Chystyakov O. Hybrid algorithms of analysis and solving algebraic problems of eigenvalues for sparse matrices

The solving of the algebraic eigenvalue problem is one of the fundamental problems of numerical modeling in many subject areas. Mathematical models of many engineering problems are described by systems of differential equations or difference equations, the solving of which consists in determining the eigenvalues and eigenvectors of matrices, which, as a rule, have a sparse structure. A characteristic feature of these matrices is superlarge orders (up to tens of millions) with a large number of nonzero elements. To solve problems of such volumes it is necessary to use modern powerful computers. Today, this problem can be solved using multi-core computers with graphics processors — hybrid computers combining MIMD and SIMD architecture, as well as efficient algorithmic software that takes into account both the mathematical properties of sparse matrices and the architectural features of computers. Thus, the problem of creating efficient hybrid algorithms for solving of eigenvalues problems of sparse matrices on computers of a hybrid architecture is quite relevant. The main results of the work are as follows: a decomposition scheme for matrices of irregular structure has been developed; developed and investigated new hybrid algorithms for solving a partial generalized algebraic eigenvalue problem of sparse symmetric positive definite matrices – the alternating-triangular method, the method of conjugate gradients, and the method of iterations on a subspace; investigated characteristics of efficiency and acceleration of hybrid algorithms. Hybrid software is created on the basis of new algorithms for hybrid computers and parallel computer whit new host-processors Intel Xeon Phi. The approbation of the created algorithmic software for solving test problems and the stability problem for a layered two-component composite material, which is reduced to solving a partial generalized eigenvalue problem, has been carried out. The developed algorithmic and software was included in the library of intelligent hybrid programs Inparlib_g – the standard software of multi-core computers with graphic processors of the Inparcom and SKIT series of Glushkov Institute of Cybernetics of NAS of Ukraine. The results of the thesis were used at the Timoshenko Institute of Mechanics of NAS of Ukraine in modeling the problem of stability of a composite material. The acceleration of the computational process is obtained up to 50 times.