MathMods :: Joint MSc

Sem3 (T)UHH Complex S.

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Applications @ TUHH or UHH 30 ECTS credits

Modelling and simulation of complex systems

The semester in Hamburg focuses on fundamental principles of contemporary computational mathematics. Concrete model scenarios arising from academic and industrial applications are used as case studies.

The portfolio of the offered courses ranges from theoretical fundamentals of mathematical modelling and numerical simulation to concrete applications in computational science and engineering. The branch "Modelling, simulation, and optimization of complex systems" will be spent at TUHH in case the programme is funded by the Erasmus Mundus scheme, at UHH otherwise. In any case, the two universities at Hamburg will work in synergy with each other for the delivery of the courses.

 

Below you can find information about the subjects for this semester.

Semester 3 @ TUHH

  • Business & Management [6 credits]

    Business & Management

    • ECTS credits 6
    • Semester 3
    • University Hamburg University of Technology

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  • Research project and seminar [18 credits]

    Research project and seminar

    • ECTS credits 18
    • Semester 3
    • University Hamburg University of Technology
    • Objectives

       

      The research project is handled in a rather flexible way at TUHH and provides excellent opportunities for an early focus on research. The student chooses an advisor and they agree on an agenda for the semester, typically preparing the ground for the Master thesis in the fourth semester.

    • Topics

      Activities in the project consist of a combination of various forms: guided reading, systematic literature review, numerical experiments, proving special cases, and also taking part in lecture courses. The standard lecture core for this project are:

      • Mathematical Systems and Control Theory (6 ECTS, Reis, UHH),
      • Optimisation of complex systems governed by ODEs and PDEs (6 ECTS, Hinze, UHH),
      • Training seminars (6 ECTS, TUHH & UHH).

      This can be adapted to the specific needs of each project if advisor and student agree.


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  • Traffic flow models [6 credits]

    Traffic flow models

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      The module introduces basic modelling concepts and the related mathematical methods for the description of vehicular traffic flow.

    • Topics

       

      Vehicular traffic flow phenomena, Modelling approaches, Cellular automata. Microscopic models: models based on ODE's. Macroscopic models: models based on PDE’s.


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  • Computer tomography [6 credits]

    Computer tomography

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      This course gives an introduction to mathematical methods of medical imaging, where particular focus is placed on computer(ized) tomography (CT). But also algebraic reconstruction techniques (ART) will be discussed. If time allows, a short overview over m...

    • Topics

       

      X-rays; the Radon transform; the back projection; the central slice theorem; the filtered back projection formula; discrete image reconstruction; algebraic reconstruction techniques; magnetic resonance imaging.


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  • Hierarchical Algorithms [6 credits]

    Hierarchical Algorithms

    • ECTS credits 6
    • Semester 3
    • University Hamburg University of Technology

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  • Matrix Analysis [6 credits]

    Matrix Analysis

    • ECTS credits 6
    • Semester 3
    • University Hamburg University of Technology

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Semester 3 @ UHH

  • Optimisation of complex systems governed by ODEs and PDEs [6 credits]

    Optimisation of complex systems governed by ODEs and PDEs

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      Optimization techniques in function space
      Applications to elliptic partial differential equations
      Optimal control with partial differential equations

    • Topics

       

      Existence of minimizers in function space
      Optimality conditions for the unconstrained case
      Elliptic PDEs as energy minimization problem
      Convex analysis and optimality conditions in the constrained case
      Optimal control problems with PDEs


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  • Mathematical systems and control theory [6 credits]

    Mathematical systems and control theory

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      This course provides mathematical tools and methods to design controls and controllers for systems described by ordinary differential equations.

    • Topics

       

      Stability, controllability, observability, Lyapunov equations, observer and controller design, Laplace transform and Hardy spaces, linear-quadratic optimal control, Riccati equations.


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  • Advanced topics in fluid dynamics [6 credits]

    Advanced topics in fluid dynamics

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      The module concerns advanced topics in fluid dynamics with a focus on applications in gasdynamics and semiconductor modelling.

    • Topics

       

      Gasdynamics: Boltzmann equation, Macroscopic limits, Euler equations of gasdynamics, Navier Stokes equations, Incompressible equations, Acoustics, Shallow water equations, Boussinesq approximation. Charged fluids: Semiconductors and plasmas, Hydrodynamic models, Energy transport models, Drift diffusion models, Quantum mechanical models. 


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  • Traffic flow models [6 credits]

    Traffic flow models

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      The module introduces basic modelling concepts and the related mathematical methods for the description of vehicular traffic flow.

    • Topics

       

      Vehicular traffic flow phenomena, Modelling approaches, Cellular automata. Microscopic models: models based on ODE's. Macroscopic models: models based on PDE’s.


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  • Computer tomography [6 credits]

    Computer tomography

    • ECTS credits 6
    • Semester 3
    • University University of Hamburg
    • Objectives

       

      This course gives an introduction to mathematical methods of medical imaging, where particular focus is placed on computer(ized) tomography (CT). But also algebraic reconstruction techniques (ART) will be discussed. If time allows, a short overview over m...

    • Topics

       

      X-rays; the Radon transform; the back projection; the central slice theorem; the filtered back projection formula; discrete image reconstruction; algebraic reconstruction techniques; magnetic resonance imaging.


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  • Model order reduction [6 credits]

    Model order reduction

    • ECTS credits 6
    • Semester 3
    • University Hamburg University of Technology
    • Objectives

       

      The module provides an introduction to mathematical methods for model reduction.

    • Topics

       

      Computational methods for model order reduction, systems theoretic methods (balanced truncation, Krylov subspace methods, interpolation methods), proper orthogonal decomposition and reduced basis methods, nonlinearity reduction methods like EIM, DEIM and QDEIM, surrogate modeling for design and optimization, model reduction methods in applications such as structural mechanics, fluid dynamics, control of PDEs, network systems. 


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  • Calculus of variation [6 credits]

    Calculus of variation

    • ECTS credits 6
    • Semester 3
    • University Hamburg University of Technology
    • Objectives

       

      The module introduces to the theory of variational minimization problems, which are mostly set in spaces of weakly differentiable functions.

    • Topics

       

      Model problems (brachistochrone, Dirichlet energy, minimal surfaces, etc.), convex integrands and generalizations, existence and uniqueness of minimizers by direct methods, necessary and sufficient (PDE) conditions for minimizers, generalized minimizers (via relaxation or Young measures), problems with constraints, variational principles and applications, duality theory, outlook on regularity. 


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University of L'Aquila, Italy (UAQ)

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Autonomous University of Barcelona, Catalonia - Spain (UAB)

Departament de Matemàtiques, Edifici Cc - Campus UAB 08193 Bellaterra – Catalonia

University of Hamburg, Germany (UHH)

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University of Nice - Sophia Antipolis, France (UNS)

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