DOI: 10.5281/zenodo.14809516
title: Supplementary material for  Adaptive Cross Approximation with Geometrical Pivot selection (ACA-GP) paper

description: Datasets for ACA-GP: an Adaptive Cross Approximation with Geometric Pivot Selection for Smooth Kernel Matrices

 

Author: Vladislav A. Yastrebov, ORCID: 0000-0002-4052-3557

Affiliation: CNRS, Centre des matériaux, Mines Paris - PSL, Evry/Paris, France

License: CC BY 4.0

 

Description

This dataset accompanies the paper "Adaptive Cross Approximation with a Geometrical Pivot Choice: ACA-GP Method" (arxiv) by V.A. Yastrebov and C. Noûs. The source code used for data generation is available on GitHub (github.com/vyastreb/ACA), and its instance used to generate the data can be accessed at: (https://archive.softwareheritage.org).

 

This dataset includes numerical results and visualization files related to (1) the genetic construction of skeleton approximations of kernel matrices and (2) the comparison of approximation methods (ACA, ACA-GP, and SVD) for two separate interacting clouds with different point distributions and configurations.





 

Keywords
Low-rank approximation, Adaptive Cross Approximation, Geometrical Pivots, Boundary Integral Method, Boundary Element Method, Hierarchical Matrices, Performance Tests, Error Structure.


 

Genetic Construction of the Skeleton Approximation of the Interaction Matrix

The folder `Genetic_Cross_Approximation/` contains datasets for analyzing the *genetic construction* of the skeleton approximation of the interaction matrix $A$. Each subfolder contains:

- .npz files with point cloud data (400 x 400 points).

- .npz files storing genetic algorithm results for skeleton approximation of the kernel matrix at different ranks (1 to 5).

- .png images depicting error distributions for randomly selected approximations for different ranks with approximate *extreme curves* (see accompanying paper for details).

- A Jupyter notebook Error_geometry.ipynb for the construction of these clouds from genetic data.



 

Generating these data is computationally intensive, requiring approximately 6-8 hours on a 32-core machine due to the high complexity of the brute-force method.

 


The subfolders are structured as follows:

- Genetic_Regular_Square: Regularly distributed points within a square.

- Genetic_Random_Square: Randomly distributed points within a square.

- Genetic_Regular_Rectangle_xi_05: Regularly distributed points within a rectangle (aspect ratio $\xi = 0.5$).

- Genetic_Random_Rectangle_xi_05: Randomly distributed points within a rectangle (aspect ratio $\xi = 0.5$).

 

Approximation of the Interaction Matrix using ACA, ACA-GP, and SVD Methods

The folder ACA_GP_cloud_performance/ contains results of approximations applied to point clouds with randomly distributed points at fixed distances, using ACA, ACA-GP, and SVD methods. The datasets cover:

- Distances: $ \text{dist} \in \{1.5, 2.5, 5.0\}$.

- Aspect ratios: $\xi \in \{1, 0.5\}$ (square and rectangular domains).

- Central subset fraction radius: $\varepsilon_r \in (0.1, 0.5)$.

 

Each dataset consists of:

- .npz files containing approximation error data for ACA, ACA-GP, and SVD methods (1000 samples for square domains, 500 samples for rectangular domains).

- .json files storing corresponding parameters.

- .pdf and .png visualization files for error distribution and performance gains.

- Python scripts for data processing and visualization (plot_data.py, plot_central_fraction_effect.py).

 

Naming Convention

Files in ACA_GP_cloud_performance/ follow the structure:

- ACA_GP_data_Dist_{dist}_distribution_uniform_xi_{xi}_cf_{cf}_ID_{ID}.npz - Approximation results.

- ACA_GP_data_Dist_{dist}_distribution_uniform_xi_{xi}_cf_{cf}_ID_{ID}.json - Corresponding parameters.

- ACA_GP_error_Dist_{dist}_distribution_uniform_xi_{xi}_cf_{cf}_ID_{ID}.pdf/.png - Approximation error visualizations.

- ACA_GP_gain_Dist_{dist}_distribution_uniform_xi_{xi}_cf_{cf}_ID_{ID}.pdf/.png - Gain analysis for ACA-GP relative to ACA.

 

+ {dist}: True distance between interacting clouds.

+ {xi}: Aspect ratio of the domain.

+ {cf}: Central subset fraction radius.

+ {ID}: Sample ID.

 

Usage

This dataset enables further analysis and reproduction of results presented in the referenced paper. The provided Jupyter notebooks and Python scripts facilitate the visualization and exploration of the approximation performance.

 

For further details, refer to:

- The paper: arXiv:2502.03886

- The GitHub repository: github.com/vyastreb/ACA

- The Software Heritage instance of the code used in the paper: archive.softwareheritage.org

 



How to cite this dataset
V.A. Yastrebov, Datasets for ACA-GP: an Adaptive Cross Approximation with Geometric Pivot Selection for Smooth Kernel Matrices, Zenodo, DOI: 10.5281/zenodo.14809517 (2025).