Approximation, Numerical Methods and Optimization
The research is organized as follows:
1. Constructive Approximation. Most of the research concentrates on the characterization of special and important classes of polynomials, introduction of some new concepts of orthogonality, constructive theory of orthogonality, extremal problems with algebraic and trigonometric polynomials in various metrics, numerical quadratures, summation processes and moment-preserving spline approximation.
2. Optimization.The research is oriented towards the selected topics in discrete as well as to both finite and infinite dimensional continuous optimization. In particular, some combinatorial optimization problems are treated.
3. Parallel Algorithms.The research is oriented in the following directions: axiomatization of investigation of parallel algorithms and systems; construction of new parallel algorithms for the solution of the problems of linear algebra; designing of new architectures of multiprocessor systems.
4. Geometric Modelling.Mathematical models playing a key role in computer aided geometric are treated. Degenerative properties of some positive linear operator, as well as properties of particular models of free-form curves and surfaces are investigated. Special attention is devoted to visualization of such models by level-sets.
5. Numerical Methods in Algebra. Some theoretical as well as practical aspects of iterative methods for solving equations are considered including convergence analysis, numerical stability, computational efficiency and implementation on sequential and parallel computers. Main attention is devoted to the development of new methods with a high order of convergence and a great computational efficiency, in ordinary complex and circular arithmetic.
6. Numerical Treatment of Differential Equations. The main field of research are finite difference schemes for solving boundary value problems for partial differential equations. Special attention is devoted to the stability and convergence analysis, as well as to computational efficiency of such schemes. In particular, the connection between convergence rate and the smoothness of input data is investigated in the framework of function spaces theory.