Lecturer
Dr. Benedikt Prifling

Teaching assistant
Tran Duc Nguyen

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Time and Place

Lecture
Tuesday, 8-10 am (220, Helmholtzstr. 18)

Excercise session
Wednesday, 4-6 pm (E60, Helmholtzstr. 18) every second week
 

News

The first lecture takes place on Tuesday, October 15th. The first exercise class takes places on Wednesday, October 30th.

Students who are interested in a master thesis covering the application of Monte Carlo methods, e.g. to problems in materials science, with a focus on stochastic geometry, spatial statistics, image processing or machine learning are kindly invited to write an e-mail to benedikt.prifling(at)uni-ulm.de.

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Type

2 hours lecture (weekly) and 2 hours excercise class (every second week). 5 ECTS credit points. The course is tought in English.

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Prerequisites

Basic knowledge of probability theory as taught for example in "Elementare Wahrscheinlichkeitsrechnung und Statistik".
Basic programming skills as thought for example in "Einführung in die Informatik 1 & 2".
 

Intended Audience

Master students in "Mathematik", "Wirtschaftsmathematik", "Finance", "Mathematical Data Science" and "Mathematische Biometrie".

Contents

Many real world problems cannot be solved analytically due to their complex nature. For this purpose, methods from Monte Carlo simulation can be used, which are based on the generation of synthetic data via stochastic simulation algorithms. In particular, due to the rapidly increasing computing capabilities and the ever growing amounts of available data, stochastic modeling approaches are becoming increasingly important. In this course, the main focus will be on the simulation of various stochastic objects such as univariate and multivariate probability distributions or Markov chains. In the first part of the lectures, the generation of pseudo-random numbers including their statistical properties is discussed, which serves as building block for the simulation of more complex stochastic objects. Next, several concepts for sampling from univariate and multivariate probability distributions such as the acceptance-rejection method or importance sampling are discussed. Moreover, time-discrete Markov chains with finite state space are introduced and fundamental mathematical properties such as stationarity are considered. This also lays the basis for methods from Markov Chain Monte Carlo (MCMC), which among others include the Gibbs sampler and the Metropolis-Hastings algorithm. Since methods from Monte Carlo simulation are also a powerful tool for parameter estimation, variance reduction techniques are discussed, which are of particular importance in applications with high computational demands. Besides a mathematical treatment of these topics, this course also includes the implementation of various algorithms in Matlab.

Moodle course

If you are interested in attending this course, please subscribe to the Moodle course, where all further information (exercise sheets, etc.) will be shared.

Requirements and Exam

In order to participate in the final exam, it is necessary to earn 50% of the points on the problem sheets.

Literature

S. Asmussen, P. Glynn. Stochastic Simulation: Algorithms and Analysis. Springer. 2007.

G. Fishman. Monte Carlo Concepts, Algorithms, and Applications. Springer. 2013.

C. Graham and D. Talay. Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation. Springer, 2013.

D. Kroese, T. Taimre, and Z. Botev. Handbook of Monte Carlo Methods. John Wiley & Sons, New York, 2011.

S. Ross. Simulation. 6th edition. Academic Press, 2022.

R. Rubinstein, D. Kroese. Simulation and the Monte Carlo Method. 3rd edition. John Wiley & Sons. 2016.