# Creating a Mesh¶

In this section, we outline how users interact with data on a mesh in nbodykit. The main ways to create a mesh include:

## Converting a CatalogSource to a Mesh¶

Users can create mesh objects from CatalogSource objects by specifying the desired number of cells per mesh side via the Nmesh parameter and using the to_mesh() function. Below, we convert a UniformCatalog to a MeshSource using a $$16^3$$ mesh in a box of side length $$1$$ $$\mathrm{Mpc}/h$$.

In [2]:

from nbodykit.lab import UniformCatalog

cat = UniformCatalog(nbar=100, BoxSize=1.0, seed=42)

mesh = cat.to_mesh(Nmesh=16)

print("mesh = ", mesh)

mesh =  (UniformCatalog(size=96, seed=42) as CatalogMesh)


Important

The to_mesh() operation does not perform any interpolation operations, which nbodykit refers to as “painting” the mesh. This function merely initializes a new object that sets up the mesh with the configuration provided by the user. For more details and examples of painting a catalog of discrete objects to a mesh, see Painting Catalogs to a Mesh.

### The Window Kernel¶

When interpolating discrete particles on to a regular mesh, we must choose which kind of interpolation kernel to use. The kernel determines which cells an object will contribute to on the mesh. In the simplest case, known as Nearest Grid Point interpolation, an object only contributes to the one cell that is closest to its position. In general, higher order interpolation schemes, which spread out objects over more cells, lead to more accurate results. See Section 2.3 of Sefusatti et al. 2015 for an introduction to different interpolation schemes.

nbodykit supports several different interpolation kernels, which can be specified using the window keyword of the to_mesh() function. The default value is cic, representing the second-order interpolation scheme known as Cloud In Cell. The third-order interpolation scheme, known as Triangular Shaped Cloud, can be specified by setting window to tsc. CIC and TSC are the most commonly used interpolation windows in the field of large-scale structure today.

Support for wavelet-based kernels is also provided. The Daubechies wavelet with various sizes can be used by specifying db6, db12, or db20. The closely related Symlet wavelet can be used by specifying sym6, sym12, or sym20. These are symmetric Daubechies wavelets and tend to perform better than the non-symmetric versions. For more information on using wavelet kernels for painting a mesh, see Cui et al. 2008.

Note that the non-traditional interpolation windows can be considerably slower than the cic or tsc methods. For this reason, nbodykit uses the cic interpolation window by default. See pmesh.window.methods for the full list of supported window kernels.

Note

Please see this cookbook recipe for a notebook exploring the accuracy of different interpolation windows.

### Interlacing¶

nbodykit provides support for the interlacing technique, which can reduce the effects of aliasing when Fourier transforming the density field on the mesh. This technique involves interpolating objects on to two separate meshes, separated by half of a cell size. When combining the complex fields in Fourier space from these two meshes, the effects of aliasing are significantly reduced on the combined field. For a more detailed discussion behind the mathematics of this technique, see Section 3.1 of Sefusatti et al. 2015.

By default this technique is turned off, but it can be turned on by the user by passing interlaced=True to the to_mesh() function.

Note

Please see this cookbook recipe for a notebook exploring the effects of interlacing on density fields in Fourier space.

### Compensation: Deconvolving the Window Kernel¶

Interpolating discrete objects on to the mesh produces a density field defined on the mesh that is convolved with the interpolation kernel. In Fourier space, the complex field is then the product of the true density field and the Fourier transform of the window kernel as given by the Convolution Theorem. For the TSC and CIC window kernels, there are well-known correction factors that can be applied to the density field in Fourier space. If we apply these correction factors, we refer to the field as “compensated”, and the use of these correction factors is controlled via the compensated keyword of the to_mesh() function.

If compensated is set to True, the correction factors that will be applied are:

 Window Interlacing Compensation Function Reference cic False CompensateCICAliasing() eq 20 of Jing et al. 2005 tsc False CompensateTSCAliasing() eq 20 of Jing et al. 2005 cic True CompensateCIC() eq 18 of Jing et al. 2005 ($$p=2$$) tsc True CompensateTSC() eq 18 of Jing et al. 2005 ($$p=3$$)

Note

If window is not equal to tsc or cic, no compensation correction is currently implemented by default in nbodykit, and if compensated is set to True, an exception will be raised. Users can implement custom compensation functions via the syntax for Applying Functions to the Mesh.

The to_mesh() function supports additional keywords for customizing the painting process. These keywords are:

• weight:

The weight keyword can be used to select a column to use as weights when painting objects to a mesh. By default, it is set to the Weight column, a default column equal to unity for all objects. In this default configuration, the density field on the mesh is normalized as $$1+\delta$$. See Painting Multiple Species of Particles for an example of using the weight keyword to represent particle masses.

• value:

The value keyword can be used to select a column to use as the field value when painting a mesh. The mesh field is a weighted average of value, with the weights given by weight. By default, it is set to the Value column, a default column equal to unity for all objects. In this default configuration, the density field on the mesh is normalized as $$1+\delta$$. See Painting the Line-of-sight Momentum Field for an example of using the value keyword to paint the momentum field.

• selection:

The selection keyword specifies a boolean column that selects a subset of the CatalogSource object to paint to the mesh. By default, the selection keyword is set to the Selection column, a default column in all CatalogSource objects that is set to True for all objects.

• position:

By default, nbodykit assumes that the Position column is the name of the column holding the Cartesian coordinates of the objects in the catalog. Thus, the to_mesh() function uses this column to paint a catalog to a mesh. The user can change this behavior by specifying the name of the desired column using the position keyword of the to_mesh() function.

## Gaussian Realizations¶

A Gaussian realization of a density field can be initialized directly on a mesh using the LinearMesh class. This class generates the Fourier modes of density field with a variance set by an input power spectrum function. It allows the user to create density fields with a known power spectrum, which is often a useful tool in large-scale structure analysis.

Users can take advantage of the built-in linear power spectrum class, LinearPower, or use their own function to specify the desired power spectrum. The function should take a single argument k, the wavenumber. Several transfer functions can be used with the LinearPower class, including from the CLASS CMB Boltzmann code, and the analytic fitting formulas from Eisenstein and Hu 1998 with and without Baryon Acoustic Oscillations.

In addition to the power spectrum function, users need to specify a mesh size via the Nmesh parameter and a box size via the BoxSize parameter. For example, to create a density field on a mesh using the 2015 Planck cosmological parameters and the Eisenstein-Hu linear power spectrum at redshift $$z=0$$, use

In [3]:

from nbodykit.lab import LinearMesh, cosmology
from matplotlib import pyplot as plt

cosmo = cosmology.Planck15
Plin = cosmology.LinearPower(cosmo, redshift=0, transfer='EisensteinHu')

# initialize the mesh
mesh = LinearMesh(Plin, Nmesh=128, BoxSize=1380, seed=42)

# preview the density field
plt.imshow(mesh.preview(axes=[0,1]))

Out[3]:

<matplotlib.image.AxesImage at 0x11fda3208>


## From In-memory Data¶

### From a RealField or ComplexField¶

If a pmesh.pm.RealField or pmesh.pm.ComplexField object is already stored in memory, they can be converted easily into a mesh object using the FieldMesh class. For example,

In [4]:

from nbodykit.lab import FieldMesh
from pmesh.pm import RealField, ComplexField, ParticleMesh

# a 8^3 mesh
pm = ParticleMesh(Nmesh=[8,8,8])

# initialize a RealField
rfield = RealField(pm)

# set entire mesh to unity
rfield[...] = 1.0

# initialize from the RealField
real_mesh = FieldMesh(rfield)

# can also initialize from a ComplexField
cfield = rfield.r2c()
complex_mesh = FieldMesh(cfield)


### From a Numpy Array¶

Given a 3D numpy array stored in memory that represents data on a mesh, users can initialize a mesh object using the ArrayMesh class. The array for the full mesh must be stored in memory on a single rank and not split in parallel across multiple ranks. After initializing the ArrayMesh object, the mesh data will be automatically spread out across the available ranks.

A common use case for this class is when a single rank handles the input/output of the mesh data in the form of numpy arrays. Then, a single rank can read in the array data from disk, and the mesh object can be initialized using the ArrayMesh class.

For example,

In [5]:

from nbodykit.lab import ArrayMesh
import numpy

# generate random data on a 128^3 mesh
data = numpy.random.random(size=(128,128,128))

# inititalize the mesh
mesh = ArrayMesh(data, BoxSize=1.0)

# preview the density mesh
plt.imshow(mesh.preview(axes=[0,1]))

Out[5]:

<matplotlib.image.AxesImage at 0x11feae7f0>