Advanced Monte Carlo Integration & Optimization Library

This project is a comprehensive C++ library designed to solve complex mathematical and engineering problems using advanced Monte Carlo integration and stochastic optimization algorithms.

The Challenge

Evaluating definite integrals over high-dimensional spaces or non-convex geometries is computationally expensive. The goal of this project was to build a highly parallelized, robust framework capable of executing multidimensional numerical integration and metaheuristic optimizations efficiently.

Architectural Overview & Core Features

The system is built with a focus on modularity and high performance, featuring runtime mathematical expression parsing via the muParserX library.

1. Integration Engines

  • Classic & Importance Sampling: Implemented baseline uniform sampling alongside Gaussian and Mixture Importance Sampling for significant variance reduction on smooth integrands.
  • Markov Chain Monte Carlo (MCMC): Utilized the Metropolis-Hastings algorithm for efficient sampling in complex, high-dimensional domains with superior $O(1/N)$ convergence.
  • Arbitrary Polytopes: Integrated with the Qhull library to support integration over arbitrary convex polytopes using half-space inequalities.

2. Stochastic Optimization

  • Particle Swarm Optimization (PSO): Engineered a fast, parallelized PSO solver to find global minima in smooth, continuous landscapes.
  • Genetic Algorithm (GA): Developed a robust GA featuring tournament selection, uniform crossover, and Gaussian mutation for non-smooth or multimodal landscapes.

Real-World Engineering Applications

To prove the framework’s capabilities, we developed two specialized physics-based applications:

  • Drone Center of Mass Optimization: Combined PSO with Monte Carlo integration to optimize the placement and size of a hole in a 3D drone arm structure, successfully shifting its center of mass to a precise target coordinate with < 1% error.
  • Wind Farm Layout Simulation: Optimized the placement of 15 wind turbines within a 1000x1000m area. The simulation uses Metropolis-Hastings integration to estimate farm power while accounting for Weibull wind distributions and aerodynamic wake effects, optimizing layouts via both PSO and GA.

Technical Highlights

  • HPC & Concurrency: Achieved significant performance scaling (~10x speedup on 8 cores) by parallelizing particle evaluations and implementing thread-safe Random Number Generation (RNG) using OpenMP.
  • Performance Profiling: Utilized the Linux perf toolkit extensively to analyze CPU cycles, cache misses, and bottleneck functions, driving critical optimization decisions.
  • Data Visualization: Automated the generation of convergence plots and 3D geometry renderings using Gnuplot scripts directly from the C++ output.

Source Code: View on GitHub