Workload: 240 hours (1 semester)
ECTS credits: 8 ECTS
Term: Winter (1^st semester)
Repeatablity: not restricted in attempts

Final assessment: Assessment folder (GPA of 2 components of the module):
• component 1: Numerics of Ordinary Differential Equations. Part A (1^st half of semester);
• component 2: Numerics of Ordinary Differential Equations. Part B: Applications in Financial Mathematics (2^nd half of semester)
or
Numerics of Ordinary Differential Equations. Part C: Applications in Technology (2^nd half of semester).

Pre-requisites for the final exam: both components should be completed.

Description of the module: Students have gained advanced knowledge in an area of numerical mathematics and can apply advanced methods. They can independently develop advanced methods and concepts of numerics and apply them to new situations.

Components of NM1a module:

• NM1a-a. Numerics of Ordinary Differential Equations (Part A)
  Teaching format: Lectures and exercises
  Weekly hours: 3 (120 hours in total)
  Contents: Analysis of ordinary differential equations: Existence and uniqueness, proper posedness. Numerical solution methods for initial value problems: one-step method, multi-step method, extrapolation method; Introduction to boundary value problems.
   
• NM1a-b. Numerics of Ordinary Differential Equations (Part B). Applications in Financial Mathematics
  Teaching format: Lectures and exercises
  Weekly hours: 3 (120 hours in total)
  Contents: Models of ordinary differential equations in finance and their numerical solution.
   
• NM1a-c. Numerics of Ordinary Differential Equations (Part C). Applications in Technology
  Teaching format: Lectures and exercises
  Weekly hours: 3 (120 hours in total)
  Contents: Models of ordinary differential equations in technical applications and their numerical solution.

Workload: 240 hours (1 semester)
ECTS credits: 8 ECTS
Term: Winter (1^st semester)
Repeatablity: not restricted in attempts

Final assessment: written 120-minutes examination or oral 30-minutes examination (The type of the final module exam is announced at the beginning of the lecture period; exam is counted as 8 ECTS)

Pre-requisites for the final exam: Core knowledge of analysis / calculus, linear algebra and univariate probability.

Description of the module: The students know basics of multivariate random variables and are able to model machine learning tasks in a statical framework backed by statistical decision theory. They are able to select and apply machine learning models for regression, classification and unsupervised learning tasks. Students know and apply core techniques to analyze the performance of developed machine learning models, while understanding the theoretical connections between model complexity, bias, variance and prediction errors. They are able to combine models, losses and training techniques to machine learning algorithms. Simple proof techniques guiding the construction of machine learning algorithms can be reproduced.

Components of MathML module:

• MathML-a. Mathematical Machine Learning
  Teaching format: Lectures and exercises
  Weekly hours: 6 (240 hours in total)
  Contents: Multivariate random variables, statistical decision theory, linear regression, (stochastic) gradient descent, estimation of prediction error, bias-variance tradeoff, linear classification, unsupervised learning, advanced regression, artificial neural networks, deep learning.

Workload: 240 hours (1 semester)
ECTS credits: 8 ECTS
Term: Summer (2^nd semester)
Repeatablity: not restricted in attempts

Final assessment: 30-minutes oral examination (exam is counted as 3 ECTS)

Pre-requisites for the final exam: Ungraded small homework and term paper for Numerical Methods in Classical Field Theory and Quantum Mechanics (5 ECTS).

Description of the module: Acquisition of knowledge of different numerical techniques to solve problems in classical field theory and quantum mechanics. The focus will be on the implementation on parallel computers. Students shall be enabled to implement the 
algorithms. Ability to prepare a documentation.

Components of NM2b module:

• NM2b-a. Numerical Methods in Classical Field Theory and Quantum Mechanics
  Teaching format: Lectures and exercises
  Weekly hours: 4 (240 hours in total)
  ECTS: 5 points
  Assessment: Ungraded small homework and term paper
  Contents: Hydrodynamics: direct simulation of Navier-Stokes for incompressible fluids, lattice-gas models; Electrodynamics: time-dependent propagation of electromagnetic fields, Yee-Weilandt discretization; Eigenvalue methods for electromagnetic cavities; Non-equilibrium thermodynamics of many-body problems; Quantum mechanics: time-dependent Schrödinger equation, quantum-spin dynamics for quantum computing, Feynman-Kac path integral.