In my dissertation for my Monte Carlo Simulations module in 3rd year, I explored the application of Monte Carlo simulation methods to solve problems in probability theory, stochastic processes, and numerical analysis, with practical applications like probability distribution sampling, Markov Chains, and solving differential equations.
Project Overview
- Random Variable Sampling & Importance Sampling: Generated samples from a heavy-tailed distribution using the inverse transform method. Applied the Kolmogorov-Smirnov test for validation and used importance sampling to estimate expectations.
- Markov Chain Monte Carlo (MCMC): Implemented MCMC algorithms to study Markov chain properties such as irreducibility, aperiodicity, and convergence to the stationary distribution, validated with simulated random walks.
- Solving Laplace & Poisson Equations: Applied Monte Carlo random walk methods to simulate electric potential between capacitor plates, comparing results to analytical solutions and analyzing error vs number of walks and resolution.
Key Results
- Validated random sampling methods against theoretical distributions.
- Confirmed the ergodic theorem through MCMC simulations.
- Numerically solved PDEs using Monte Carlo approaches with accurate error analysis.
- Analyzed convergence rates and trade-offs in simulation precision vs computation time.
Tools & Skills
- Programming: R
- Techniques: Monte Carlo Simulation, Importance Sampling, Markov Chains, Random Walks
- Mathematics: Probability Theory, Stochastic Processes, Numerical Analysis
- Validation: Kolmogorov-Smirnov Test, Error Metrics
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