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Direction Field Visualization with Python

The scope of this project is mathematics, numerical / scientific computing and high performance computing and the case study differential equation is the differential equation of a falling object. It is an extension of my How to Plot a Direction Field with Python article posted on medium. You can find it here.

Objectives

The overall objective of this project is to demonstrate the visualization of a direction field with Python. Specifically, the project aims to address the following objectives:

O1. To use the differential equation of a falling object as a case study.

O2. To use an open source application softwar in developing programs for the direction field plot.

O3. To explore the numerical package of the python software (numpy) in developing programs for the direction field plot.

O4. To extend the functionality of the programs developed in O3. with packages provided in the Intel’s Distribution for Python* (IDP*) in order to demonstrate heterogeneous computing.

O5. To explore the possibilities of increasing performance of the python programs developed in 03. with the data parallel package offered by Intel.

Approach

In this project, the implementation of a direction field plot with python is based on two methods:

  1. The matplotlib.pyplot.quiver() (MPQ) method.

  2. The straight line equation (SLE) method.

I consider these two methods for the purpose of comparison of performance based on the time of execution as well as for the purpose of exploring the following packages in the Intel® Distribution of Python* (IDP*):

The following packages in DPPY are explored :

1. Numpy - an optimized Python numerical package.

2. The numba.prange expression provided by Numba* - an open-source, NumPy-aware optimizing compiler for Python developed by Anaconda, 
Inc in collaboration with an open-source community.

3. SYCL-based XPU programming provided by Data Parallel Python (DPPY). The following packages in DPPY are explored :

   • Data Parallel Control (dpctl) - A package for controlling execution on SYCL devices and for SYCL USM data management.
   • Numba_dppy - A standalone extension to Numba adding SYCL kernel program-ming to Numba*.

Technologies Used

Development Environment: Intel devcloud - for developing, testing and running the project.

Intel Hardware: Intel® Xeon® E-2176G.

oneAPI Toolkits: oneAPI Base Toolkit.

Programming Support: Intel® Distribution for Python®.

Build source files

cd build_src

cd build_quiver

./*.sh

cd build_line_equation

./*.sh

Read the Full documentation and performance results

  1. Read on Github.

  2. Read on Intel DevMesh.

References

  1. W. E Boyce and R. C DiPrima, Elementary Differential Equations and Boundary Value Problems, eigth edition. USA: John Wiley & Sons, Inc. 2005.
  2. Steven Holzner, Differential Equations For Dummies. Indiana: Wiley Publishing, Inc. 2008.
  3. B. D Bunday and H Mulholland, Pure Mathematics for Advanced Level, second edition. Nigeria: Heinemann Eductional Books (Nigeria) Plc. 2004.
  4. P. Howard, Ordinary Differential Equations in MATLAB. 2013.
  5. https://medium.com/@olutosinbanjo/how-to-plot-a-direction-field-with-python-1fd022e2d8f8
  6. https://www.r-bloggers.com/2014/09/generate-slope-fields-in-r-and-python/amp/
  7. https://github.com/IntelPython/dpnp
  8. https://github.com/IntelPython/dpctl
  9. https://numpy.org/doc/
  10. https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.html