Main research areas
MODELLING WITH NONLINEAR DIFFERENTIAL AND DIFFERENCE EQUATIONS
Our research concerns the field of differential and difference equations, which forms the theoretical basis of mathematical modelling of natural and technological processes and significantly affects many areas of physics, chemistry, ecology and social sciences. The effort to describe as precisely as possible the rich dynamic behaviour of real processes and phenomena leads to the use of complex nonlinear mathematical models. Our team is mainly concerned with the qualitative and quantitative analysis of nonlinear ordinary and partial differential equations, i.e., questions of existence, uniqueness or multiplicity and bifurcations of solutions, which are crucial for the analysis of modelled systems and the setting of their parameters. The research team also studies systems in discrete networks reflecting spatial heterogeneity.
STRUCTURES AND METHODS OF
We are mainly engaged in research in the field of discrete mathematics and theoretical computer science, which forms the theoretical basis of modern information technologies. The research has a strongly multidisciplinary character with an overlap into a number of mathematical fields: algebra (group theory, linear algebra), probability theory (probabilistic methods in combinatorics, probabilistic analysis of algorithms), geometry (combinatorial geometry, algebraic geometry), topology and others. The research is mainly focused on structural issues in graph theory (Hamiltonian graph theory, closure-type graph operations, special classes of graphs, graph factorization, chromatic graph theory, and general issues of the colourings of combinatorial structures), including the development of relevant algorithms, as well as on problems of computational complexity and on optimization problems of scheduling and control.
MODERN METHODS OF GEOMETRIC MODELLING AND THEIR APPLICATIONS
The research is focused on the study of new mathematical methods of geometric modelling in the area of description, data representation, modification and presentation of curves and surfaces with regard to their applicable properties in engineering practice. The team is involved in the recognition of significant curves and surfaces given exactly or approximately, and the development and implementation of algorithms based on a hybrid approach combining symbolic and numerical computations, which are then used to solve actual complex geometric modelling problems. As a significantly innovative area of research and development with high application potential, the use of isogeometric analysis methods for numerical solution of equations describing real-life problems in engineering practice and natural sciences is studied.
COMPUTER AND NUMERICAL
MODELLING AND SIMULATION
We are mainly devoted to the study, design, analysis and implementation of numerical methods for the approximate solution of boundary value and initial-boundary value problems for partial differential equations. In particular, attention is paid to the finite volume method, the finite element method and, in collaboration with the geometry group, isogeometric analysis (including algorithmization and parallelization). In the area of applications, the team focuses on numerical modelling of compressible and incompressible fluid flows (recently with a focus on water turbines and turbulence modelling). The research team is also involved in numerical modelling of biomechanics problems (currently focusing mainly on stress-strain analysis of the temporomandibular joint with various impairments and subsequent reconstruction, e.g. using artificial replacements).
STATISTICS AND FINANCE
We deal with the creation and analysis of statistical, reliability and statistical-economic models and methodologies both at the general level and at the application level according to the needs of a specific customer. Our basic research is focused on the investigation of bivariate probabilistic models with strong and weak dependence, which find application in forecasting and default-situation prevention. Our applied research focuses on the use of standard and newly developed statistical and statistical-economic methods in various industries (e.g., including medical diagnostics). Another area is applications where statistical and economic perspectives are combined.