Semester2 Numerics UHH

Semester2 Numerics UHH

Numerics  @  UHH  30 ECTS credits

The second semester will focus on Numerics and will be spent from March to August at the 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.
The branch 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 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.



  • Scientific computing [6 credits]

    Scientific computing

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

       

      ...

    • Topics

       

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  • Modelling camp [3 credits]

    Modelling camp

    • ECTS credits 3
    • 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) [3 credits]

    German language and culture for foreigners (level A1)

    • ECTS credits 3
    • 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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Two courses from the list below

  • 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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  • 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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One course from the list below

(Note that "Probability theory" is compulsory if you're spending Year 2 at UCA)

  • Calculus of variations [6 credits]

    Calculus of variations

    • 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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  • Optimisation [6 credits]

    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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  • Probability theory [6 credits]

    Probability theory

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

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University of L'Aquila, Italy (UAQ)

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

University of Hamburg , Germany (UHH)

Department of Mathematics
Bundesstr. 55
20146 Hamburg - Germany

University of Côte d'Azur, Nice - France (UCA)

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