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.

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.Num-a (compulsory module) Numerics of ordinary differential equations
• Wei.Num-b (elective module) Applications in financial mathematics
• Wei.Num-c (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
  •     One-step methods
  •     Multistep methods
  •     Extrapolation methods
• Introduction to boundary value problems
• Depending on your choice, applications in financial mathematics or engineering

Course Type: Lecture with exercise (4+2 SWS)

Tonus: Winterterm

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’)

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
• One-Step methods and extrapolation methods
• Multi-step methods
• Numerical methods for stiff systems
• Application-oriented models and schemes
• Boundary Value Problems
• Methods for Differential Algebraic Equations
• Geometric integrators

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 well-posedness 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

Course Type: Lecture with exercise (4+2 SWS)

Tonus: Summerterm

Language: English

Prior Knowledge: Analysis I-II, Linear Algebra I-II, 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 Black-Scholes 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 time-continuous (not time-discrete) 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, Monte-Carlo 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
• risk-neutral 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. 525-546.
• P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992

Course Type: Lecture with exercise (4+2 SWS)

Tonus: Winterterm

Language: English

Prior Knowledge: Analysis I-II, Linear Algebra I-II, 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. 525-546.
• P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992

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