TITLE:
Designing a Parallel Fractal Dimension Estimator
AUTHORS:
Athanasios I. Margaris
KEYWORDS:
Fractal Dimensions, Lyapunov Exponent, Open MP, MPI, Henon Map
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.13 No.12,
December
16,
2025
ABSTRACT: This paper presents a theoretical framework for parallelizing the FD3 algorithm, which estimates the capacity, information, and correlation dimensions of chaotic time series using the box-counting method. We propose a hybrid parallel implementation leveraging MPI as well as OpenMP to achieve scalable performance on modern high-performance computing architectures. The serial FD3 algorithm is analyzed in detail, followed by a comprehensive parallelization strategy for its key phases, including data retrieval, trajectory reconstruction, sorting, and dimension estimation. Theoretical models based on Amdahl’s and Gustafson’s Laws predict significant speedup, supported by complexity analysis and communication cost models. While experimental validation is planned as future work, this study provides a robust foundation for parallel fractal dimension estimation, with potential applications in chaotic system analysis and computational physics. The motivation for this research is that, although the FD3 algorithm is relatively old and more advanced and efficient algorithms have since been developed, its structure and operation allow for parallelization, which, by providing significant acceleration, could render it competitive once again and suitable for efficient use.