Weekly syllabus

week        topics
1 Overview of Monte Carlo, applications, methods, problems. Discussion of projects.
       Assignment 1, part 1 (Python and C++ setup) and part 2 (Monte Carlo) given
2 Direct sampling methods, mapping, rejection, weighted sampling. Central limit theorem, error bars, and MC computing practice. Variance reduction methods, antithetic variates, control variates. Rare events and importance sampling.
       Assignment 1, part 1 due. Preliminary project proposals due.
3 Markov chain Monte Carlo (MCMC): Definition of Markov chains and invariant distributions. The Perron Frobenius theorem and the Markov chain Ergodic theorem. Detailed balance and Metropolis sampling, Partial resampling/heat bath/Gibbs sampler.
       Assignment 1, part 2 due. Feedback on project proposals.
       Assignment 2 given.
4 MCMC error bars. Markov chain central limit theorem and the Kubo formula. Estimating the autocorrelation time. Burn-in and checking for steady state. Computating practice. Linear Markov chains.
       Final project proposal due.
5 Bayesian statistics and uncertainty quantification. Definition of prior distribution and likelihod model. Large sample theory, Bernstein theorem, and Fisher information. Parameter identification in dynamical models. Hierarchical models.
       Assignment 2 due, feedback on project proposal.
       Assignment 3 given.
6 Field and critical point problems in statistical physics. Some field models, including the Ising model and the lattice Gaussian free field. Models of fluids, Leonard Jones and other interactions. Thermodynamic quantities of interest, such as specific heat. Critical phenomena, critical exponents, critical slowing down.
       Project literature plan due.
7 Sampling using continuous time stochastic dynamics. Langevin equation. Hamiltonian sampler. Metropolizing.
       Assignment 3 due, feedback on project literature plan.
       Assignment 4 given.
8 Multi-scale samplers for statistical physics. Multigrid Monte Carlo. Swendsen Wang.
       Project code design due.
9 Optimizing functions evaluated by Monte Carlo -- stochastic approximation. Robbins Munro/stochastic gradient descent. Methods for stochastic sensitivity analysis, score functions and same paths approximation. Stochastic sample approximation (SSA).
       Assignment 4 due. Feedback on project code design.
       Assignment 5 given.
10 Variable temperature and umbrella sampling. Simulated annealing for optimization. Simulated and parallel tempering. Multi-histogram and thermodynamic integration.
       Project midway progress report due.
11 Strategies for rare event simulation. Large deviation theory, variational principles for the most likely rare event. Checkpoint methods. Nested sampling.
       Assignment 5 due. Feedback progress report.
       Assignment 6 given.
12 Theory I, capacitance methods. Bottlenecks in Markov chains. Capacitance and Cheeger's inequality. The convergence estimate of Lovasz and Simonovitz. Sampling in a convex set (sketch).
       First projet results due -- at least one graph with some results.
13 Theory II, potential functions and Poincare and Sobolev inequalities. Potential/Lyapounov functions . Spectral gaps and Poincare inequalities. Sampling a log-convex function, sampling random permutations.
       Feedback on first results.
14 Project presentations. Project report due.