Parallelized Mandelbrot Set Generator with OpenMP

Introduction

A Mandelbrot set is defined as the set of all complex numbers ${c}$ for which the generator function:

$f_c(z) = z^2 + c$

remains bounded (i.e., does not diverge to $\pm \infty$). We start with $z=0$ and iteratively evaluate $f_c$ as follows:

$[f_c(0), f_c(f_c(0)), f_c(f_c(f_c(0))),\dots]$

Defining Divergence

Traditionally, divergence is defined by a threshold of the magnitude of the current element in the sequence. For example, if we choose a threshold of $f_{max} = 10$, then the sequence diverges at the $i^{th}$ iteration if $|f_c(i)| > f_{max} = 10$.

Methods

Approach A: Fine-Grained Parallelization

• Pixel-level parallelization, each thread evolves the Mandelbrot set for differnt starting values $c$ in the complex plane.
• Poor scaling due to critical access of shared data buffer.

Approach B: Parallel Frame Generation for Zoom Animation

• Frame-by-frame parallelization when generating multiple frames of the Mandelbrot set at different scales to create a movie (i.e. zooming into one point of the plane).
• Each frame is assigned to a different thread for parallel processing.
• The generated frames are then compiled into a video.
• Much more efficient scaling, each thread works on independent chunck of data

Generated Outputs

Mandelbrot Image

Mandelbrot Zoom Animation

Generated animation of the Mandelbrot set zooming into specific regions:

• Seahorse Valley $(-0.743643887037151 + 0.131825904205330i)$: Download Seahorse.mp4

  

• Elephant Valley $(0.282 + 0.5307i)$: Download Elephant.mp4

  

• Feigenbaum Point $(-1.401155 + 0i)$: Download Feigenbaum.mp4

  

The colormap uses the Escape-Time Algorithm to provide a cool visualization of the complex plane.

Running the Project

Dependencies

• g++ with OpenMP support
• ffmpeg (for compiling frames into a video)

Compiling and Running (Frame Generation)

1. Compile the Mandelbrot generator:

   g++ -fopenmp -o mandelbrot_generator main.cpp src/*cpp -I ./include -O3

   Note: if program crashes try reducing compiler optimizations by removing -O3 flag.

2. Run the generator with parallel processing:

   ./mandelbrot_generator 8 100 "Seahorse" true "./figures/frames/" 1.025 0.001

   • 8 → Number of processors
   • 100 → Number of frames
   • "Seahorse" → Region of interest
   • true → Save frames
   • "./figures/frames/" → Directory to save frames
   • 1.05 → The zoom factor, set to 5.0% zoom between frames
   • 0.001 → Standard resolution (defined as distance between grid points so smaller value is more resolution) for a frame without zoom, scales with dimensions

   Note: resolution significantly increses the computational load with $O(r^2)$, and computing 100 frames at the above resolution takes ~10min with 32 threads of execution. Try changing resolution to 0.005 or 0.01 for quicker execution.

3. Convert frames into an MP4 video:

   python3 makeVideo.py ./figures/frames ./seahorse-100.mp4 --fps 30

Future Improvements

• Implement GPU acceleration using CUDA.
• Optimize frame generation with better load balancing.
• Enable dynamic thresholding to improve visualization quality.
• Reuse computation between frames.