Overview

SemiDiscreteOT is a high-performance C++ library for solving regularized semi-discrete optimal transport problems. It provides efficient numerical strategies for computing optimal transport plans between continuous and discrete probability measures, with a focus on large-scale applications.

The methods implemented in this library are detailed in the publication: **"Efficient Numerical Strategies for Entropy-Regularized Semi-Discrete Optimal Transport"** by Moaad Khamlich, Francesco Romor, and Gianluigi Rozza (2025).

Features

Efficient Solvers: Fast and scalable algorithms for regularized semi-discrete optimal transport.
Performance Optimizations: Includes distance-based truncation based on R-tree spatial queries.
Multilevel Acceleration: Supports multilevel techniques for both source and target measures to handle large-scale problems.
Flexible: Integrates with Deal.II for finite element computations and supports both structured and unstructured meshes.

Dependencies

Deal.II (9.5.0 or later)
Geogram
VTK
CMake (3.13.0 or later)
A C++17 compatible compiler

Quick Start

Clone the repository: bash git clone https://github.com/SemiDiscreteOT/SemiDiscreteOT.git cd SemiDiscreteOT
Build the library: bash mkdir build && cd build cmake .. make
Run an example: Check out the examples directory for detailed use cases.

Documentation

API Documentation

Online Documentation: https://semidiscreteot.github.io/SemiDiscreteOT/ (Auto-generated with Doxygen)

Build Documentation Locally

doxygen Doxyfile

xdg-open docs/doxygen/html/index.html

How to Cite

If you use SemiDiscreteOT in your research, please cite the following publication:

@misc{khamlich2025efficientnumericalstrategiesentropyregularized,
      title={Efficient Numerical Strategies for Entropy-Regularized Semi-Discrete Optimal Transport}, 
      author={Moaad Khamlich and Francesco Romor and Gianluigi Rozza},
      year={2025},
      eprint={2507.23602},
      archivePrefix={arXiv},
      primaryClass={math.NA},
      url={https://arxiv.org/abs/2507.23602}, 
}

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

License

This project is licensed under the GNU Lesser General Public License v3.0 (LGPL-3.0) - see the [LICENSE](LICENSE) file for details. For more information about LGPL-3.0, visit the GNU website.

Authors

Moaad Khamlich (PhD student at SISSA)
Francesco Romor (Postdoc at Weierstrass Institute, Berlin)