- Conversion of sparse matrices
- New algorithms for reducing sparse matrices to triangular and block triangular forms with the original elements;
- New formulas showing dependencies of matrix transform to triangular form versus the number of non-zero elements and matrix size;
- New criteria for determining matrix rows that prevent a matrix from being reduced to block diagonal form;
- New software for dealing with sparse matrices.
- Relational database organization
- Deduction of new functional dependencies using declared ones;
- New methods of automatic checking of a relational database for the third normal form membership and Boyce-Kodd memebership;
- New techniques of relational expression decomposition for reducing redundancy based on the graph theory and sparse matrices;
- New techniques of primary key deduction;
- New techniques of different database merging in a single database;
- Software for relational database design with relations satisfying the normal Boyce-Kodd form.
- Parallel algorithm design for distributed computational systems
- New mathematical formulas estimation algorithm parallelization using information dependencies;
- New techniques based on information graphs for obtaining full groups of mutually-independent algorithm operations, algorithm optimization using the number of processors, computation density, and algorithm execution time;
- Software for manual and automatic construction of information graphs, temporal diagrams.
Present mathematical studies
- Optimization of parallel algorithms taking into account inter-processor data transmission, architecture of a computational system, and communications in a system;
- Solving systems of polynomial equations by forming tables of monomial multipliers and finding the solution as a linear combination of polynomials;
- Development of techniques for automatic parallelization of queries to relational databases using the existing SQL-servers;
- Application of sparse matrices for building decision trees for increasing internal parallelism of heuristic algorithms.
Participation in projects
1) Federal Grant-in-Aid Program "Human Capital for Science and Education in Innovative Russia in 2009-2013" with the project "Creation of high-performance computational technologies for intellectual systems of operative processing and visualization of hydroacoustic information".
2) Fundamental research program of the President of the Russian Academy of Sciences "Problems of creating the national academic distributed information and computational environment on the basis of GRID-technologies, cloud computing, and modern telecommunication networks".