Note: It's best to view this readme in the openConv documentation.
The main goal of openConv is to provide fast and efficient numerical convolutions of symmetric and smooth kernels and data of equispaced data in Python with all actual calculations done in Cython. It is intended to work in conjunction with openAbel to calculate 2D convolutions of radially symmetric functions in atomic collisions. The most useful methods implemented in my module for that purpose use the Fast Multipole Method combined with arbitrary order end correction of the trapezoidal rule to achieve both fast convergence and linear run time. Other methods are implemented for comparisons.
In most cases this should be pretty simple:
- Clone the repository:
git clone https://github.com/oliverhaas/openConv.git - Install:
sudo python setup.py install - Run example:
python example000_exponential.py
This assumes dependencies are already met and you run a more or less similar system to mine (see Dependencies).
The code was run on several Ubuntu systems without problems. More specific I'm running Ubuntu 16.04 and the following libraries and
Python modules, which were all installed the standard way with either sudo apt install libName or
sudo pip install moduleName.
- Python 3.5.2
- Numpy 1.18.1
- Scipy 1.4.1
- Cython 0.29.14
- Matplotlib 3.0.3
- FFTW3 3.3.4
As usual newer versions of the libraries should work as well, and many older versions will too. I'm sure it's possible to get openConv to run on vastly different systems, like e.g. Windows systems, but obviously I haven't extensively tested different setups.
In contrast to other codes I made available, openConv has as of now only very specific use-cases I actually needed, thus implemented and debugged. I strongly recommend every user to thourougly check if the methods work as intended for their specific problem. For most people openConv will thus not be a useable code as is, but more a starting point or inspriration for their own code. If there are any issues, bugs or feature request just let me know. Gaps in the implementation might be filled by me if requested.
It is fairly common to use directly the discrete convolution to approximate the convolution integral, often with smaller improvements like using trapezoidal rule instead of rectangle rule. This yields usually neither good order of convergence (second order with trapezoidal rule), nor fast calculation (quadratic computational complexity). openConv intends to provide methods to calculate these convolutions efficiently, fast, and with high accuracy. Beside the common "fast convolution" algorithm based on the Fast Fourier Transform we provide methods based on the Fast Multipole Method and high order end correction, which outclass common methods in many cases in most aspects (convergence order, error, computational complexity, etc.), as long as the kernel is smooth.
For the most important methods of we adapted the Chebyshev interpolation Fast Multipole Method (FMM) as described by Tausch and calculated end corrections for smooth functions similar to Kapur. If data points outside of the integration interval can be provided these end corrections are arbitrary order stable and we provide coefficients up to 20th order, otherwise it's recommended to use at most 5th order. The FMM leads to an linear O(N) computational complexity algorithm. For approximately exponentially decaying functions, like e.g. often encountered in atomic physics, we introduced an exponential shift into the Chebyshev interpolation to get relative errors of the convolution result of up to machine precision.
In both error and computational complexity there is no better existing method for the intended purpose to my knowledge.
In the documentation and the examples more details are discussed and mentioned; in general both are a good way to learn how to understand and use the code.
Copyright 2016-2020 Oliver Sebastian Haas.
The code openConv is published under the GNU GPL version 3. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
For more information see the GNU General Public License copy provided in this repository LICENSE.