Foundations of Monte Carlo and quasi-Monte Carlo methods.

Syllabus

• Introduction. Reminders about the convergence of moments estimators. Uniform law generators. Other law generators: distribution function inversion method, rejection method and conditional laws, transformation method with application to the generation of Gaussians, correlated variables and mixture of laws and conditioning approach.
• Error control and the variance reduction method. Reminders of techniques for evaluating estimation error. Antithetic control. Control variable. Importance sampling. Stratification and post-stratification. Latin hypercube sampling.
• Quasi-Monte Carlo method. Uniform sequences over the unit cube and discrepancy. Functions of limited variation in the sense of measures, Koksma-Hlawka inequality and numerical integration. Example of low-discrepancy sequences. Randomized deterministic sequences. Concepts of effective dimension and reduction of total variation.

Link to the webpage of the course.