A. N. Khimich

• Head of Department at National Academy of Sciences of Ukraine

The paper considers the intellectual computer mathematics system InparSolver, which is designed to automatically explore and solve basic classes of computational mathematics problems on multi-core computers with graphics accelerators. The problems of results reliability of solving problems with approximate input data are outlined. The features of t...

У статті простежено еволюцію математичних і технічних засобів комп’ютерного моделювання від найперших комп’ютерів до сучасних суперкомп’ютерів та від вузькоспеціалізованих розрахункових задач до міждисциплінарних моделей, які відповідають потребам широкого кола галузей вітчизняної економіки. Коротко схарактеризовано математичні напрями, в яких Інст...

An experimental and theoretical study of the dehydration of natural gas using microporous silica beds for motor fuel technology in extreme winter climates is described. Analytical solutions to the problem of non-isothermal adsorption and desorption are based on Heaviside's operational method and Laplace integral transform, but the development of ca...

The authors consider high-performance computational algorithms for computers of hybrid architecture to solve problems of predicting the stress–strain state of critical welded structures with regard for initiation and development of subcritical fracture of metal in accordance with the characteristic problems of serviceability analysis of welded pipe...

We present the definition of weighted pseudoinverse matrices with nondegenerate weights of indefinite sign and study these matrices. The theorems on existence and uniqueness for these matrices are proved. The weighted pseudoinverse matrices with indefinite weights are represented in terms of the coefficients of characteristic polynomials of symmetr...

У доповіді висвітлено сучасний стан проблеми використання суперкомп’ютерних технологій для математичного моделювання складних систем і розв’язування великорозмірних надскладних обчислювальних задач. Представлено створену в Інституті кібернетики ім. В.М. Глушкова НАН України інфраструктуру та запропоновано сучасний підхід до математичного моделюванн...

The mathematical model of adsorption and desorption of the main component of emissions of incomplete combustion products in nanoporous zeolite-medium for the cold start conditions is formulated. Using the method of Heveside's operating calculus, the model analytical solution, which provides high-speed modeling of concentration dependences in the ga...

The problem of numerical investigation of the stability of composite materials under compression along the reinforced elements using multicore computers with graphic processors is considered. The problem of the three-dimensional theory of stability of composites with the using the "finite-size fibers" model and mathematical methods for its solution...

Hybrid algorithms for solving the partial generalized eigenvalue problem for symmetric positive definite sparse matrices of different structures by hybrid computers with graphic processors are proposed, coefficients for the efficiency of the algorithms are obtained, and approbation of the developed algorithms for test and practical problems is carr...

Визначаються та досліджуються зважені псевдообернені матриці з невиродженими знаконевизначеними вагами. Доведено теорему існування та єдиності цих матриць. Дано зображення зважених псевдообернених матриць зі знаконевизначеними вагами в термінах коефіцієнтів характеристичних многочленів матриць, що симетризуються, одержано розвинення зазначених матр...

We obtain the asymptotic estimates of the accuracy of eigenvalues of the fourth-order operator with mixed boundary conditions on the boundary of a rectangle. Knowledge of the main part of eigenvalues error allows us to reasonably specify eigenvalues on a sequence of grids, obtain discrete analogs of high accuracy, and construct discrete analogs who...

The paper establishes the exact error energy compression ratio on each step of the two-step method, which shows a better convergence rate compared with the one-step method with optimal parameter.

Consideration is given to algorithm for solving on computers of hybrid architecture (with multi-core processors and graphics accelerators) the systems of linear algebraic equations with band symmetrical positive definite matrices arising in the numerical modeling of 3D building constructions using the finite element method. Some results of the prop...

The authors formulate the initial–boundary-value problem of step pulse propagating from input to output on the basis of a hyperbolic equation with dissipation, which describes the propagation of waves with a finite speed. Exact analytical solution is obtained in the Laplace transform space. Numerical inversion of the Laplace transform by using Eule...

We consider a new parallel algorithm to solve systems of linear algebraic equations with band symmetric positive definite matrix, which combines computations on multi-core processors and graphics processing units. We present the results of simulation of spatial filtration of groundwater for Chernihiv region with Nadra-3D finite-element solver.

Methods and information technologies based on parallel computations for the mathematical modeling of kinetics of stress-strain state as well as processes of critical constructions structure or their elements are dealt with. While solving the problems on modeling of processes related to the ductile fracture of parts of thick-walled pipelines with th...

We want our readers to have more examples of programs using GMP and MPFR library. Chapter 8 consists of two parallel algorithms and appropriate parallel programs for solving SLAE by LU-decomposition and SVD method. These are parallel MPI-programs using MPIGMP and GMP library. Considered programs were tested on supercomputer with different numbers o...

One of the most important uses of computers is their use for solving scientific and engineering problems, most of which finally reduced or have their intermediate stage solution of problems in computational mathematics. MPI interface for a long time is the unofficial standard for building distributed computing systems. We consider organization and...

The algorithm of joint using GMP (MPFR) library and MPI interface in parallel programming is described. The most suitable way to implement GMP in a parallel program with MPI is the usage of MPIGMP library, which can transport some multiple presicion variables of MPFR and GMP by packing them into one memory area. MPIGMP is a part of MPIBNCpack.

MPFR library (MultiPrecision Float with Rounding) deserves special attention, as the programmer who uses it creates their own floating-point format, indicating the bit order of the number and the mantissa. Thus, you can create formats that are not described in modern standards, but to perform arithmetic operations according to the classical algorit...

Gauss method is one of the most effective methods for solving linear systems with nonsingular matrices of general form. As you know, from Chapter 1 double digits for solution of SLAEs (1.1) was not enough to get the correct solutions of the system. For this case there is a need for increasing precision. It is possible with GMP library. In this Chap...

Systems of linear algebraic equations (SLAEs) can arise during the solving of many problems-scientific, engineering, economic etc., for example, in: data processing problems, discretization of linear differential problems by projection difference methods, the solving of linear problems by least squares method; calculation of electric circuits and c...

We present algorithm for using GMP library in your program. The Main function of GMP for floating point numbers are described here. After learning this Chapter you can start to write your program with multiple precision. Most of described functions are accompanied by examples.

Parallel algorithms for direct methods of analysis and solution of linear algebra problems with sparse symmetric irregularly structured matrices are considered. The performance of such algorithms is investigated. Upper estimates of the speedup and efficiency factors are obtained for a parallel algorithm for triangular decomposition of sparse matric...

The general case of incompatible systems of linear algebraic equations with matrices of arbitrary rank is considered. The estimates for total errors are obtained for all the considered cases under conditions of approximate data. In solving systems by iterative methods, the conditions of completion of iterative processes that provide solutions with...

This article analyzes mathematical and technological problems that arise in performing computational experiments on modern high-performance computers (supercomputers). As a means of overcoming the difficulties associated with the analysis and solution of computer model problems under conditions of approximate initial data on computers with parallel...

Parallel algorithms and programs for solving so-called problems of transcomputational complexity are addressed. Quality requirements to both IT program component and development tools for parallel MPI-programs are formulated. An intelligent IT for research automation and solution of scientific and engineering problems is proposed and used to formul...

Abstract The weighted least squares problem is considered. Given a generally inconsistent system of linear algebraic equations, error estimates are obtained for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix.

This paper deals with game problems with impulse and geometric constraints on the players' controls. To analyze conflict-controlled processes with discontinuous trajectories, the method of resolving functions is used. This makes it possible to derive sufficient conditions of the game termination in a finite guaranteed time.

A weighted least squares problem {ie863-01} with positive definite weights M and N is considered, where A ∈ Rm×n is a rank-deficient matrix, b ∈ Rm. The hereditary, computational, and global errors of a weighted normal pseudosolution are estimated for perturbed initial data, including the case where the rank of the perturbed matrix varies.

The error of solution of Cauchy problems for systems of ordinary differential equations is estimated in the case where the input data are approximate. It is shown how to prepare a program for computing the right-hand sides of the system automatically and simultaneously. Diagrams are presented to illustrate the efficiency of parallelization.

The conception of MIMD computers is considered from the viewpoint of automatic analysis of characteristics of engineering and scientific problems with approximate initial data, development of parallel computational algorithms and programs, solution of problems, and estimation of the reliability of computer solutions. The conception was experimental...

Estimates are derived for the total error in solving linear algebraic systems, including non-consistent ones, by one- and two-step iterative methods, with input data being approximate. Criteria for terminating iterative processes are constructed providing a given accuracy of the solution.

The stress concentration around a two-layered spherical inclusion located near the surface of an elastic half-space under uniform compression is considered. A numerical solution algorithm is elaborated, application software is created, and numerical experiments demonstrating the capabilities of the approach proposed are conducted.

The problem of a stress concentration in the neighborhood of two-layered spherical inclusion located near the surface of an elastic semi-space under action of a uniform compression is considered. The numerical solution algorithm is elaborated, the application software is created, and numerical experiments demonstrating the capabilities of the appro...

Without Abstract

An algorithm is proposed for computing some of the eigenvalues and the corresponding eigenvectors of the generalized eigenvalue problem for symmetric positive-definite matrices stored in profile form on disk.

A study is made of the spatial-temporal dynamics of two-dimensional wave beams described by a system including a stationary Schroedinger equation and a dynamic equation for a perturbed medium (potential). Despite the asymptotic transition to a stationary structure with a strictly spatial evolution, systems of this kind exhibit a complex behavior wh...

The dynamics in time and space of a Gaussian beam of high power free electron laser (FEL) radiation, polarized in the ordinary mode, propagating in a magnetized plasma, is presented within a paraxial approximation. The process of self-focusing of radiation and the subsequent formation of a focus point, running back towards the incoming high frequen...

The ability of plasma wave solitons to emit electromagnetic waves at the frequency of the natural oscillations of the field is studied. It is shown that this radiation is the main channel for the dissipation of the energy of strong plasma turbulence in a magnetoactive plasma. An interpretation is presented for the artificial radio waves arising in...

An estimate of the accuracy of a conservative difference scheme in an energy net metric is established, when the solution of the initial problem belongs to Sobolev spaces.The accuracy of difference schemes for fourth-order elliptic differential operators when free-boundary type boundary conditions are imposed were investigated in [1–4] assuming tha...

The accuracy of the net method when the principal and natural boundary conditions are specified at the boundary is established. The interchange points of the conditions type are found at the corners of the rectangle. Estimates of the error of the solution are obtained under the assumption that the eigen functions belong to the Sobolev space .

The theoretical problems related to the stability of reflection transformations and the convergence of the conjugate gradient method have been studied in [4, 6] (in the case of the above method of parallel calculation it is not necessary to study them additionally). Let us note that the effectiveness of parallel calculations depends on both the alg...