In this unit you will learn the main theoretical aspects and numerical methods employed for Monte Carlo calculations on the lattice.
Basic knowledge of fundamentals of statistics.
Review of probability and statistics: Laws of large numbers, CLT, sampling and properties of sample averages. Variance and covariance.
Introducing Monte Carlo: Why use Monte Carlo? Variance reduction and importance sampling. Examples.
Drawing from simple distributions: Transformation and rejection methods. Box-Muller, SU(2) heatbath.
Markov chains: Properties and convergence. Detailed Balance. Metropolis.
Local (bosonic) theories: Gibbs samplers (Cabbibo-Marinari). Over-relaxation. Example: SU(2) Yang-Mills.
Autocorrelations: Definitions and examples. Connection to physics - correlation lengths and scaling, topology freezing.
The fermion determinant: Path integral and Grassmann integral. Gaussian representation and pseudofermions. Non-locality problem.
Hamiltonian dynamics: Equations of motion. Dynamics on Lie manifolds. Symplectic integrators (leap-frog) and step-size errors. Metropolis, reversibility and area preservation.
Fermions and pseudofermions: Equations of motion. The resulting linear systems.
Advanced methods: Higher-order and Omelyan. Multiple time-scales. Force-gradient, Hasenbusch.
Hadronic two-point functions: Wick contractions. Point and smeared sources.
Disconnected diagrams: All-to-all techniques and stochastic representations.
Introduction and examples: Factorising path integrals. Fermion methods.
Krylov spaces: conjugate gradient and BiCG. Preconditioning (even-odd).
Eigensolvers: Lanczos and Arnoldi. Tridiagonal forms. re-orthogonalisation and round-off.
Deflation and Multigrid