Stochastic Methods in Scientific Computing

1. Random Numbers. 1.1. Random numbers and probability distribution. 1.2. Central limit theorem. 1.3. Beyond the Normal distribution. 1.4. Exercises. 2. Random walks. 2.1. Random walk as a Markov process. 2.2. Random walks in 1 and 2 dimensions. 2.3. Levy flight. 2.4. Random walks with potentials. 2.5. Exercises. 3. Monte Carlo methods. 3.1. Objectives and concepts. 3.2. Monte-Carlo integration. 3.3. Markov Chain Monte-Carlo. 3.4. Advanced Error Analysis Techniques. 3.5. Error estimate in the presence of autocorrelation. 3.6. Error estimate for non-trivial estimators: The Jackknife, and the Bootstrap. 3.7. Biased Estimators. 3.8. Exercises. 4. Statistical models. 4.1. An introduction to thermodynamics. 4.2. From thermodynamics to statistical mechanics. 4.3. Phase transitions. 4.4. The Ising model. 4.5. An overview of other models. 4.6. Exercises. 5. Advanced Monte-Carlo simulation techniques. 5.1. Hamiltonian (Hybrid) Monte-Carlo (HMC) simulations. 5.2. Non-local Monte-Carlo update. 5.3. Micro-canonical simulations. 5.4. Flat histogram methods. 5.5. The Linear Logarithmic Relaxation (LLR) method. 5.6. Exercises. 6. From Statistical Systems to Quantum Field Theory. 6.1. Invitation: The O(2) model. 6.2. The Bridge to QFT: the Feynman path-integral. 6.3. Gauge Theories. 6.4. Adding fermion fields. 6.5. Exercises. 7. Current challenges in Monte-Carlo Simulations. 7.1. Sign and overlap problems. 7.2. Introduction to overlap problems. 7.3. Estimating probability density functions. 8. Data Analytics and Statistical Systems. 8.1. Model regression - L2 norm. 8.2. Gaussian Process. 8.3. Machine learning with graphs. 8.4. Emulation of statistical systems with Machine Learning. 8.5. Categorisation in statistical physics: Naive Bayes. 8.6. Machine learning classification of phase transitions.