Adds spectral functions to Mesh TensorFlow

[EiffL]

This PR adds spectral operations needed for the flowpm project in Mesh TensorFlow, which was the subject of this blogpost: https://blog.tensorflow.org/2020/03/simulating-universe-in-tensorflow.html .

These operations are useful for lots of applications including N-body simulations and MRI reconstructions. For now, we have only added the implementation of 3D FFTs.

The implementation is based on applying a series of 1D FFTs along the trailing dimensions of the input tensors, then using all2all communications and local transpose operations, to transpose the tensor until all 3 dimensions have been transformed.

The algorithm is illustrated here:

from https://www-user.tu-chemnitz.de/~potts/workgroup/pippig/paper/PFFT_SIAM_88588.pdf

2 things to note:

• The user needs to specify the Fourier dimensions, which will be referenced in the mesh layout, to make sure the output of the FFT remains distributed.
• The output of the FFT is transposed, to save on extra all2all operations.
  These two things could be avoided but require at least 2 more all2all operations to transpose and reshape the output array back to the original memory layout of the input array... this could be provided as a option to the user probably.

A minimal example for would be:

batch_dim = mtf.Dimension("batch", batch_size)
x_dim = mtf.Dimension("nx", nc)
y_dim = mtf.Dimension("ny", nc)
z_dim = mtf.Dimension("nz", nc)

kx_dim = mtf.Dimension("kx", nc)
ky_dim = mtf.Dimension("ky", nc)
kz_dim = mtf.Dimension("kz", nc)

# Create field
field = mtf.random_uniform(mesh, [batch_dim, x_dim, y_dim, z_dim])
# Apply FFT
fft_field = mtf.signal.fft3d(mtf.cast(field, tf.complex64), [kx_dim, ky_dim, kz_dim])
# fft_field as shape: [batch, ky, kz, kx]
# Inverse FFT
recfield = mtf.cast(mtf.signal.ifft3d(fft_field, [x_dim, y_dim, z_dim]), tf.float32)

# The  following layout would be appropriate
mesh_shape = [("row", nblockx), ("col", nblocky)]
layout_rules = [("nx", "row"), ("ny", "col"),
      ("ky", "row"), ("kz", "col"),]   # Note  that the Fourier dimensions are aslo split differently

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[EiffL]

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[zaccharieramzi]

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[zaccharieramzi]

An important note is that this PR in addition to the spectral ops adds complex support for the gradient (otherwise the gradient will not flow through the spectral ops) and complex manipulation operations if you need to navigate between complex-valued tensors and float-valued tensors.