Использование рядов по сферическим функциям для аппроксимации гравитационного поля Земли в локальном районе

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Gravity and Low-Frequency Geodynamics

The article discusses a field geophysical complex that includes the Earth's gravity and magnetic fields parameters meters — an digital zenith camera (DZC), a relative gravimeter, onboard quantum magnetometer and ground magnetovariation station. This complex allows determining such parameters of the Earth's gravity field as acceleration and anomalies of gravity, plumb line deviations, horizontal components of the acceleration of gravity and gravitational gradients, and parameters of the Earth's magnetic field — absolute and anomalous values of induction. Maps of the Earth's gravity and magnetic fields can be used in various fields, in particular, in the creation and testing of integrated autonomous navigation systems using the parameters of these fields. The paper presents examples of parameter maps created using this complex. The measurement error of acceleration and gravity anomaly is 10 µGal, plumb line deviation is ≈ 0.2″, horizontal components of gravity acceleration are ≈ 1 mGal, gravitational gradients are 10 Eötvös at a distanc

One of the main tasks of modern physical geodesy is to determine the high-frequency part of the Earth’s gravity field (EGF), which inevitably depends on the individual characteristics of a particular computational area. It is well known that in global EGF modeling it is especially convenient to have a spherical function series. However, these series forming a reliable basis for global modeling are unsuitable for approximation in a local area because they lose orthogonality, the most important property of basis functions. The authors describe the possibility of imparting this property to spherical functions in the necessary part of the sphere, which enables further using the habitual apparatus of harmonic series for approximation of EGF in local areas. Of course, the foregoing is only concerned to the most basic concepts of the new direction in modeling the EGF in a local region

Abstract—The paper addresses issues related to constructing the analytical approximations of geopotential fields based on a modified method of S-approximations which approximates the initial field by a sum of potentials of a simple and a double layer distributed on a set of carriers of mass located below a given relief. Special attention is paid to the application of new highly efficient methods for solving the systems of linear algebraic equations (SLAE) of large and super-large dimensions to which the inverse geophysical and geodetic problems are reduced. An improved block contrasting method is proposed for solving these SLAEs. The underlying idea of the method is to divide the initial domain into the subdomains that are most contrasting by their properties and to subsequently deform the blocks. The results of mathematical experiments on constructing the analytical approximations of anomalous gravity and magnetic fields and separating the fields created by the different sets of the sources are presented.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

Radial Basis Functions (RBF) are popular for interpolating scattered data. In this context, the solution of the system of linear algebraic equations (SLAE) is the most time-consuming operation. Techniques fail with large point sets consisting of more than several thousands of points when direct methods and global support are used. In this paper we demostrate that the solution of the SLAE in the wavelet domain is suitable for the problem of image interpolation by means of Compactly-Supported Radial Basis Functions (CSRBF). The iterative solution of SLAE with highly irregular matrices cannot be accelerated by wavelet transformation and subsequent sparcification if the transformed matrix is still highly irregular. To solve the SLAE in the wavelet domain, the ordering of the samples defines the spacial relationship and the energy of the coefficients in the low frequency domain. Two sorting algorithms for the wavelet domain solution are tested and compared with the spacial solution of the SLAE. Examples of image interpolation by means of CSRBF demostrate the superiority of the solution in the wavelet domain using GMRES iterative method.

Summary The computational costs required to calculate three-dimensional fields in geophysical problems can be huge. Vector finite element methods allows constructing mathematical models directly in terms of vector variables, since vector basis functions are used. When using the finite element method, the main computational load, as a rule, falls on the solution of systems of linear algebraic equations obtained as a result of the finite element approximation of the boundary value problem. This paper investigates the efficiency of using various methods for solving linear systems obtained as a result of using the vector finite element method for the direct problem of induction logging.

The article deals with high-performance information technology (HPC) for the problems of stress-strain analysis at all stages of the life cycle of buildings and structures: construction, operation and reconstruction. The results of numerical simulation of high buildings using software as a processor component based on a new hybrid algorithm for solving systems of linear algebraic equations [1] with a symmetric positive-definite matrix that combines computation on multi-core processors and graphs. It has been found that to accelerate the calculations, hybrid systems that combine multi-core CPUs with accelerator coprocessors, including GPUs, are promising [5]. To test the effectiveness of the proposed parallel algorithm for solving systems of linear algebraic equations [1], numerical experiments were carried out at the most dangerous loads of a 27-story building. Results of numerical researches with use for preprocessor (input of initial data) and postprocessor (output of results of calculations) of processing of the LIRA-SAPR software complex are presented [2, 4, 6]. The results of numerical studies of the behavior of structures of high buildings have shown a multiple reduction in the time of solving systems of linear algebraic equations with symmetric matrices on multiprocessor (multi-core) computers with graphical accelerators using the proposed hybrid algorithms [1]. High-performance technologies based on parallel calculations give more effect than more complex processes: modeling of life cycle of high buildings, bridges, especially complex structures of NPPs, etc. for static and dynamic loads, including emergencies in normal and difficult geological conditions, which make up 70% of Ukraine's territories.

Conditionality in the problem of the minimum of a quadratic functional

According to statistical research, a violation of license agreements annually causes huge losses to software companies. On the one hand, illegal copies of the software product are created, on the other hand, some fragments of the programs are used by third parties unauthorized. Another important problem is the violation of the program integrity, for example, in terms of blocking functions of the license key checking. In this regard, the task of construction methods for protecting intellectual property in software applications is highly relevant. Previously known methods solve this problem by means of fragile digital watermarks. Below is presented a method for constructing a fragile digital watermark used in executable files. A model of a developers team creating software product protected by DWM is considered. The application of this method will allow to reveal the fact of the container integrity violation, on the one hand, and, on the other hand, will allow the author, if it is necessary, to confirm his participation in the development and embedding of the DWM. In this method we use mathematical properties of systems of linear algebraic equations, digital signature and cryptographic hash functions. The scheme is based on the Kronecker – Capelli theorem. To find the group password the co-author who is in the group finds one root of the consistent system of linear algebraic equations. The indicated system consists of n equations and contains n + 1 variables. For an outsider who is not in the group, the system of equations does not have a unique solution. The co-author of the group is able to calculate one variable in system based on their passport data. Therefore, the system of equations for such co-author has a unique solution. Various attacks on a protected by the new method are analyzed, and it is shown its efficiency. The constructed method can be applied in companies with a large team of developers.

One of the aims of the Earth Explorer Gravity Field and Steady-State Ocean Circulation (GOCE) mission is to provide global and regional models of the Earth's gravity field and of the geoid with high spatial resolution and accuracy. Using the GOCE error model, simulation studies were performed in order to estimate the accuracy of datum transfer in different areas of the Earth. The results showed that with the GOCE error model, the standard deviation of the height anomaly differences is about one order of magnitude better than the corresponding value with the EGM96 error model. As an example, the accuracy of the vertical datum transfer from the tide gauge of Amsterdam to New York was estimated equal to 57 cm when the EGM96 error model was used, while in the case of GOCE error model this accuracy was increased to 6 cm. The geoid undulation difference between the two places is about 76.5 m. Scaling the GOCE errors to the local gravity variance, the estimated accuracy varied between 3 and 7 cm, depending on the scaling model.

GOCE closed-loop simulation