MathMods :: Joint MSc

Semester2 Numerics

Numerics  @  TUW   OR TUHH   OR UHH  30 ECTS credits

The second semester will focus on Numerics and will be spent from March to August in one of the following universities:

  1. Vienna University of Technology (TUW)
  2. Hamburg University of Technology (TUHH) OR University of Hamburg (UHH)

Semester 2 is designed with the goal of providing the student with advanced skills in both the design of fast and efficient numerical schemes and their implementation. Both skills are essential in the formation of the modern applied mathematician, and are nowadays considered essential in the implementation of mathematical models arising in modern applications of mathematics such as social sciences, urban sciences, economics.
Both optional branches (Vienna and Hamburg) will offer courses centered on numerical methods for ordinary and partial differential equations as main subject. Optional units covering modern scientific computing, optimization, statistics, computer science, and industrial applications will complete this semester.

The Consortium will address each student to either Vienna or Hamburg depending on the student's background and will provide assistance in the choice of the optional courses in such a way to design the best study plan to approach the chosen Year 2 option.

 

Below you can find more information about the two branches.

when  WHEN

what  WHAT

where  WHERE

Semester 2 Numerics  
     •  Numerics - TUW TUW, Austria
     •  Numerics - (T)UHH TUHH or UHH, Germany

 
  

Semester 2: Numerics @ TUW, Vienna

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TUW smallSemester 2: Numerics @ TUW, Vienna University of Technology

The second semester at TUW provides a stronger background on advanced methods for the implementation of numerical codes (computer programming). Moreover, the student will have the opportunity to acquire basic skills in parallel computing and high performance computing. This semester will also include optional units in Numerical Optimization and Statistics.

 

  • Computer programming [5 credits]

    Computer programming

    • ECTS credits 5
    • Semester 2
    • University Vienna University of Technology
    • Objectives

       

      This course aims at providing the students with a minimal knowledge of Matlab and C++ needed to tackle the numerical methods courses of this semester. It will be concentrated in the first weeks.

    • Topics

       

      Introduction to Computer Programming: introductory knowledge of Matlab and C++ as needed for the parallel numerical methods courses. MATLAB syntax (command- and object-oriented), graphical representations, toolboxes, selected problems from engineering and statistics.


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  • Numerics of differential equations [15 credits]

    Numerics of differential equations

    • ECTS credits 15
    • Semester 2
    • University Vienna University of Technology
    • Objectives

       

      Knowledge of standard numerical methods for the approximation of solutions of ordinary and partial differential equations (discretization methods). Finite element methods. Discontinuous Galerkin methods. Instationary PDEs.

    • Topics

       

      Initial- and boundary value problems for ordinary and partial differential equations: One-step and multi-step methods, adaptivity. Introduction to numerical methods for partial differential equations of elliptic, parabolic, and hyperbolic type. Variational formulation of PDEs and function spaces. Finite element convergence theory. Discontinuous Galerkin methods for convection dominated problems. Mixed methods and applications in fluid mechanics. Nonlinear equations and applications in solid mechanics. Vectorial function spaces and applications in electromagnetics. Instationary PDEs and time-stepping methods. Analysis of iterative solvers and preconditioners. A posteriori error estimates and adaptivity. Exercises: Apply these methods to solve pde problems; exercises with NGSolve-Python; verify qualitative and quantitative properties.


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  • German language and culture for foreigners (level A1) [4 credits]

    German language and culture for foreigners (level A1)

    • ECTS credits 4
    • Semester 2
    • University Vienna University of Technology
    • Objectives

       

      TBD

    • Topics

       

      TBD


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  • Introduction to parallel computing [6 credits]

    Introduction to parallel computing

    • ECTS credits 6
    • Semester 2
    • University Vienna University of Technology
    • Objectives

       

      Motivation, goals of parallel computing. Parallel architectures, programming models, performance measurement and analysis. Problems in parallel algorithms. Introduction to MPI (Message-Passing interface), hreads and OpenMP. Task-parallel models and interfaces (Cilk). Other languages for multi-core processors.

    • Topics

       

      Basic understanding of motivation and goals of parallel computing, basic knowledge of parallel architectures, programming models, languages and interfaces (concrete examples OpenMP, Cilk, MPI), performance analysis and modeling, pitfalls, basic programming skills in the discussed parallel interfaces (C or C++ with MPI and OpenMP; pthreads and Cilk or related).


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  • Numerical optimization [6 credits]

    Numerical optimization

    • ECTS credits 6
    • Semester 2
    • University Vienna University of Technology
    • Objectives

       

      Learn the basic concepts and methods of unconstrained and constrained optimization.

    • Topics

       

      Unconstrained optimisation: gradient methods, classical Newton method, quasi-Newton method (e.g. BFGS method). Constrained optimisation: trust region methods, linear programming (simplex method, interior point method), quadratic programming (inner point method, active sets method), sequential quadratic programming.


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  • Stationary processes and time series analysis [6 credits]

    Stationary processes and time series analysis

    • ECTS credits 6
    • Semester 2
    • University Vienna University of Technology
    • Objectives

       

      Introduction to the theory of stationary processe and time series analysis.

    • Topics

       

      Stationary processes, basics, autocovariance function, spectral representation, spectrum, linear filters, transfer function, AR/ARMA processes, forecasting, estimation.


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Semester 2: Numerics @ (T)UHH, Hamburg

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T UHH logoSemester 2: Numerics @ (T)UHH, Hamburg

The second semester in Hamburg is characterized by a stronger "engineering oriented" perspective, in that it provides optional units at the interface of Computer Science and case studies of industrial applications of mathematics. The Hamburg semester 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 semester 2 courses.

Semester 2 @ TUHH

TUHH logo smallSemester 2 @ TUHH, Hamburg University of Technology

To be activated in case the programme is funded by the Erasmus Mundus scheme

 

  • Numerical approximation of PDEs by finite differences and finite volumes [6 credits]

    Numerical approximation of PDEs by finite differences and finite volumes

    • ECTS credits 6
    • Semester 1
    • University University of Hamburg
    • Objectives

       

      The overall emphasis is on studying the mathematical tools that are essential in developing, analyzing, and successfully using numerical methods for non-linear systems of conservation laws, particularly for problems involving shock waves.

    • Topics

       

      Numerical methods for linear equations. Computing discontinuous solutions. Conservative methods for non-linear problems. Godunov's methods. Approximate Riemann solvers. Nonlinear stability. High resolution methods. Semi-discrete methods.


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  • Numerics Treatment of Ordinary Differential Equations [6 credits]

    Numerics Treatment of Ordinary Differential Equations

    • ECTS credits 6
    • Semester 2
    • University Hamburg University of Technology
    • Objectives

       

      The course covers numerical methods for the solution of ordinary differential equations, including convergence properties and aspects regarding the practical execution of a method. Students learn to select the appropriate numerical method for concrete problems, implement the numerical algorithms efficiently and interpret the numerical results.

    • Topics

       

      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, variational methods).

    • Prerequisites

       

      TBD


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  • German language and culture for foreigners (level A1) [4 credits]

    German language and culture for foreigners (level A1)

    • ECTS credits 4
    • Semester 2
    • University University of Hamburg
    • Objectives

       

      - 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.

    • Topics

       

      - To introduce sb to sb + 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"

       


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  • Intercultural communication [2 credits]

    Intercultural communication

    • ECTS credits 2
    • Semester 2
    • University Hamburg University of Technology
    • Objectives

       

      In this workshop you will learn how to increase and express your cultural competence. You will apply cultural knowledge and images in order to positively influence communicative situations. You will learn how to add character and interest to your talks, papers and publications by referring to your own and European Cultural background. You will find out the basics of communicating professionally and convincingly by showing personality and by referring to your own cultural knowledge. You will get hands-on experience both in preparing and in conducting such communicative situations.

    • Topics

       

      How to enrich the personal character of your presentations by referring to European and your own culture. How to properly arrange content and structure. How to use PowerPoint for visualization. How to be well-prepared and convincing when delivering your thoughts to your audience.


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  • Approximation and stability [6 credits]

    Approximation and stability

    • ECTS credits 6
    • Semester 2
    • University Hamburg University of Technology
    • Objectives

       

      Recall and apply basic results from functional analysis; understand operator approximation methods, their stability and convergence for the approximate solution of linear equations and spectral problems.

    • Topics

       

      Recall basic functional analysis (Hilbert spaces, operators, main theorems); stability and convergence of projection methods (Galerkin, collocation, spline interpolation); condition numbers and regularization of approximation methods; approximation of spectrum and pseudospectra.


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  • Optimisation [6 credits]

    Optimisation

    • ECTS credits 6
    • Semester 2
    • University Hamburg University of Technology
    • Objectives

       

      TBD

    • Topics

       

      TBD


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  • Randomised Algorithms and Random Graphs [6 credits]

    Randomised Algorithms and Random Graphs

    • ECTS credits 6
    • Semester 2
    • University Hamburg University of Technology
    • Objectives

       

      The course is intended to develop an understanding of the fundamental concepts in randomised algorithms and the theory of random graphs. The primary focus is on the ability to determine average or typical behaviour of algorithms and structures, and to analyze how this behaviour evolves when the underlying random distribution is changing.

    • Topics

       

      Randomised search, random walks, text search with fingerprinting. parallel and distributed algorithms, online algorithms. Typical properties of random graphs, first and second moment method, tail bounds, thresholds and phase transitions, probabilistic method, models for complex networks.


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Semester 2 @ UHH

uhh smallSemester 2 @ UHH, University of Hamburg

To be activated in case the programme is NOT funded by the Erasmus Mundus scheme

 

  • Numerical approximation of PDEs by finite differences and finite volumes [6 credits]

    Numerical approximation of PDEs by finite differences and finite volumes

    • ECTS credits 6
    • Semester 1
    • University University of Hamburg
    • Objectives

       

      The overall emphasis is on studying the mathematical tools that are essential in developing, analyzing, and successfully using numerical methods for non-linear systems of conservation laws, particularly for problems involving shock waves.

    • Topics

       

      Numerical methods for linear equations. Computing discontinuous solutions. Conservative methods for non-linear problems. Godunov's methods. Approximate Riemann solvers. Nonlinear stability. High resolution methods. Semi-discrete methods.


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  • Numerical methods for PDEs - Galerkin Methods [6 credits]

    Numerical methods for PDEs - Galerkin Methods

    • ECTS credits 6
    • Semester 2
    • University University of Hamburg
    • Objectives

       

      In this course the basic principles of the FE element method applied to the numerical treatment of elliptic PDEs are considered. Successful applicants will be able to use the FE method as analytic and numerical tool for the mathematical investigation of l

    • Topics

       

      Topics include construction of FE spaces, derivation and numerical solution of the resulting linear algebra systems, error analysis in Sobolev spaces including interpolation estimates, basic principles of residual based a posteriori error analysis.


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  • Advanced Optimisation [6 credits]

    Advanced Optimisation

    • ECTS credits 6
    • Semester 2
    • University University of Hamburg
    • Objectives

       

      Minimization of nonlinear functionals on infinite dimensional spaces subject to constraints. Solution algorithms for constrained and unconstrained nonlinear optimization problems. Aspects of numerical approximation and implementation. Convex optimization. Nonsmooth optimization.

    • Topics

       

      Existence and uniqueness of solutions. Necessary and sufficient optimality conditions. Constraint qualifications. Kuhn-Tucker theorems. Steepest descent and Newton-type methods for unconstrained optimization, SQP methods, penalty methods and interior point methods for constrained optimization. Semismooth Newton and Primal-Dual Active set methods for nonsmooth problems.


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  • Scientific computing [6 credits]

    Scientific computing

    • ECTS credits 6
    • Semester 2
    • University University of Hamburg
    • Objectives

       

      ...

    • Topics

       

      ...


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  • Modelling camp [2 credits]

    Modelling camp

    • ECTS credits 2
    • Semester 2
    • University University of Hamburg
    • Objectives

       

      In this seminar we face the students with real world problems. We discuss the interface between the real world  industrial problems and applied mathematics.

    • Topics

       

      A series of 22 small real world industrial problems where mathematics has been applied successfully.


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  • German language and culture for foreigners (level A1) [4 credits]

    German language and culture for foreigners (level A1)

    • ECTS credits 4
    • Semester 2
    • University University of Hamburg
    • Objectives

       

      - 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.

    • Topics

       

      - To introduce sb to sb + 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"

       


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Connect with us

Our partners' addresses

University of L'Aquila, Italy (UAQ)

Department of Information Engineering, Computer Science and Mathematics, via Vetoio (Coppito), 1 – 67100 L’Aquila (Italy)

Autonomous University of Barcelona, Catalonia - Spain (UAB)

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

Hamburg University of Technology, Germany (TUHH)

Institute of Mathematics
Schwarzenberg-Campus 3, Building E-10
D-21073 Hamburg - Germany

University of Nice - Sophia Antipolis, France (UNS)

Laboratoire J.A.Dieudonné
Parc Valrose, France-06108 NICE Cedex 2

Vienna Univ. of Technology, Austria (TUW)

Technische Universität Wien
Institute of Analysis & Scientific Computing
Wiedner Hauptstr. 8, 1040 Vienna - Austria

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