Distinctly Useful Code Collection (DUCC)

This is a collection of basic programming tools for numerical computation, including Fast Fourier Transforms, Spherical Harmonic Transforms, non-equispaced Fourier transforms, as well as some concrete applications like 4pi convolution on the sphere and gridding/degridding of radio interferometry data.

The code is written in C++17, but provides a simple and comprehensive Python interface.

Requirements

• Python >= 3.7
• only when compiling from source: pybind11
• only when compiling from source: a C++17-capable compiler, e.g.
  • g++ 7 or later
  • clang++
  • MSVC 2019 or later
  • Intel icpx (oneAPI compiler series). (Note that the older icpc compilers are not supported.)

Sources

The latest version of DUCC can be obtained by cloning the repository via

git clone https://gitlab.mpcdf.mpg.de/mtr/ducc.git

Licensing terms

• All source code in this package is released under the terms of the GNU General Public License v2 or later.
• Some files (those constituting the FFT component and its internal dependencies) are also licensed under the 3-clause BSD license. These files contain two sets of licensing headers; the user is free to choose under which of those terms they want to use these sources.

Documentation

Online documentation of the most recent Python interface is available at https://mtr.pages.mpcdf.de/ducc.

The C++ interface is documented at https://mtr.pages.mpcdf.de/ducc/cpp. Please note that this interface is not as well documented as the Python one, and that it should not be considered stable.

Installation

For best performance, it is recommended to compile DUCC from source, optimizing for the specific CPU on the system. This can be done using the command

pip3 install --no-binary ducc0 --user ducc0

NOTE: compilation requires the appropriate compilers to be installed (see above) and can take a significant amount of time (several minutes).

Alternatively, a simple

pip3 install --user ducc0

will install a pre-compiled binary package, which makes the installation process much quicker and does not require any compilers to be installed on the system. However, the code will most likely perform significantly worse (by a factor of two to three for some functions) than a custom built version.

Additionally, pre-compiled binaries are distributed for the following systems.

Installing multiple major versions simultaneously

The interfaces of the DUCC components are expected to evolve over time; whenever an interface changes in a manner that is not backwards compatible, the DUCC major version number will increase. As a consequence it might happen that one part of a Python code may use an older version of DUCC while at the same time another part requires a newer version. Since DUCC's major version number is included in the module name itself (the module is not called ducc, but rather ducc<X>), this is not a problem, as multiple DUCC versions can be installed simultaneously. The latest patch levels of a given DUCC version will always be available at the HEAD of the git branch with the respective name. In other words, if you need the latest incarnation of DUCC 0, this will be on branch "ducc0" of the git repository, and it will be installed as the package "ducc0". Later versions will be maintained on new branches and will be installed as "ducc1" and "ducc2", so that there will be no conflict with potentially installed older versions.

DUCC components

ducc.fft

This package provides Fast Fourier, trigonometric and Hartley transforms with a simple Python interface. It is an evolution of pocketfft and pypocketfft which are currently used by numpy and scipy.

The central algorithms are derived from Paul Swarztrauber's FFTPACK code.

Features

• supports fully complex and half-complex (i.e. complex-to-real and real-to-complex) FFTs, discrete sine/cosine transforms and Hartley transforms
• achieves very high accuracy for all transforms
• supports multidimensional arrays and selection of the axes to be transformed
• supports single, double, and long double precision
• makes use of CPU vector instructions when performing 2D and higher-dimensional transforms
• supports prime-length transforms without degrading to O(N**2) performance
• has optional multi-threading support for multidimensional transforms

Design decisions and performance characteristics

• there is no explicit plan management to be done by the user, making the interface as simple as possible. A small number of plans is cached internally, which does not consume much memory, since the storage requirement for a plan only scales with the square root of the FFT length for large lengths.
• 1D transforms are significantly slower than those provided by FFTW (if FFTW's plan generation overhead is ignored)
• multi-D transforms in double precision perform fairly similar to FFTW with FFTW_MEASURE; in single precision ducc.fft can be significantly faster.

ducc.sht

This package provides efficient spherical harmonic transforms (SHTs). Its code is derived from libsharp, but has been significantly enhanced.

Noteworthy features

• support for any grid based on iso-latitude rings with equidistant pixels in each of the rings
• support for accurate spherical harmonic analyis on certain sub-classes of grids (Clenshaw-Curtis, Fejer-1 and McEwen-Wiaux) at band limits beyond those for which quadrature weights exist. For details see this note.
• substantially improved transformation speed (up to a factor of 2) on the above mentioned grid geometries for high band limits
• accelerated recurrences as presented in Ishioka (2018)
• vector instruction support
• multi-threading support

The code for rotating spherical harmonic coefficients was taken (with some modifications) from Mikael Slevinsky's FastTransforms package.

ducc.healpix

This library provides Python bindings for the most important functionality related to the HEALPix tesselation, except for spherical harmonic transforms, which are covered by ducc.sht.

The design goals are - similarity to the interface of the HEALPix C++ library (while respecting some Python peculiarities) - simplicity (no optional function parameters) - low function calling overhead

ducc.totalconvolve

Library for high-accuracy 4pi convolution on the sphere, which generates a total convolution data cube from a set of sky and beam a_lm and computes interpolated values for a given list of detector pointings. This code has evolved from the original totalconvolver algorithm via the conviqt code.

Algorithmic details:

• the code uses ducc.sht SHTs and ducc.fft FFTs to compute the data cube
• shared-memory parallelization is provided via standard C++ threads.
• for interpolation, the algorithm and kernel described in https://arxiv.org/abs/1808.06736 are used. This allows very efficient interpolation with user-adjustable accuracy.

ducc.wgridder

Library for high-accuracy gridding/degridding of radio interferometry datasets (code paper available at https://arxiv.org/abs/2010.10122). This code has also been integrated into wsclean (https://arxiv.org/abs/1407.1943) as the wgridder component.

Programming aspects

• shared-memory parallelization via standard C++ threads.
• kernel computation is performed on the fly, avoiding inaccuracies due to table lookup and reducing overall memory bandwidth

Numerical aspects

• uses the analytical gridding kernel presented in https://arxiv.org/abs/1808.06736
• uses the "improved W-stacking method" described in https://arxiv.org/abs/2101.11172
• in combination these two aspects allow extremely accurate gridding/degridding operations (L2 error compared to explicit DFTs can go below 1e-12) with reasonable resource consumption

ducc.nufft

Library for non-uniform FFTs in 1D/2D/3D (currently only supports transform types 1 and 2). The goal is to provide similar or better performance and accuracy than FINUFFT, making use of lessons learned during the implementation of the wgridder module.

ducc.misc

Various unsorted functionality which will hopefully be categorized in the future.

This module contains an efficient algorithm for the computation of abscissas and weights for Gauss-Legendre quadrature. For degrees up to 100, the solutions are computed in the standard iterative fashion; for higher degrees Ignace Bogaert's FastGL algorithm is used.