Stochastic Modeling and Simulation

Upon completing the module, the students master the basics of stochastic modelling and simulation. We first discuss discrete-time models, followed by two classic examples, and then continuous-time models.

Conditional probabilities, normal distributions, and scale-free distributions; Markov chains and their matrix representation, mixing times and Perron-Frobenius theory; Applications of Markov chains, such as the PageRank algorithm; Monte Carlo Methods: Convergence, Law of Large Numbers, Variance Reduction, Importance Sampling, Markov Chains Monte-Carlo Using Metropolis-Hastings & Gibbs Samplers; Random processes and Brownian motion: properties in 2, 3 and more dimensions, connection to the diffusion equation, Levy processes and anomalous diffusion; Stochastic differential equations (SDEs): Nonlinear transformations of Brownian motion (Ito calculus), Ornstein-Uhlenbeck process and other solvable equations; Examples from population dynamics, genetics, protein kinetics, etc.; Numerical simulation of SDEs: strong and weak error, Euler-Maruyama scheme, Milstein scheme.

Topic Prerequisites

Working knowledge of computer programming in any language (e.g. Matlab, Python, Java), basic knowledge of classical physics, and solid undergraduate knowledge in calculus, probabilities and statistics.

Program / Module

M.Sc. Computational Modeling and Simulation
Module: CMS-COR-SAP - Stochastics and Probability

Format

2 SWS lecture, 1 SWS exercise, 1 SWS tutorial, self-study
5 credits

Registration to the course

For students of the Master program "Computational Modeling and Simulation: via CampusNet SELMA

Teachers

Lecture: Dr. Abhishek Behera, Mr. Serhii Yaskovets & Prof. Ivo F. Sbalzarini
Exercises: Mr. Mohammad-Hadi Salehi & Dr. Nandu Gopan

Instruction language: ENGLISH

Script

Lecture notes are available as PDF here.

Suggested Literature

Feller - an introduction to probability theory and its applications, Wiley+Sons, 1957.
Robert & Casella - Monte Carlo statistical methods, Springer, 2004.

Time/Place

Winter Term 2024/25
First Lecture on 21/10/2024
First Tutorial on 24/10/2024

Lecture: 4 DS (13:00- 14:30), Mondays at HSZ/403
Exercises / Tutorials: 4 DS (13:00- 14:30), Thursdays at FOE/0244

LECTURES AND EXERCISES WILL BE IN PRESENCE FOR THE WHOLE SEMESTER, BUT SUPPORTED WITH ONLINE VIDEO RECORDINGS

Link to the videos in OPAL: https://bildungsportal.sachsen.de/opal/auth/RepositoryEntry/32365445134

Exam 2024/25

Please refer to the OPAL page of the course for exam related information.

Lecture syllabus/Handouts

Oct 09, 2023: Lecture 0 - Introduction to the Course and Organization
Oct 16, 2023: Lecture 1 - Probability refresher, conditional probabilities, Bayes' rule, random variables, discrete and continuous probability distributions, scale-free distributions
Oct 23, 2023: Lecture 2 - Transformation of random variables, pseudo- and quasi-random numbers, low discrepancy sequences, transformation algorithms: inversion, Box-Muller, accept-reject method, composition-rejection method
Oct 30, 2023: Lecture 3 - Discrete-time stochastic processes, discrete Markov chains and their matrix
Nov 06, 2023: Lecture 4 - Law of large numbers, Monte Carlo methods, example: MC integration, importance sampling
Nov 13, 2023: Lecture 5 - Monitoring variance, variance reduction
Nov 20, 2023: Lecture 6 - Rao-Blackwell, Markov Chain Monte Carlo (MCMC), detailed balance, convergence criteria, acceleration methods
Nov 27, 2023: Lecture 7 - Classic MCMC samplers 1: Gibbs sampling
Dec 04, 2023: Lecture 8 - Classic MCMC samplers 2: Metropolis-Hastings, convergence diagnostics, stopping conditions
Dec 11, 2023: Lecture 9 - Random Walks, Brownian motion in 1,2,3,n-dim, connection to diffusion, continuum limit of random walks
Jan 8, 2024: Lecture 10 - Monte-Carlo optimization: stochastic gradient descent, simulated annealing, evolution strategies, CMA-ES
Jan 15, 2024: Lecture 11 - Stochastic calculus, Ito calculus, Ornstein-Uhlenbeck process (analytical)
Jan 22, 2024: Lecture 12 - Numerical methods for SDE: Euler-Maruyama, Milstein, strong and weak convergence
Jan 29, 2024: Lecture 13 - Master equation, Fokker-Planck, Kolmogorov forward, Example: chemical kinetics