MathMods :: Joint MSc

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.



  • Modelling camp [3 credits]

    Modelling camp

    • ECTS credits 3
    • Code DT0064
    • University University of Hamburg
    • Semester 2
    • Lecturer 1 Ingenuin Gasser
    • 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
    • Code DT0669
    • University University of Hamburg
    • Semester 2
    • 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 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".
    • Prerequisites

       

      Will be announced in lectures.

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  • Machine Learning [6 credits]

    Machine Learning

    • ECTS credits 6
    • Code DT0373
    • University University of Hamburg
    • Semester 2
    • Lecturer 1 Y. Saleh
    • Lecturer 2 Dr. T. M. Wenzel
    • Objectives

       

      ...
    • Topics

       

      ...

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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
    • University University of Hamburg
    • Semester 2
    • Lecturer 1 Christian KAHLE
    • 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
    • Code I0064
    • University University of Hamburg
    • Semester 2
    • Lecturer 1 Jens Struckmeier
    • 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
    • Code DT0651
    • University Hamburg University of Technology
    • Semester 2
    • Lecturer 1 Prof. S. Le Borne
    • Objectives

       

      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.
    • 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
    • Prerequisites

       

      Analysis, Linear Algebra, Basic MATLAB knowledge
    • Books

       

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

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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
    • Code DT0402
    • University Hamburg University of Technology
    • Semester 2
    • Lecturer 1 T. Schmidt
    • Objectives

       

      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.
    • 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. 
    • Prerequisites

       

      A solid background in analysis and linear algebra is necessary.

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

       

      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.

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  • Optimisation [6 credits]

    Optimisation

    • ECTS credits 6
    • Code DT0217
    • University University of Hamburg
    • Semester 2
    • Lecturer 1 J. Oberle
    • 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
    • Code DT0654
    • University Hamburg University of Technology
    • Semester 2
    • Lecturer 1 Matthias Schulte
    • Objectives

       

      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.
    • Topics

       

      • 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
    • Prerequisites

       

      Familiarity with the basic concepts of probability
    • Books
       
      • 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.

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