Here you can find all the lectures that our department offers on a regular basis. As always, you can register via Moodle, where you can also see the lecturers responsible for each semester. If you have any questions, please do not hesitate to contact us!
Numerical Mathematics I
This lecture introduces you to numerical mathematics. You may achieve 9 credit points and another 3 credit points for the programming lab.
Course Type: Lecture with exercise (4+2 SWS)
Tonus: Summerterm
Language: German
Prior Knowledge: Analysis I+II, Linear Algebra I+II
Module: E.Num (9ECTS)
Examination: in order complete the course, you have to pass the written exam
Complementary Course: Programming lab
This course introduces students to numerical mathematics and
treates the topics:
 Error analysis
 Solution of linear systems of equations
 Linear equation calculus
 Interpolation with polynomials and splines
 Numerical quadrature
 Solution of nonlinear systems of equations
For these topics, algorithms are presented that are of central importance in many applications in science and technology. In addition to theoretical issues such as convergence analyses, practical aspects (implementation, linking with applications, visualization of results, use of modern software tools). The exercises serve to apply and deepen the lecture material.
Literature
 J. Stoer: Numerische Mathematik 1. Springer.
 A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 1. Springer.
 A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 2. Springer.
 P. Deuflhard, A. Hohmann: Numerische Mathematik I. de Gruyter.
 J. Stoer, R. Bulirsch: Introduction to Numerical Analysis. Springer.
 A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics. Springer.
Numerical Mathematics II
According to the course name, you will deepen your knowledge of numerics in this course. You can earn 9 credit points.
Course Type: Lecture with exercise
Tonus: Winterterm
Language: German
Prior Knowledge: Analysis I+II, Linear Algebra I+II, Introduction to Numerical Mathematics (Numerics I), knowledge of MATLAB (e.g. from Tools and Working Techniques), Python or Julia
Module: Wei.Num (9ECTS)
Examination:
 Wei.Numa (compulsory module) Numerics of ordinary differential equations
 Wei.Numb (elective module) Applications in financial mathematics
 Wei.Numc (elective module) Applications in technology
Complementary Course: Programming lab
 Brief introduction to MATLAB
 Modelling
 Analysis of ordinary differential equations: Existence and uniqueness
 Numerical solution methods for initial value problems
 Onestep methods
 Multistep methods
 Extrapolation methods
 Introduction to boundary value problems
 Depending on your choice, applications in financial mathematics or engineering
Numerical Analysis and Simulation I: ODE
During this course you will get to know models for ordinary differential equations (ODE).
Course Type: Lecture with exercise (4+2 SWS)
Tonus: Winterterm
Language: English
Prior Knowledge: Basic knowledge in mathematics (Analysis III, Linear Algebra III or similar), Numerical Mathematics I (or: the block course ’Introduction to Numerical Methods for Computer Simulation’)
Module: NM1 (9ECTS), NumAna1 (8ECTS)
Examination: In order complete the course, you have to successfully deliver at least 50% of the exercises as well as pass the final exam. Depending on the course, the final exam may be oral (30min) or written (120min).
Complementary Course: Lab Exercise for ODE
 Ordinary Differential Equations (ODE) models in Science, Economics and Engineering
 Short synopsis on the theory of ODEs
 OneStep methods and extrapolation methods
 Multistep methods
 Numerical methods for stiff systems
 Applicationoriented models and schemes
 Boundary Value Problems
 Methods for Differential Algebraic Equations
 Geometric integrators
Numerical Analysis and Simulation II: PDE
In addition to the ODEs, you will deepen your knowledge of partial differential equations (PDEs) in this course.
Course Type: Lecture with exercise (4+2 SWS)
Tonus: Summerterm
Language: English
Prior Knowledge: Basic knowledge in mathematics (Analysis I+II, Linear Algebra I+II or similar), Numerical Mathematics I (or the block course 'Introduction to Numerical Methods for Computer Simulation') as well as Numerical Analysis and Simulation I: ODE.
Module: NM2 (9ECTS), NumAna2 (8ECTS)
Examination: In order complete the course, you have to successfully deliver at least 50% of the exercises as well as pass the final exam. Depending on the course, the final exam may be oral (30min) or written (120min).
Complementary Course: Lab Exercise for PDE
 PDE models in science, economics and engineering
 Classification and wellposedness of PDEs; basic principles: derivation and discretization of PDEs;
 elliptic problems (maximum principle and finite differences, variational formulation and Sobolev spaces, finite elements); numerical solutions of discretized problems
 hyperbolic systems, especially conservation laws (weak formulation, theory of characteristics, entropy, conservative schemes)
 parabolic problems (evolution equations, method of lines, Rothemethod and convergence)
 mixed systems (models of heterogeneous systems, splitting schemes)
 case studies
Computational Finance I
In this course you will get to know the basics of computational finance, for which you can receive 89 credit points depending on your study programme.
Course Type: Lecture with exercise (4+2 SWS)
Tonus: Summerterm
Language: English
Prior Knowledge: Analysis III, Linear Algebra III, Introduction to Numerical Mathematics
Module: CompFi1 (9ECTS), CompFin1 (8ECTS), SKap.WM (9ECTS), SKap.NAaA
(9ECTS)
Examination: In order complete the course, you have to successfully deliver at least 50% of the exercises as well as pass the final exam. Depending on the course, the final exam may be oral (30min) or written (120min).
Complementary Course: Lab Exercise Computational Finance I
Financial derivatives have become an essential tool for the control and hedging of risks. The crucial problem consist in the determination of the fair price of a financial derivative, which is based on mathematical methods. Simple models apply the BlackScholes equation, where an explicit formula of the corresponding solution exists. More complex models do not allow for an analytic solution. Thus numerical methods are required to solve the models. Both the mathematical modelling and the numerical simulation of financial derivatives is discussed in this lecture. We focus on timecontinuous (not timediscrete) models given by stochastic differential equations or partial differential equations. A rough classification of the methods for the determination of the price of a financial derivative yields three types: binomial methods, MonteCarlo simulations and techniques for partial differential equations possibly including free boundary conditions. We explain each type of method extensively. Corresponding algorithms are implemented in lab exercises using the software package MATLAB.
 Modeling of financial markets, options
 Binomial method and its extensions
 riskneutral valuation, stochastic processes
 Geometric Brownian Motion, Ito Lemma
 exotic options
 stochastic differential equations (SDEs)
 Calibration, jump models
 generating random numbers with specified distributions
 Monte Carlo Methods, variance reduction approaches
Literature
 R. Seydel, Tools for Computational Finance, 5th edition, Springer, 2012.
 D. Higham, An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge University Press, 2004.
 M. Günther and A. Jüngel, Finanzderivate mit MATLAB, Vieweg, 2nd ed., 2010.
 B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, 6th edition, Springer, 2003.
 D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, Vol. 43, No. 3, pp. 525546.
 P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992
Computational Finance II
In addition to Computational Finance I, you will deepen your knowledge in this course, for which you can receive 89 credits depending on your study programme.
Course Type: Lecture with exercise (4+2 SWS)
Tonus: Winterterm
Language: English
Prior Knowledge: Analysis III, Linear Algebra III, Introduction to Numerical Mathematics
Module: CompFi2 (9ECTS), CompFin2 (8ECTS), SKap.NAaA (9ECTS), SKap.WM (9ECTS)
Examination: In order complete the course, you have to successfully deliver at least 50% of the exercises as well as pass the final exam. Depending on the course, the final exam may be oral (30min) or written (120min).
Complementary Course: Lab Exercise in Computational Finance II
 Partial Differentail Equations arising in finance
 Finite difference methods
 Finite Element Methods
 Numerical solution of initial boundary value problems
Literature
 R. Seydel, Tools for Computational Finance, 5th edition, Springer, 2012.
 D. Higham, An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation, Cambridge University Press, 2004.
 M. Günther and A. Jüngel, Finanzderivate mit MATLAB, Vieweg, 2nd ed., 2010.
 B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, 6th edition, Springer, 2003.
 D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, Vol. 43, No. 3, pp. 525546.
 P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992
Introduction to Data Science
In this course you will learn how to extract knowledge from large, possibly incomplete and highdimensional data to identify relationships, patterns and predictions to support decisionmaking processes and the development of recommender systems.
Type: Lecture with exercises (4+2 SWS / 9 LP)
Tonus: Summer semester
Language: Englisch
Background knowledge: 
Performance assessment: Written exam, 2 hours

Motivation and Foundations

Supervised Techniques for Inference and Prediction

Classification Algorithms
 Unsupervised Learning and Dimensionality Reduction