- Code: DT0653
- Unit Coordinator: Thomas Schmidt
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: Hamburg University of Technology
- Language: English
- Aims:
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.
- Content:
A selection out of the following:
- Model problems and examples (Dirichlet energy, isoperimetric and brachistochrone problems, minimal surfaces, Bolza and Weierstrass examples, …),
- Existence and uniqueness of minimizers by direct methods,
- Weak lower semicontinuity of (quasi)convex variational integrals,
- Necessary and sufficient (PDE) conditions for minimizers,
- Problems with constraints (obstacles, capacities, manifold and volume constraints, ...),
- Generalized minimizers (relaxation, Young measures, ...),
- Variational principles and applications,
- Duality theory,
- Outlook on regularity.
- Pre-requisites:
A solid background in analysis and linear algebra is necessary.
Familiarity with functional analysis, Sobolev spaces, and PDEs can be advantageous.
- Reading list:
- 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.
- Unit Coordinator: Janusz Morawiec
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of Silesia in Katowice
- Language: English
- Content:
The main goal of the lecture is to present basic properties of wavelet transforms and some methods of construction of wavelet bases. We will pay special attention to these wavelet transforms which have used to the analysis and the synthesis of sound signals. We also will pay special attention to structures of bases with special properties which have used to the data compression in digital transmissions.
- Reading list:
[1] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.
[2] I. Daubechies, The wavelet transform, Time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961-1005.
[3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philidelphia, 1992.
[4] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628-666.
[5] G. Kaiser, A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
[6] D. Kozlow, Wavelets. A tutorial and a bibliography, Rendiconti dell’Instituto di Matematica dell’Universita di Trieste, 26, supplemento (1994).
- Unit Coordinator: Thomas Zürcher
- ECTS Credits: 2
- Semester: 1
- Year: 2
- Campus: University of Silesia in Katowice
- Language: English
- Aims:
The main aim of the module Problem Workshops is to acquaint students with chosen branches of mathematics with applications to knowledge domains such as: economics, biology, physics, chemistry, and computer science.
Additional aims are: training analytical skills (for example, constructing mathematical models of chosen problems from applied sciences), training methodological skills (for example, use of available technology to prepare a project or analysis), training cognitive skills (for example, an analysis of data or source content given in a form of articles or manuals, also in a foreign language) and training skills of team-work (for example, work in small groups during and outside the workshop).
