comm)
real, stats = self.painter.paint(pmpaint, self.datasource)
if self.writeFourier:
result = ComplexField(pm)
else:
result = RealField(pm)
real.resample(result)
# reuses the memory
result.sort(out=result)
result = result.ravel()
# return all the necessary results
return result, stats
def read(self, real):
import bigfile
if self.comm.rank == 0:
self.logger.info("Reading from Nmesh = %d to Nmesh = %d" %
(self.Nmesh, real.Nmesh[0]))
if any(real.Nmesh != self.Nmesh):
pmread = ParticleMesh(BoxSize=real.BoxSize,
Nmesh=(self.Nmesh, self.Nmesh, self.Nmesh),
dtype='f4',
comm=self.comm)
else:
pmread = real.pm
f = bigfile.BigFileMPI(self.comm, self.path)
with f[self.dataset] as ds:
if self.isfourier:
if self.comm.rank == 0:
self.logger.info("reading complex field")
complex2 = ComplexField(pmread)
assert self.comm.allreduce(complex2.size) == ds.size
start = sum(
self.comm.allgather(complex2.size)[:self.comm.rank])
end = start + complex2.size
complex2.unsort(ds[start:end])
complex2.resample(real)
else:
if self.comm.rank == 0:
self.logger.info("reading real field")
real2 = RealField(pmread)
start = sum(self.comm.allgather(real2.size)[:self.comm.rank])
end = start + real2.size
real2.unsort(ds[start:end])
real2.resample(real)
def _compute_multipoles(self, kedges):
"""
Compute the window-convoled power spectrum multipoles, for a data set
with non-trivial survey geometry.
This estimator builds upon the work presented in Bianchi et al. 2015
and Scoccimarro et al. 2015, but differs in the implementation. This
class uses the spherical harmonic addition theorem such that
only :math:`2\ell+1` FFTs are required per multipole, rather than the
:math:`(\ell+1)(\ell+2)/2` FFTs in the implementation presented by
Bianchi et al. and Scoccimarro et al.
References
----------
* Bianchi, Davide et al., `Measuring line-of-sight-dependent Fourier-space clustering using FFTs`,
MNRAS, 2015
* Scoccimarro, Roman, `Fast estimators for redshift-space clustering`, Phys. Review D, 2015
"""
# clear compensation from the actions
for source in [self.first, self.second]:
source.actions[:] = []
source.compensated = False
assert len(source.actions) == 0
# compute the compensations
compensation = {}
for name, mesh in zip(['first', 'second'], [self.first, self.second]):
compensation[name] = get_compensation(mesh)
if self.comm.rank == 0:
if compensation[name] is not None:
args = (compensation[name]['func'].__name__, name)
self.logger.info(
"using compensation function %s for source '%s'" %
args)
else:
self.logger.warning(
"no compensation applied for source '%s'" % name)
rank = self.comm.rank
pm = self.first.pm
# setup the 1D-binning
muedges = numpy.linspace(-1, 1, 2, endpoint=True)
edges = [kedges, muedges]
# make a structured array to hold the results
cols = ['k'] + ['power_%d' % l
for l in sorted(self.attrs['poles'])] + ['modes']
dtype = ['f8'] + ['c8'] * len(self.attrs['poles']) + ['i8']
dtype = numpy.dtype(list(zip(cols, dtype)))
result = numpy.empty(len(kedges) - 1, dtype=dtype)
# offset the box coordinate mesh ([-BoxSize/2, BoxSize]) back to
# the original (x,y,z) coords
offset = self.attrs['BoxCenter'] + 0.5 * pm.BoxSize / pm.Nmesh
# always need to compute ell=0
poles = sorted(self.attrs['poles'])
if 0 not in poles:
poles = [0] + poles
assert poles[0] == 0
# spherical harmonic kernels (for ell > 0)
Ylms = [[get_real_Ylm(l, m) for m in range(-l, l + 1)]
for l in poles[1:]]
# paint the 1st FKP density field to the mesh (paints: data - alpha*randoms, essentially)
rfield1 = self.first.compute(Nmesh=self.attrs['Nmesh'])
meta1 = rfield1.attrs.copy()
if rank == 0:
self.logger.info("%s painting of 'first' done" %
self.first.resampler)
# store alpha: ratio of data to randoms
self.attrs['alpha'] = meta1['alpha']
# FFT 1st density field and apply the resampler transfer kernel
cfield = rfield1.r2c()
if compensation['first'] is not None:
cfield.apply(out=Ellipsis, **compensation['first'])
if rank == 0: self.logger.info('ell = 0 done; 1 r2c completed')
# monopole A0 is just the FFT of the FKP density field
# NOTE: this holds FFT of density field #1
volume = pm.BoxSize.prod()
A0_1 = ComplexField(pm)
A0_1[:] = cfield[:] * volume # normalize with a factor of volume
# paint second mesh too?
if self.first is not self.second:
# paint the second field
rfield2 = self.second.compute(Nmesh=self.attrs['Nmesh'])
meta2 = rfield2.attrs.copy()
if rank == 0:
self.logger.info("%s painting of 'second' done" %
self.second.resampler)
# need monopole of second field
if 0 in self.attrs['poles']:
# FFT density field and apply the resampler transfer kernel
A0_2 = rfield2.r2c()
A0_2[:] *= volume
if compensation['second'] is not None:
A0_2.apply(out=Ellipsis, **compensation['second'])
else:
rfield2 = rfield1
meta2 = meta1
# monopole of second field is first field
if 0 in self.attrs['poles']:
A0_2 = A0_1
# ensure alpha from first mesh is equal to alpha from second mesh
# NOTE: this is mostly just a sanity check, and should always be true if
# we made it this far already
if not numpy.allclose(
rfield1.attrs['alpha'], rfield2.attrs['alpha'], rtol=1e-3):
msg = (
"ConvolvedFFTPower cross-correlations currently require the same"
" FKPCatalog (data/randoms), such that only the weight column can vary;"
" different ``alpha`` values found for first/second meshes")
raise ValueError(msg)
# save the painted density field #2 for later
density2 = rfield2.copy()
# initialize the memory holding the Aell terms for
# higher multipoles (this holds sum of m for fixed ell)
# NOTE: this will hold FFTs of density field #2
Aell = ComplexField(pm)
# the real-space grid
xgrid = [
xx.astype('f8') + offset[ii]
for ii, xx in enumerate(density2.slabs.optx)
]
xnorm = numpy.sqrt(sum(xx**2 for xx in xgrid))
xgrid = [x / xnorm for x in xgrid]
# the Fourier-space grid
kgrid = [kk.astype('f8') for kk in cfield.slabs.optx]
knorm = numpy.sqrt(sum(kk**2 for kk in kgrid))
knorm[knorm == 0.] = numpy.inf
kgrid = [k / knorm for k in kgrid]
# proper normalization: same as equation 49 of Scoccimarro et al. 2015
for name in ['data', 'randoms']:
self.attrs[name + '.norm'] = self.normalization(
name, self.attrs['alpha'])
if self.attrs['randoms.norm'] > 0:
norm = 1.0 / self.attrs['randoms.norm']
# check normalization
Adata = self.attrs['data.norm']
Aran = self.attrs['randoms.norm']
if not numpy.allclose(Adata, Aran, rtol=0.05):
msg = "normalization in ConvolvedFFTPower different by more than 5%; "
msg += ",algorithm requires they must be similar\n"
msg += "\trandoms.norm = %.6f, data.norm = %.6f\n" % (Aran,
Adata)
msg += "\tpossible discrepancies could be related to normalization "
msg += "of n(z) column ('%s')\n" % self.first.nbar
msg += "\tor the consistency of the FKP weight column for 'data' "
msg += "and 'randoms';\n"
msg += "\tn(z) columns for 'data' and 'randoms' should be "
msg += "normalized to represent n(z) of the data catalog"
raise ValueError(msg)
if rank == 0:
self.logger.info(
"normalized power spectrum with `randoms.norm = %.6f`" %
Aran)
else:
# an empty random catalog is provides, so we will ignore the normalization.
norm = 1.0
if rank == 0:
self.logger.info(
"normalization of power spectrum is neglected, as no random is provided."
)
# loop over the higher order multipoles (ell > 0)
start = time.time()
for iell, ell in enumerate(poles[1:]):
# clear 2D workspace
Aell[:] = 0.
# iterate from m=-l to m=l and apply Ylm
substart = time.time()
for Ylm in Ylms[iell]:
# reset the real-space mesh to the original density #2
rfield2[:] = density2[:]
# apply the config-space Ylm
for islab, slab in enumerate(rfield2.slabs):
slab[:] *= Ylm(xgrid[0][islab], xgrid[1][islab],
xgrid[2][islab])
# real to complex of field #2
rfield2.r2c(out=cfield)
# apply the Fourier-space Ylm
for islab, slab in enumerate(cfield.slabs):
slab[:] *= Ylm(kgrid[0][islab], kgrid[1][islab],
kgrid[2][islab])
# add to the total sum
Aell[:] += cfield[:]
# and this contribution to the total sum
substop = time.time()
if rank == 0:
self.logger.debug("done term for Y(l=%d, m=%d) in %s" %
(Ylm.l, Ylm.m, timer(substart, substop)))
# apply the compensation transfer function
if compensation['second'] is not None:
Aell.apply(out=Ellipsis, **compensation['second'])
# factor of 4*pi from spherical harmonic addition theorem + volume factor
Aell[:] *= 4 * numpy.pi * volume
# log the total number of FFTs computed for each ell
if rank == 0:
args = (ell, len(Ylms[iell]))
self.logger.info('ell = %d done; %s r2c completed' % args)
# calculate the power spectrum multipoles, slab-by-slab to save memory
# NOTE: this computes (A0 of field #1) * (Aell of field #2).conj()
for islab in range(A0_1.shape[0]):
Aell[islab, ...] = norm * A0_1[islab] * Aell[islab].conj()
# project on to 1d k-basis (averaging over mu=[0,1])
proj_result, _ = project_to_basis(Aell, edges)
result['power_%d' % ell][:] = numpy.squeeze(proj_result[2])
# summarize how long it took
stop = time.time()
if rank == 0:
self.logger.info(
"higher order multipoles computed in elapsed time %s" %
timer(start, stop))
# also compute ell=0
if 0 in self.attrs['poles']:
# the 3D monopole
for islab in range(A0_1.shape[0]):
A0_1[islab, ...] = norm * A0_1[islab] * A0_2[islab].conj()
# the 1D monopole
proj_result, _ = project_to_basis(A0_1, edges)
result['power_0'][:] = numpy.squeeze(proj_result[2])
# save the number of modes and k
result['k'][:] = numpy.squeeze(proj_result[0])
result['modes'][:] = numpy.squeeze(proj_result[-1])
# compute shot noise
self.attrs['shotnoise'] = self.shotnoise(self.attrs['alpha'])
# copy over any painting meta data
if self.first is self.second:
copy_meta(self.attrs, meta1)
else:
copy_meta(self.attrs, meta1, prefix='first')
copy_meta(self.attrs, meta2, prefix='second')
return result
def test_cgetitem(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
for i in numpy.ndindex((4, 4)):
complex = RealField(pm)
complex[...] = 0
v2 = complex.csetitem(i, 100.)
v1 = complex.cgetitem(i)
assert v2 == 100.
assert_array_equal(v1, v2)
for i in numpy.ndindex((4, 3)):
complex = ComplexField(pm)
complex[...] = 0
v2 = complex.csetitem(i, 100. + 10j)
complex.c2r(out=Ellipsis).r2c(out=Ellipsis)
v1 = complex.cgetitem(i)
if i == (0, 0):
assert v2 == 100.
assert comm.allreduce(complex.value.sum()) == 100.
elif i == (0, 2):
assert v2 == 100.
assert comm.allreduce(complex.value.sum()) == 100.
elif i == (2, 0):
assert v2 == 100.
assert comm.allreduce(complex.value.sum()) == 100.
elif i == (1, 0):
assert v2 == 100 + 10j
assert comm.allreduce(complex.value.sum()) == 200.
elif i == (3, 0):
assert v2 == 100 + 10j
assert comm.allreduce(complex.value.sum()) == 200.
elif i == (3, 2):
assert v2 == 100 + 10j
assert comm.allreduce(complex.value.sum()) == 200.
elif i == (1, 2):
assert v2 == 100 + 10j
assert comm.allreduce(complex.value.sum()) == 200.
elif i == (2, 2):
assert v2 == 100.
assert comm.allreduce(complex.value.sum()) == 100.
else:
assert v2 == 100. + 10j
assert_array_equal(comm.allreduce(complex.value.sum()), 100. + 10j)
assert_array_equal(v1, v2)
for i in numpy.ndindex((4, 3, 2)):
complex = ComplexField(pm)
complex[...] = 0
v2 = complex.csetitem(i, 100.)
complex.c2r(out=Ellipsis).r2c(out=Ellipsis)
v1 = complex.cgetitem(i)
if i == (0, 0, 0):
assert v2 == 100.
if i == (0, 0, 1):
assert v2 == 0.
elif i == (0, 2, 0):
assert v2 == 100.
elif i == (0, 2, 1):
assert v2 == 0.
elif i == (2, 0, 0):
assert v2 == 100.
elif i == (2, 0, 1):
assert v2 == 0.
elif i == (2, 2, 0):
assert v2 == 100.
elif i == (2, 2, 1):
assert v2 == 0.
else:
assert v2 == 100.
assert_array_equal(v1, v2)
def test_ctol(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
complex = ComplexField(pm)
value, local = complex._ctol((3, 3))
assert local is None
def test_fupsample(comm):
pm1 = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
pm2 = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
numpy.random.seed(3333)
truth = numpy.fft.rfftn(numpy.random.normal(size=(8, 8)))
complex1 = ComplexField(pm1)
for ind in numpy.ndindex(*complex1.cshape):
complex1.csetitem(ind, truth[ind])
if any(i == 4 for i in ind):
complex1.csetitem(ind, 0)
else:
complex1.csetitem(ind, truth[ind])
if any(i >= 2 and i < 7 for i in ind):
complex1.csetitem(ind, 0)
assert_almost_equal(complex1[...], complex1.c2r().r2c())
complex2 = ComplexField(pm2)
for ind in numpy.ndindex(*complex2.cshape):
newind = tuple([i if i <= 2 else 8 - (4 - i) for i in ind])
if any(i == 2 for i in ind):
complex2.csetitem(ind, 0)
else:
complex2.csetitem(ind, truth[newind])
tmpr = RealField(pm1)
tmp = ComplexField(pm1)
complex2.resample(tmp)
assert_almost_equal(complex1[...], tmp[...], decimal=5)
complex2.c2r().resample(tmp)
assert_almost_equal(complex1[...], tmp[...], decimal=5)
complex2.resample(tmpr)
assert_almost_equal(tmpr.r2c(), tmp[...])
complex2.c2r().resample(tmpr)
assert_almost_equal(tmpr.r2c(), tmp[...])
def test_fupsample(comm):
pm1 = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
pm2 = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
numpy.random.seed(3333)
truth = numpy.fft.rfftn(numpy.random.normal(size=(8, 8)))
complex1 = ComplexField(pm1)
for ind in numpy.ndindex(*complex1.cshape):
complex1.csetitem(ind, truth[ind])
if any(i == 4 for i in ind):
complex1.csetitem(ind, 0)
else:
complex1.csetitem(ind, truth[ind])
if any(i >= 2 and i < 7 for i in ind):
complex1.csetitem(ind, 0)
assert_almost_equal(complex1[...], complex1.c2r().r2c())
complex2 = ComplexField(pm2)
for ind in numpy.ndindex(*complex2.cshape):
newind = tuple([i if i <= 2 else 8 - (4 - i) for i in ind])
if any(i == 2 for i in ind):
complex2.csetitem(ind, 0)
else:
complex2.csetitem(ind, truth[newind])
tmpr = RealField(pm1)
tmp = ComplexField(pm1)
complex2.resample(tmp)
assert_almost_equal(complex1[...], tmp[...], decimal=5)
complex2.c2r().resample(tmp)
assert_almost_equal(complex1[...], tmp[...], decimal=5)
complex2.resample(tmpr)
assert_almost_equal(tmpr.r2c(), tmp[...])
complex2.c2r().resample(tmpr)
assert_almost_equal(tmpr.r2c(), tmp[...])
def test_ctol(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[4, 4], comm=comm, dtype='f8')
complex = ComplexField(pm)
value, local = complex._ctol((3, 3))
assert local is None
def test_shape_complex(comm):
pm = ParticleMesh(BoxSize=8.0, Nmesh=[8, 8], comm=comm, dtype='f8')
comp = ComplexField(pm)
assert (tuple(comp.cshape) == (8, 5))
def gaussian_complex_fields(pm,
linear_power,
seed,
unitary_amplitude=False,
inverted_phase=False,
compute_displacement=False):
r"""
Make a Gaussian realization of a overdensity field, :math:`\delta(x)`.
If specified, also compute the corresponding 1st order Lagrangian
displacement field (Zel'dovich approximation) :math:`\psi(x)`,
which is related to the linear velocity field via:
.. math::
v(x) = f a H \psi(x)
Notes
-----
This computes the overdensity field using the following steps:
#. Generate complex variates with unity variance
#. Scale the Fourier field by :math:`(P(k) / V)^{1/2}`
After step 2, the complex field has unity variance. This
is equivalent to generating real-space normal variates
with mean and unity variance, calling r2c() and dividing by :math:`N^3`
since the variance of the complex FFT (with no additional normalization)
is :math:`N^3 \times \sigma^2_\mathrm{real}`.
Furthermore, the power spectrum is defined as V * variance.
So a normalization factor of 1 / V shows up in step 2,
cancels this factor such that the power spectrum is P(k).
The linear displacement field is computed as:
.. math::
\psi_i(k) = i \frac{k_i}{k^2} \delta(k)
.. note::
To recover the linear velocity in proper units, i.e., km/s,
from the linear displacement, an additional factor of
:math:`f \times a \times H(a)` is required
Parameters
----------
pm : pmesh.pm.ParticleMesh
the mesh object
linear_power : callable
a function taking wavenumber as its only argument, which returns
the linear power spectrum
seed : int
the random seed used to generate the random field
compute_displacement : bool, optional
if ``True``, also return the linear Zel'dovich displacement field;
default is ``False``
unitary_amplitude : bool, optional
if ``True``, the seed gaussian has unitary_amplitude.
inverted_phase: bool, optional
if ``True``, the phase of the seed gaussian is inverted
Returns
-------
delta_k : ComplexField
the real-space Gaussian overdensity field
disp_k : ComplexField or ``None``
if requested, the Gaussian displacement field
"""
if not isinstance(seed, numbers.Integral):
raise ValueError(
"the seed used to generate the linear field must be an integer")
# use pmesh to generate random complex white noise field (done in parallel)
# variance of complex field is unity
# multiply by P(k)**0.5 to get desired variance
delta_k = pm.generate_whitenoise(seed,
mode='complex',
unitary=unitary_amplitude)
if inverted_phase: delta_k[...] *= -1
# initialize the displacement fields for (x,y,z)
if compute_displacement:
disp_k = [ComplexField(pm) for i in range(delta_k.ndim)]
for i in range(delta_k.ndim):
disp_k[i][:] = 1j
else:
disp_k = None
# volume factor needed for normalization
norm = 1.0 / pm.BoxSize.prod()
# iterate in slabs over fields
slabs = [delta_k.slabs.x, delta_k.slabs]
if compute_displacement:
slabs += [d.slabs for d in disp_k]
# loop over the mesh, slab by slab
for islabs in zip(*slabs):
kslab, delta_slab = islabs[:2] # the k arrays and delta slab
# the square of the norm of k on the mesh
k2 = sum(kk**2 for kk in kslab)
zero_idx = k2 == 0.
k2[zero_idx] = 1. # avoid dividing by zero
# the linear power (function of k)
power = linear_power((k2**0.5).flatten())
# multiply complex field by sqrt of power
delta_slab[...].flat *= (power * norm)**0.5
# set k == 0 to zero (zero config-space mean)
delta_slab[zero_idx] = 0.
# compute the displacement
if compute_displacement:
# ignore division where k==0 and set to 0
with numpy.errstate(invalid='ignore'):
for i in range(delta_k.ndim):
disp_slab = islabs[2 + i]
disp_slab[...] *= kslab[i] / k2 * delta_slab[...]
disp_slab[zero_idx] = 0. # no bulk displacement
# return Fourier-space density and displacement (which could be None)
return delta_k, disp_k
def _compute_multipoles(self):
"""
Compute the window-convoled power spectrum multipoles, for a data set
with non-trivial survey geometry.
This estimator builds upon the work presented in Bianchi et al. 2015
and Scoccimarro et al. 2015, but differs in the implementation. This
class uses the spherical harmonic addition theorem such that
only :math:`2\ell+1` FFTs are required per multipole, rather than the
:math:`(\ell+1)(\ell+2)/2` FFTs in the implementation presented by
Bianchi et al. and Scoccimarro et al.
References
----------
* Bianchi, Davide et al., `Measuring line-of-sight-dependent Fourier-space clustering using FFTs`,
MNRAS, 2015
* Scoccimarro, Roman, `Fast estimators for redshift-space clustering`, Phys. Review D, 2015
"""
# clear compensation from the actions
for source in [self.first, self.second]:
source.actions[:] = []; source.compensated = False
assert len(source.actions) == 0
# compute the compensations
compensation = {}
for name, mesh in zip(['first', 'second'], [self.first, self.second]):
compensation[name] = get_compensation(mesh)
if self.comm.rank == 0:
if compensation[name] is not None:
args = (compensation[name]['func'].__name__, name)
self.logger.info("using compensation function %s for source '%s'" % args)
else:
self.logger.warning("no compensation applied for source '%s'" % name)
rank = self.comm.rank
pm = self.first.pm
# setup the 1D-binning
muedges = numpy.linspace(0, 1, 2, endpoint=True)
edges = [self.edges, muedges]
# make a structured array to hold the results
cols = ['k'] + ['power_%d' %l for l in sorted(self.attrs['poles'])] + ['A_%d' %l for l in sorted(self.attrs['poles'])] + ['modes']
dtype = ['f8'] + ['c8']*len(self.attrs['poles'])*2 + ['i8']
dtype = numpy.dtype(list(zip(cols, dtype)))
result = numpy.empty(len(self.edges)-1, dtype=dtype)
# offset the box coordinate mesh ([-BoxSize/2, BoxSize]) back to
# the original (x,y,z) coords
offset = self.attrs['BoxCenter'] + 0.5*pm.BoxSize / pm.Nmesh
# always need to compute ell=0
poles = sorted(self.attrs['poles'])
if 0 not in poles:
poles = [0] + poles
assert poles[0] == 0
# spherical harmonic kernels (for ell > 0)
Ylms = [[get_real_Ylm(l,m) for m in range(-l, l+1)] for l in poles[1:]]
# paint the 1st FKP density field to the mesh (paints: data - alpha*randoms, essentially)
rfield1 = self.first.paint(Nmesh=self.attrs['Nmesh'])
vol_per_cell = (pm.BoxSize/pm.Nmesh).prod()/rfield1.attrs['num_per_cell'] #to compensate the default normalization by num_per_cell
rfield1[:] /= vol_per_cell
meta1 = rfield1.attrs.copy()
if rank == 0: self.logger.info("%s painting of 'first' done" %self.first.window)
# FFT 1st density field and apply the paintbrush window transfer kernel
cfield = rfield1.r2c()
if compensation['first'] is not None:
cfield.apply(out=Ellipsis, **compensation['first'])
if rank == 0: self.logger.info('ell = 0 done; 1 r2c completed')
# monopole A0 is just the FFT of the FKP density field
# NOTE: this holds FFT of density field #1
volume = pm.BoxSize.prod()
A0_1 = ComplexField(pm)
A0_1[:] = cfield[:] * volume # normalize with a factor of volume
# paint second mesh too?
if self.first is not self.second:
# paint the second field
rfield2 = self.second.paint(Nmesh=self.attrs['Nmesh'],normalize=False)
rfield2[:] /= vol_per_cell
meta2 = rfield2.attrs.copy()
if rank == 0: self.logger.info("%s painting of 'second' done" %self.second.window)
# need monopole of second field
if 0 in self.attrs['poles']:
# FFT density field and apply the paintbrush window transfer kernel
A0_2 = rfield2.r2c()
A0_2[:] *= volume
if compensation['second'] is not None: A0_2.apply(out=Ellipsis, **compensation['second'])
else:
rfield2 = rfield1
meta2 = meta1
# monopole of second field is first field
if 0 in self.attrs['poles']:
A0_2 = A0_1
# save the painted density field #2 for later
density2 = rfield2.copy()
# initialize the memory holding the Aell terms for
# higher multipoles (this holds sum of m for fixed ell)
# NOTE: this will hold FFTs of density field #2
Aell = ComplexField(pm)
# the real-space grid
xgrid = [xx.astype('f8') + offset[ii] for ii, xx in enumerate(density2.slabs.optx)]
xnorm = numpy.sqrt(sum(xx**2 for xx in xgrid))
xgrid = [x/xnorm for x in xgrid]
# the Fourier-space grid
kgrid = [kk.astype('f8') for kk in cfield.slabs.optx]
knorm = numpy.sqrt(sum(kk**2 for kk in kgrid)); knorm[knorm==0.] = numpy.inf
kgrid = [k/knorm for k in kgrid]
# loop over the higher order multipoles (ell > 0)
start = time.time()
for iell, ell in enumerate(poles[1:]):
# clear 2D workspace
Aell[:] = 0.
# iterate from m=-l to m=l and apply Ylm
substart = time.time()
for Ylm in Ylms[iell]:
# reset the real-space mesh to the original density #2
rfield2[:] = density2[:]
# apply the config-space Ylm
for islab, slab in enumerate(rfield2.slabs):
slab[:] *= Ylm(xgrid[0][islab], xgrid[1][islab], xgrid[2][islab])
# real to complex of field #2
rfield2.r2c(out=cfield)
# apply the Fourier-space Ylm
for islab, slab in enumerate(cfield.slabs):
slab[:] *= Ylm(kgrid[0][islab], kgrid[1][islab], kgrid[2][islab])
# add to the total sum
Aell[:] += cfield[:]
# and this contribution to the total sum
substop = time.time()
if rank == 0:
self.logger.debug("done term for Y(l=%d, m=%d) in %s" %(Ylm.l, Ylm.m, timer(substart, substop)))
# apply the compensation transfer function
if compensation['second'] is not None:
Aell.apply(out=Ellipsis, **compensation['second'])
# factor of 4*pi from spherical harmonic addition theorem + volume factor
Aell[:] *= 4*numpy.pi*volume
# log the total number of FFTs computed for each ell
if rank == 0:
args = (ell, len(Ylms[iell]))
self.logger.info('ell = %d done; %s r2c completed' %args)
# project on to 1d k-basis (averaging over mu=[0,1])
proj_result, _ = project_to_basis(Aell, edges)
result['A_%d' %ell][:] = numpy.squeeze(proj_result[2])
# calculate the power spectrum multipoles, slab-by-slab to save memory
# NOTE: this computes (A0 of field #1) * (Aell of field #2).conj()
for islab in range(A0_1.shape[0]):
Aell[islab,...] = A0_1[islab] * Aell[islab].conj()
# project on to 1d k-basis (averaging over mu=[0,1])
proj_result, _ = project_to_basis(Aell, edges)
result['power_%d' %ell][:] = numpy.squeeze(proj_result[2])
# summarize how long it took
stop = time.time()
if rank == 0:
self.logger.info("higher order multipoles computed in elapsed time %s" %timer(start, stop))
# also compute ell=0
if 0 in self.attrs['poles']:
# the 1D monopole
proj_result, _ = project_to_basis(A0_2, edges)
result['A_0'][:] = numpy.squeeze(proj_result[2])
# the 3D monopole
for islab in range(A0_1.shape[0]):
A0_1[islab,...] = A0_1[islab]*A0_2[islab].conj()
# the 1D monopole
proj_result, _ = project_to_basis(A0_1, edges)
result['power_0'][:] = numpy.squeeze(proj_result[2])
# save the number of modes and k
result['k'][:] = numpy.squeeze(proj_result[0])
result['modes'][:] = numpy.squeeze(proj_result[-1])
# copy over any painting meta data
if self.first is self.second:
copy_meta(self.attrs, meta1)
else:
copy_meta(self.attrs, meta1, prefix='first')
copy_meta(self.attrs, meta2, prefix='second')
return result