Short Presentation of the Unit of Engineering Mathematics
As part of an engineering faculty, the Unit of Engineering Mathematics focusses on mathematical methods in the engineering sciences, applied differential calculus, stochastics and numerics, both in teaching and research.
Teaching
The unit delivers basic courses in mathematics, computer science (including programming languages) and probability/statistics in the bachelor and masters curricula. Higher mathematical training is offered in numerical mathematics, advanced calculus, numerics of finite elements, optimization and scientific programming. Emphasis is put on the mathematical methods that are of vital importance in civil and environmental engineering, mechatronics and electrical engineering. Further topics in operations research and risk analysis are taught in collaboration with the unit of project and construction management. In the doctoral program for engineering sciences further specialized lectures are offered as necessary in data analysis, stochastics and advanced optimization.
▸ Courses (all courses at the Institute of Basic Sciences in Engineering Science).
Research
The main focus is on mathematical modelling in engineering. However, theoretically oriented research in analysis and stochastics and practical research in biomechanics is undertaken as well.
The two main research subjects of the unit are:
▸ Partial differential equations and dynamical systems.
The research topics comprise:
- Generalized solutions to nonlinear partial differential equations, propagation of singularities in linear partial differential equations with discontinuous coefficients, stochastic partial differential equations, algebras of generalized functions, Fourier integral operators. → Michael Oberguggenberger
- Inverse problems, ill-posed problems, imaging and optimization, kinetic theory and statistical mechanics, shock and delta wave solutions to quasilinear hyperbolic systems. → Lukas Neumann
- Linear partial differential equations and systems, fundamental solutions and Green's functions, linear theory of generalized functions, functional analysis. → Peter Wagner
- Multibody simulations, coupled systems of ordinary and partial differential equations. → Robert Eberle
In all these fields, contributions cover theory (development of new mathematical concepts and methods) and applications in physics and in the engineering sciences, especially in elasticity.
▸ Stochastic, probabilistic and generalized probabilistic methods.
The research topics cover the development of new mathematical methods and applications in engineering sciences, risk analysis, operations research and biomechanics. They comprise:
- Classical probabilistic methods and Monte-Carlo simulation.
- Imprecise probabilities such as families of probability measures or random sets.
- Non-probabilistic methods using intervals and fuzzy sets.
The unit is well imbedded in the international research communities in the mentioned fields. The unit is also actively engaged in the research center Computational Engineering. An additional important activity of the unit is advice and consultations of other units of the faculty on mathematical questions arising in engineering research. The resulting collaboration frequently leads to joint publications. Traditionally, the unit’s research has been funded by the Austrian Science Fund (FWF). In recent years collaboration with industrial partners has been intensified, which resulted in a number of research projects financed by the Austrian Research Promotion Agency (FFG) as well as third-party funded industrial projects.
Some typical examples of recent research results include: wave propagation in discontinuous continua [1], multibody simulations with stochastic effects [2], imprecise probability and Monte Carlo simulation with applications in reliability [3], inverse problems and optimization [4], strongly singular solutions in the systems of gas kinetics [5].
[1] Deguchi, M. Oberguggenberger, Propagation of singularities for generalized solutions to wave equations with discontinuous coefficients. SIAM J. Math. Analysis 48 (2016), 397-442, https://doi.org/10.1137/15M1032661.
[2] Eberle, P. Kaps, M. Oberguggenberger, A multibody simulation study of alpine ski vibrations caused by random slope roughness. Journal of Sound and Vibration 446 (2019), 225-237, https://doi.org/10.1016/j.jsv.2019.01.035.
[3] Fetz, M. Oberguggenberger: Imprecise random variables, random sets, and Monte Carlo simulation. International Journal of Approximate Reasoning 78 (2016), 252-264, http://dx.doi.org/10.1016/j.ijar.2016.06.012.
[4] Rabanser, L. Neumann, M. Haltmeier, Analysis of the Block Coordinate Descent Method for Linear Ill-Posed Problems. SIAM Journal on Imaging Sciences 12/4 (2019), 1808-1832, https://doi.org/10.1137/19M1243956.
[5] M. Nedeljkov, L. Neumann, M. Oberguggenberger, M. R. Sahoo, Radially symmetric shadow wave solutions to the system of pressureless gas dynamics in arbitrary dimensions. Nonlinear Analysis 163 (2017), 104-126, https://doi.org/10.1016/j.na.2017.07.006