# Teaching

## What is Numerical Mathematics?

What is Scientific Computing?

**Numerical mathematics **deals with the development of numerical computational methods, their analysis, and their implementation in computational programs for digital computing machines. It is not limited to the discussion of the existence and uniqueness of solutions to mathematical problems, but rather seeks to provide a concrete calculation of the solutions. Approximations for the (unknown) exact solution are accepted as equivalent if their errors can be made arbitrarily small by increasing the technical effort. For example, although it is known that solutions of polynomial equations exist, there is generally no possibility (see Abel-Ruffini's theorem) to calculate them from a polynomial degree of at least 5 by a finite sequence of arithmetic operations and roots (example: x^{5}=x-1).

For this reason, even the simplest problems cannot be treated without use of numerical methods. New powerful and sophisticated numerical methods enable researchers to solve mathematical problems based on increasingly complex and realistic models from concrete applications. Today it is possible to understand and master entire technical processes by numerical simulation on the computer prior to the actual production. The mathematical prediction of technical processes is of immense importance for numerous key areas of the economy. Once you have mastered the simulation of technical processes, you want to design them in an intuitive way. Control and optimization of technical processes then follow simulation.

At the interface between Mathematics, Computer Science, and Engineering, a new field has developed: **Scientific Computing**. Here, numerical methods are used to solve the most difficult problems from the natural sciences and engineering, and also more and more from other scientific fields. Starting with the creation of a mathematical model to describe a concrete application problem as accurately as possible, through the analysis of the mathematical problem and its numerical solution (if necessary using modern supercomputers and with the help of numerical methods which must usually be developed anew or tailored to the given problem), the spectrum extends to the retranslation of the calculated results into the language of the user. The aim is always to replace expensive, real experiments with cheaper computer experiments, at least in part. Often virtual experiments can be carried out on the computer, even if they would not be feasible in reality.

## List of courses

In addition to the following lectures and seminars in mathematics degree programs

Bachelor mathematics:

- Introduction to Numerical Mathematics
- Higher Skills in Numerical Mathematics (bachelor and master mathematics)
- Numerical Methods for Differential Equations (bachelor and master mathematics)
- Applied functional analysis (bachelor and master mathematics)
- Seminar Numerical Mathematics and Scientific Computing (bachelor and master mathematics)

Master mathematics:

- Numerical Methods of General Types of PDEs
- Nonlinear Optimization
- Numerical Methods of Optimal Control
- Modeling with Differential Equations
- Mathematical Modeling for Climate and Environment
- Optimization of Differential Equations
- Efficient Treatment of non-local Operators
- Fast Methods for Differential and Integral Equations
- Efficient Numerical Treatment of Multiscale Problems
- Modeling and Status Seminar
- Practical Course on Parallel Numerical Methods

Special courses (master mathematics):

- High-dimensional Approximation
- Boundary Element Methods
- Interior Point Methods for Optimization
- Biomathematics - Mathematical Methods of Dynamical Biochemical Processes
- Singularly Perturbed Differential Equations
- Optimal Control of Ordinary Differential and Algebro-Differential Equations

the chair is responsible for the mathematical education of engineers and computer scientists (lecture series *Ingenieurmathematik 1-3).*