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Advanced Monte Carlo Methods

Recommended for: 5-6th year of Specialist’s program
1-2nd year of Master’s program
Start: September 23, 2021, 4:45 pm

Lectures
Day: Thursday Time: 4:45-6:20pm Language: English Format: online
Seminars
Day: Thursday Time: 8:10-9:40pm Language: Russian Format: online
Summary
This course will cover topics in the general area of Monte Carlo methods and their application domains. The topics include Markov chain Monte Carlo and Sequential Monte Carlo methods, Quantum and Diffusion Monte Carlo techniques, as well as branching and interacting particle methodologies. The lectures cover discrete and continuous time stochastic models, starting from traditional sampling techniques (perfect simulation, Metropolis-Hasting, and Gibbs-Glauber models) to more refined methodologies such as gradient flows diffusions on constraint state space and Riemannian manifolds, ending with the more recent and rapidly developing Branching and mean field type Interacting Particle Systems techniques (forward/backward particle filters, Ensemble Kalman filers, interacting Kalman filters, Sequential Monte Carlo, genealogical tree
based samplers, and many others).
The course offers a pedagogical introduction to the theoretical foundations of these advanced stochastic models, combined with a series of concrete illustrations taken from different application domains. The applications considered in these lectures will range from Bayesian statistical learning (hidden Markov chain, statistical machine learning), risk analysis and rare event sampling (mathematical finance, and industrial risk assessment), operation research (global optimization, combinatorial counting and ranking), advanced signal processing (stochastic nonlinear filtering and control, and data association and multiple objects tracking), computational and statistical physics (Feynman-Kac formulae on path spaces, molecular dynamics, Schredinger's ground states, Boltzmann-Gibbs distributions, and free energy computation). Approximately the first half of the course will be concerned with linear type Markov chain Monte Carlo methods, and the second part to nonlinear particle type methodologies, including interacting diffusions, interacting jump processes and genealogical tree based samplers. A list of topics intended to be covered is attached.

Course Schedule
PART 1: Linear Monte Carlo methods
Markov chains
Continuous time models
Markov chain Monte Carlo models
Advanced Markov chain monte Carlo models

PART 2: Illustrations
Computational physics
Bayesian inference
Signal processing

PART 3: Nonlinear Monte Carlo methods
Nonlinear Markov chains models and their mean field particle interpretations
Branching and interacting particle interpretations of Feynman-Kac models
Particle Markov chain Monte Carlo methodologies
Analysis toolbox

PART 4: Illustrations
Bayesian statistical inference
Signal processinge
Computational physics
Operation research
Mathematical finance

See full course outline.