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This article is devoted to some problems of declaring and storing information for objects' visualization. The storage structure and resources control technology can be applied for real-time visualization of three-dimensional scenes. Such instruments as Sample Framework from DirectX SDK and Direct3D Extension Library (D3DX) were used in the implementation.

A new grid and cloud services simulation for NICA accelerator complex data storage and processing system are described. This system is focused on improving the efficiency of the grid-cloud systems development by using work quality indicators of some real system to design and predict its evolution. For these purpose the simulation program are combined with real monitoring system of the grid-cloud service through a special database. An example of the program usage to simulate a sufficiently general cloud structure, which can be used for more common purposes, is given.

We consider the approaches to the construction of methods for solving four-dimensional programming problems for calculating directions for multiple minimizations of smooth functions on a set of a given set of linear equalities. The approach consists of two stages.

At the first stage, the problem of quadratic programming is transformed by a numerically stable direct multiplicative algorithm into an equivalent problem of designing the origin of coordinates on a linear manifold, which defines a new mathematical formulation of the dual quadratic problem. For this, a numerically stable direct multiplicative method for solving systems of linear equations is proposed, taking into account the sparsity of matrices presented in packaged form. The advantage of this approach is to calculate the modified Cholesky factors to construct a substantially positive definite matrix of the system of equations and its solution in the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made in the position of the next processed row of the matrix, which allows the use of static data storage formats.

At the second stage, the necessary and sufficient optimality conditions in the form of Kuhn–Tucker determine the calculation of the direction of descent — the solution of the dual quadratic problem is reduced to solving a system of linear equations with symmetric positive definite matrix for calculating of Lagrange's coefficients multipliers and to substituting the solution into the formula for calculating the direction of descent.

It is proved that the proposed approach to the calculation of the direction of descent by numerically stable direct multiplicative methods at one iteration requires a cubic law less computation than one iteration compared to the well-known dual method of Gill and Murray. Besides, the proposed method allows the organization of the computational process from any starting point that the user chooses as the initial approximation of the solution.

Variants of the problem of designing the origin of coordinates on a linear manifold, a convex polyhedron and a vertex of a convex polyhedron are presented. Also the relationship and implementation of methods for solving these problems are described.

Small practical value of many numerical methods for solving single-ended systems of linear equations with ill-conditioned matrices due to the fact that these methods in the practice behave quite differently than in the case of precise calculations. Historically, sustainability is not enough attention was given, unlike in numerical algebra ‘medium-sized’, and emphasis is given to solving the problems of maximal order in data capabilities of the computer, including the expense of some loss of accuracy. Therefore, the main objects of study is the most appropriate storage of information contained in the sparse matrix; maintaining the highest degree of rarefaction at all stages of the computational process. Thus, the development of efficient numerical methods for solving unstable systems refers to the actual problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving systems of linear equations, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach consists in minimization of filling the main lines of the multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats. The storage format of sparse matrices has been studied and the advantage of this format consists in possibility of parallel execution any matrix operations without unboxing, which significantly reduces the execution time and memory footprint.

Direct multiplier methods for solving systems of linear equations are best suited for solving problems of large size on a computer — sparse matrix systems allow you to get multipliers, the main row of which is also sparse, and the operation of multiplication of a vector-row of the multiplier according to the complexity proportional to the number of nonzero elements of this multiplier.

As a direct continuation of this work is proposed in the basis for constructing a direct multiplier algorithm of linear programming to put a modification of the direct multiplier algorithm for solving systems of linear equations based on integration of technique of linear programming for methods to select the host item. Direct multiplicative methods of linear programming are best suited for the construction of a direct multiplicative algorithm set the direction of descent Newton methods in unconstrained optimization by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

Multiplicative methods for sparse matrices are best suited to reduce the complexity of operations solving systems of linear equations performed on each iteration of the simplex method. The matrix of constraints in these problems of sparsely populated nonzero elements, which allows to obtain the multipliers, the main columns which are also sparse, and the operation of multiplication of a vector by a multiplier according to the complexity proportional to the number of nonzero elements of this multiplier. In addition, the transition to the adjacent basis multiplier representation quite easily corrected. To improve the efficiency of such methods requires a decrease in occupancy multiplicative representation of the nonzero elements. However, at each iteration of the algorithm to the sequence of multipliers added another. As the complexity of multiplication grows and linearly depends on the length of the sequence. So you want to run from time to time the recalculation of inverse matrix, getting it from the unit. Overall, however, the problem is not solved. In addition, the set of multipliers is a sequence of structures, and the size of this sequence is inconvenient is large and not precisely known. Multiplicative methods do not take into account the factors of the high degree of sparseness of the original matrices and constraints of equality, require the determination of initial basic feasible solution of the problem and, consequently, do not allow to reduce the dimensionality of a linear programming problem and the regular procedure of compression — dimensionality reduction of multipliers and exceptions of the nonzero elements from all the main columns of multipliers obtained in previous iterations. Thus, the development of numerical methods for the solution of linear programming problems, which allows to overcome or substantially reduce the shortcomings of the schemes implementation of the simplex method, refers to the current problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving problems in linear programming, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach is to reduce dimensionality and minimize filling of the main rows of multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats.

As a direct continuation of this work is the basis for constructing a direct multiplicative algorithm set the direction of descent in the Newton methods for unconstrained optimization is proposed to put a modification of the direct multiplier method, linear programming by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

We consider a numerically stable direct multiplicative algorithm of solving linear equations systems, which takes into account the sparseness of matrices presented in a packed form. The advantage of the algorithm is the ability to minimize the filling of the main rows of multipliers without losing the accuracy of the results. Moreover, changes in the position of the next processed row of the matrix are not made, what allows using static data storage formats. Linear system solving by a direct multiplicative algorithm is, like the solving with $LU$-decomposition, just another scheme of the Gaussian elimination method implementation.

In this paper, this algorithm is the basis for solving the following problems:

Problem 1. Setting the descent direction in Newtonian methods of unconditional optimization by integrating one of the known techniques of constructing an essentially positive definite matrix. This approach allows us to weaken or remove additional specific difficulties caused by the need to solve large equation systems with sparse matrices presented in a packed form.

Problem 2. Construction of a new mathematical formulation of the problem of quadratic programming and a new form of specifying necessary and sufficient optimality conditions. They are quite simple and can be used to construct mathematical programming methods, for example, to find the minimum of a quadratic function on a polyhedral set of constraints, based on solving linear equations systems, which dimension is not higher than the number of variables of the objective function.

Problem 3. Construction of a continuous analogue of the problem of minimizing a real quadratic polynomial in Boolean variables and a new form of defining necessary and sufficient conditions of optimality for the development of methods for solving them in polynomial time. As a result, the original problem is reduced to the problem of finding the minimum distance between the origin and the angular point of a convex polyhedron, which is a perturbation of the $n$-dimensional cube and is described by a system of double linear inequalities with an upper triangular matrix of coefficients with units on the main diagonal. Only two faces are subject to investigation, one of which or both contains the vertices closest to the origin. To calculate them, it is sufficient to solve $4n – 4$ linear equations systems and choose among them all the nearest equidistant vertices in polynomial time. The problem of minimizing a quadratic polynomial is $NP$-hard, since an $NP$-hard problem about a vertex covering for an arbitrary graph comes down to it. It follows therefrom that $P = NP$, which is based on the development beyond the limits of integer optimization methods.

A numerically stable direct multiplicative method for solving systems of linear equations that takes into account the sparseness of matrices presented in a packed form is considered. The advantage of the method is the calculation of the Cholesky factors for a positive definite matrix of the system of equations and its solution within the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made to the position of the next processed row of the matrix, which allows using static data storage formats. The solution of the system of linear equations by a direct multiplicative algorithm is, like the solution with LU-decomposition, just another scheme for implementing the Gaussian elimination method.

The calculation of the Cholesky factors for a positive definite matrix of the system and its solution underlies the construction of a new mathematical formulation of the unconditional problem of quadratic programming and a new form of specifying necessary and sufficient conditions for optimality that are quite simple and are used in this paper to construct a new mathematical formulation for the problem of quadratic programming on a polyhedral set of constraints, which is the problem of finding the minimum distance between the origin ordinate and polyhedral boundary by means of a set of constraints and linear algebra dimensional geometry.

To determine the distance, it is proposed to apply the known exact method based on solving systems of linear equations whose dimension is not higher than the number of variables of the objective function. The distances are determined by the construction of perpendiculars to the faces of a polyhedron of different dimensions. To reduce the number of faces examined, the proposed method involves a special order of sorting the faces. Only the faces containing the vertex closest to the point of the unconditional extremum and visible from this point are subject to investigation. In the case of the presence of several nearest equidistant vertices, we investigate a face containing all these vertices and faces of smaller dimension that have at least two common nearest vertices with the first face.

The report presents a number of news and updates of the ALICE computing for RUN2 and RUN3.

This includes:

– implementation in production of a new system EOS;

– migration to the file system CVMFS to be used for storage of the software;

– the plan for solving the problem of “Long-Term Data Preservation”;

– overview of the concept of “O square”, combining offline and online data processing;

– overview of the existing models to use the virtual clouds for ALICE data processing. Innovations are shown on the example of the Russian sites.

The problem of accumulation and the statistical analysis of computer experiment results are solved. The main experiment program is considered as the data source. The results of main experiment are collected on specially prepared sheet Excel with pre-organized structure for the accumulation, statistical processing and visualization of the data. The created method and the program are used at efficiency research of the scientific researches which are carried out by authors.

The paper is devoted to the secondary use of spectrum in telecommunication networks. It is emphasized that one of the solutions to this problem is the use of cognitive radio technologies and dynamic spectrum access for the successful functioning of which a large amount of information is required, including the parameters of base stations and network subscribers. Storage and processing of information should be carried out using a radio environment map, which is a spatio-temporal database of all activity in the network and allows you to determine the frequencies available for use at a given time. The paper presents a two-level model for forming a map of the radio environment of a cellular communication system LTE, in which the local and global levels are highlighted, which is described by the following parameters: a set of frequencies, signal attenuation, signal propagation map, grid step, current time count. The key objects of the model are the base station and the subscriber unit. The main parameters of the base station include: name, identifier, cell coordinates, range number, radiation power, numbers of connected subscriber devices, dedicated resource blocks. For subscriber devices, the following parameters are used: name, identifier, location, current coordinates of the device cell, base station identifier, frequency range, numbers of resource blocks for communication with the station, radiation power, data transmission status, list of numbers of the nearest stations, schedules movement and communication sessions of devices. An algorithm for the implementation of the model is presented, taking into account the scenarios of movement and communication sessions of subscriber devices. A method for calculating a map of the radio environment at a point on a coordinate grid, taking into account losses during the propagation of radio signals from emitting devices, is presented. The software implementation of the model is performed using the MatLab package. The approaches are described that allow to increase the speed of its work. In the simulation, the choice of parameters was carried out taking into account the data of the existing communication systems and the economy of computing resources. The experimental results of the algorithm for the formation of a radio environment map are demonstrated, confirming the correctness of the developed model.