ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0651

Unit Coordinator: Daniel Ruprecht

Aims:

Students are able to list numerical methods for the solution of ordinary differential equations and explain their core ideas, repeat convergence statements for the treated numerical methods (including the prerequisites tied to the underlying problem), explain aspects regarding the practical execution of a method, select the appropriate numerical method for concrete problems, implement the numerical algorithms efficiently and interpret the numerical results.

Content:

• Numerical methods for Initial Value Problems: single step methods, multistep methods, stiff problems, differential algebraic equations (DAE) of index 1;
• Numerical methods for Boundary Value Problems: multiple shooting method, difference methods

Pre-requisites:

Analysis, Linear Algebra, Basic MATLAB knowledge

Reading list:

• E. Hairer, S. Noersett, G. Wanner: Solving Ordinary Differential Equations I: Nonstiff Problems
• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0654

Unit Coordinator: Matthias Schulte

Aims:

This course provides an introduction to probability theory and stochastic processes with special emphasis on applications and examples.

The first part covers some important concepts from measure theory, stochastic convergence and conditional expectation, while the second part deals with some important classes of stochastic processes. 

Content:

• Measure and probability spaces
• Integration and expectation
• Types of stochastic convergence
• Law of large numbers
• Central limit theorem
• Radon-Nikodym theorem
• Conditional expectation
• Martingales
• Markov chains
• Poisson processes 

Pre-requisites:

Familiarity with the basic concepts of probability

Reading list:

• H. Bauer, Probability theory and elements of measure theory, second edition, Academic Press, 1981.
• A. Klenke, Probability Theory: A Comprehensive Course, second edition, Springer, 2014.
• G. F. Lawler, Introduction to Stochastic Processes, second edition, Chapman & Hall/CRC, 2006.
• A. N. Shiryaev, Probability, second edition, Springer, 1996. 

ECTS Credits: 9   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0652

Unit Coordinator: Sabine Le Borne, Daniel Ruprecht

Aims:

Students can list classical and modern iteration methods and their interrelationships, repeat convergence statements for iterative methods and explain aspects regarding the efficient implementation of iteration methods.

They will learn the fundamental concepts of parallel programming and how to translate them into efficient, parallel code.

They will learn how to compile parallel code and how to model and measure performance of parallelized software. 

Content:

• Sparse systems: orderings and storage formats, direct solvers;
• Classical methods: basic notions, convergence;
• Projection methods;
• Rylov space methods;
• Preconditioning (e.g. ILU);
• Multigrid methods;
• Domain decomposition methods, shared versus distributed memory parallelization;
• Message Passing Interface;
• OpenMP;
• Threads;
• Processes;
• HPC architecture;
• Performance models;
• Speedup and parallel efficiency 

Pre-requisites:

• Analysis,
• Linear Algebra,
• Programming experience in C and C++, FORTRAN, Python or a similar programming language.

Reading list:

• Y. Saad. Iterative methods for sparse linear systems
• M. Olshanskii, E. Tyrtyshnikov. Iterative methods for linear systems: theory and applications
• Thomas Rauber, Parallel Programming : for Multicore and Cluster Systems, Berlin [u.a.] Springer 2010
• Bertil Schmidt, Parallel programming : concepts and practice, Amsterdam Morgan Kaufmann 201

Additional info:

Parallel programming for interdisciplinary mathematics (DT0857)

Scientific programming for interdisciplinary mathematics (DT0858)

ECTS Credits: 6   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |   Language: English   |   Code: DT0653

Unit Coordinator: Thomas Schmidt

Aims:

The module introduces to variational minimization problems and/or variational methods for PDEs.

It may cover problems in a classical smooth setting as well as theory in Sobolev spaces.

Content:

A selection out of the following:

• Model problems and examples (Dirichlet energy, isoperimetric and brachistochrone problems, minimal surfaces, Bolza and Weierstrass examples, …),
• Existence and uniqueness of minimizers by direct methods,
• Weak lower semicontinuity of (quasi)convex variational integrals,
• Necessary and sufficient (PDE) conditions for minimizers,
• Problems with constraints (obstacles, capacities, manifold and volume constraints, ...),
• Generalized minimizers (relaxation, Young measures, ...),
• Variational principles and applications,
• Duality theory,
• Outlook on regularity.

Pre-requisites:

A solid background in analysis and linear algebra is necessary.

Familiarity with functional analysis, Sobolev spaces, and PDEs can be advantageous.

Reading list:

• H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MOS-SIAM Series on Optimization 17, Philadelphia, 2014.
• G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, An Introduction, Oxford Lecture Series in Mathematics and its Applications 15, Clarendon Press, Oxford, 1998.
• B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, London, 2014.
• B. Dacorogna, Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Springer, Berlin, 2008.
• I. Ekeland, R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics 28, SIAM, Philadelphia, 1999.
• M. Giaquinta, S. Hildbrandt, Calculus of Variations 1, The Lagrangian Formalism, Grundlehren der Mathematischen Wissenschaften 310, Springer, Berlin, 1996.
• E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.
• F. Rindler, Calculus of Variations, Universitext, Springer, Cham, 2018.
• F. Santambrogio, Optimal Transport for Applied Mathematicians, Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications 87, Birkhäuser/Springer, Cham, 2015.
• M. Struwe, Variational Methods, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 34, Springer, Berlin, 2008.

ECTS Credits: 3   |   Semester: 2   |   Year: 1   |   Campus: Hamburg University of Technology   |      |   Code: DT0668

Unit Coordinator: Kathrin Heuking

Aims:

• To understand sentences and frequently used expressions related to familiar everyday expressions, university life and its requirements.
• To describe your place of residence and the city of Hamburg.
• To complete simple forms.
• To impart basic grammar structures and vocabulary.
• To enable to read and understand.

Content:

• To introduce somebody to somebody,
• Talking about yourself;
• Alphabet,
• Spelling,
• So called "w-questions",
• Numbers (1-1 Mio),
• Time,
• Days of the week,
• Pronunciation.
• Simple role plays.
• Grammar: word order, construction of German sentences, affirmative sentence, questions, regular verbs → present tense, auxiliary verbs "haben" and "sein", pronouns (Nominativ), negative answer to a question "nicht" and "kein". 

Reading list:

Will be announced in lectures.