Parallel Box-Counting Method for Evaluating the Fractal Dimension of Analytically Defined Curves

Abstract

© 2020, Springer Nature Switzerland AG. Serial and parallel implementations of the algorithm intended for calculating the number of boxes for evaluating the fractal dimension of analytical curves are considered. The algorithm contains four stages: (1) preparing the data for calculations; (2) determining the boxes into which the curve fell; (3) counting the boxes that have an intersection with a curve; (4) counting the boxes of a larger size that intersect with a curve. The acceleration of computations performed by a parallel code (on OpenMP and CUDA) with respect to calculations performed by a serial code depending on the size of the box is investigated. Numerical experiments are carried out, the results of which exhibit a significant increase in performance for GPU calculations in the case of a large number of segments of the curve. A 100-fold increase in the computational speed is obtained for a curve containing a million segments with a billion boxes (box size is 2 - 15). The graphs depicting an increase in acceleration of parallel code performance with decreasing the box size and increasing the number of curve segments are shown. It is concluded that the efficiency of using the GPU begins with three million boxes and grows with an increase in the number of curve points.