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Semester 2 in Hamburg;

Numerical – Modelling Training

Mathematical modelling and simulation heavily rely on scientific computing, seen as a scientific area encompassing numerical analysis, finite-element methods, numerical optimization, parallel computing. These keywords are the core or applied mathematics in that they train the use of advanced computing capabilities to solve complex models in a wide set of disciplines. The second semester at TU Hamburg provides a selected list of subjects in this framework, plus some additional subjects that complement the student’s knowledge to continue with Year 2 studies.

The Institute of Mathematics at TU Hamburg puts into place a group combining four chairs: Applied Analysis, Computational Mathematics, Numerical Mathematics, and Stochastics.

The computational part of this semester is taught by experts in the fields such as Sabine Le Borne (Professor in numerical mathematics with longstanding experience with computational mathematics programs) and Daniel Ruprecht (an expert in the parallelization of numerical methods).

Two additional courses are offered in this semester. A first one on Probability Theory provides a sound theoretical basis on stochastic modelling and an overview on its applications. This course is taught by Prof. Matthias Schulte, an internationally acknowledged expert in the field with wide range of expertise on probability, stochastics, large deviations, and random graphs (this course prepares for the specialization branch taught in Nice). A second complementary course on Variational Calculus provides a sound basis to the topics of Semester 3 at TUHH on biomedical imaging. This course is taught by Prof. Thomas Schmidt, an expert of calculus of variations and geometric PDEs, and by Prof. Ingenuin Gasses, who is an internationally acknowledged applied mathematician and a longstanding expert in managing international MSc programmes as well.

#Semester 2 in Hamburg EMJMD InterMaths Study Track

Numerical – Modelling Training;

Campus

Hamburg University of Technology

Year

1

Semester

2

ECTS Credits

30

Language

English
Numerical treatment of ordinary differential equations

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

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

Variational calculus

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

Unit Coordinator: Thomas Schmidt, Ingenuin Gasser,

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.

#Consortium InterMaths EMJMD;

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