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A hybrid cluster implies the use of computational devices with radically different architectures. Usually, these are conventional CPU architecture (e.g. x86_64) and GPU architecture (e. g. NVIDIA CUDA). Creating and exploiting such a cluster requires some experience: in order to harness all computational power of the described system and get substantial speedup for computational tasks many factors should be taken into account. These factors consist of hardware characteristics (e.g. network infrastructure, a type of data storage, GPU architecture) as well as software stack (e.g. MPI implementation, GPGPU libraries). So, in order to run scientific applications GPU capabilities, software features, task size and other factors should be considered.

This report discusses opportunities and problems of hybrid computations. Some statistics from tests programs and applications runs will be demonstrated. The main focus of interest is open source applications (e. g. OpenFOAM) that support GPGPU (with some parts rewritten to use GPGPU directly or by replacing libraries).

There are several approaches to organize heterogeneous computations for different GPU architectures out of which CUDA library and OpenCL framework are compared. CUDA library is becoming quite typical for hybrid systems with NVIDIA cards, but OpenCL offers portability opportunities which can be a determinant factor when choosing framework for development. We also put emphasis on multi-GPU systems that are often used to build hybrid clusters. Calculations were performed on a hybrid cluster of SPbU computing center.

Protection of aquatic biological resources in the coastal area has significant features (a large number of small fishing vessels, the dynamism of the situation, the use of coastal protection), by virtue of which stands in a class of applications. A mathematical model of protection designed for the determination of detection equipment and means of violators of the situation in order to ensure the function of deterrence of illegal activities. Resolves a tactical game-theoretic problem - find the optimal line patrol (parking) means of implementation (guard boats) and optimal removal of seats from the shore fishing violators. Using the methods of the theory of experimental design, linear regression models to assess the contribution of the main factors affecting the results of the simulation.

In order to enhance the sustainability and adequacy of the model is proposed to use the mechanism of rankings means of protection, based on the borders and the rank and Pareto allows to take into account the principles of protection and further means of protection. To account for the variability of the situation offered several scenarios in which it is advisable to perform calculations.

Execution or violation of the principle of competitive exclusion in communities is the subject of many studies. The principle of competitive exclusion means that coexistence of species in community is impossible if the number of species exceeds the number of controlling mutually independent factors. At that time there are many examples displaying the violations of this principle in the natural systems. The explanations for this paradox vary from inexact identification of the set of factors to various types of spatial and temporal heterogeneities. One of the factors breaking the principle of competitive exclusion is intraspecific competition. This study holds the model of community with two species and one influencing factor with density-dependent mortality and spatial heterogeneity. For such models possibility of the existence of stable equilibrium is proved in case of spatial homogeneity and negative effect of the species on the factor. Our purpose is analysis of possible variants of dynamics of the system with spatial heterogeneity under the various directions of the species effect on the influencing factor. Numerical analysis showed that there is stable coexistence of the species agreed with homogenous spatial distributions of the species if the species effects on the influencing factor are negative. Density-dependent mortality and spatial heterogeneity lead to violation of the principle of competitive exclusion when equilibriums are Turing unstable. In this case stable spatial heterogeneous patterns can arise. It is shown that Turing instability is possible if at least one of the species effects is positive. Model nonlinearity and spatial heterogeneity cause violation of the principle of competitive exclusion in terms of both stable spatial homogenous states and quasistable spatial heterogeneous patterns.

The paper considers the problem of estimating the parameters of time series described by regression models with Markov switching of two regimes at random instants of time with independent Gaussian noise. For the solution, we propose a variant of the EM algorithm based on the iterative procedure, during which an estimation of the regression parameters is performed for a given sequence of regime switching and an evaluation of the switching sequence for the given parameters of the regression models. In contrast to the well-known methods of estimating regression parameters in the models with Markov switching, which are based on the calculation of a posteriori probabilities of discrete states of the switching sequence, in the paper the estimates are calculated of the switching sequence, which are optimal by the criterion of the maximum of a posteriori probability. As a result, the proposed algorithm turns out to be simpler and requires less calculations. Computer modeling allows to reveal the factors influencing accuracy of estimation. Such factors include the number of observations, the number of unknown regression parameters, the degree of their difference in different modes of operation, and the signal-to-noise ratio which is associated with the coefficient of determination in regression models. The proposed algorithm is applied to the problem of estimating parameters in regression models for the rate of daily return of the RTS index, depending on the returns of the S&P 500 index and Gazprom shares for the period from 2013 to 2018. Comparison of the estimates of the parameters found using the proposed algorithm is carried out with the estimates that are formed using the EViews econometric package and with estimates of the ordinary least squares method without taking into account regimes switching. The account of regimes switching allows to receive more exact representation about structure of a statistical dependence of investigated variables. In switching models, the increase in the signal-to-noise ratio leads to the fact that the differences in the estimates produced by the proposed algorithm and using the EViews program are reduced.

The volume and structure of accumulated credit debt to the banking system depends on many factors, the most important of which is the level of interest rates. The correct assessment of borrowers’ reaction to the changes in the monetary policy allows to develop econometric models, representing the structure of the credit portfolio in the banking system by terms of lending. These models help to calculate indicators characterizing the level of interest rate risk in the whole system. In the study, we carried out the identification of four types of models: discrete linear model based on transfer functions; the state-space model; the classical econometric model ARMAX, and a nonlinear Hammerstein –Wiener model. To describe them, we employed the formal language of automatic control theory; to identify the model, we used the MATLAB software pack-age. The study revealed that the discrete linear state-space model is most suitable for short-term forecasting of both the volume and the structure of credit debt, which in turn allows to predict trends in the structure of accumulated credit debt on the forecasting horizon of 1 year. The model based on the real data has shown a high sensitivity of the structure of credit debt by pay back periods reaction to the changes in the Ñentral Bank monetary policy. Thus, a sharp increase in interest rates in response to external market shocks leads to shortening of credit terms by borrowers, at the same time the overall level of debt rises, primarily due to the increasing revaluation of nominal debt. During the stable falling trend of interest rates, the structure shifts toward long-term debts.

A simplified implicit Euler method was analyzed as an alternative to the explicit Euler method, which is a commonly used method in numerical modeling in electrophysiology. The majority of electrophysiological models are quite stiff, since the dynamics they describe includes a wide spectrum of time scales: a fast depolarization, that lasts milliseconds, precedes a considerably slow repolarization, with both being the fractions of the action potential observed in excitable cells. In this work we estimate stiffness by a formula that does not require calculation of eigenvalues of the Jacobian matrix of the studied ODEs. The efficiency of the numerical methods was compared on the case of typical representatives of detailed and conceptual type models of excitable cells: Hodgkin–Huxley model of a neuron and Aliev–Panfilov model of a cardiomyocyte. The comparison of the efficiency of the numerical methods was carried out via norms that were widely used in biomedical applications. The stiffness ratio’s impact on the speedup of simplified implicit method was studied: a real gain in speed was obtained for the Hodgkin–Huxley model. The benefits of the usage of simple and high-order methods for electrophysiological models are discussed along with the discussion of one method’s stability issues. The reasons for using simplified instead of high-order methods during practical simulations were discussed in the corresponding section. We calculated higher order derivatives of the solutions of Hodgkin-Huxley model with various stiffness ratios; their maximum absolute values appeared to be quite large. A numerical method’s approximation constant’s formula contains the latter and hence ruins the effect of the other term (a small factor which depends on the order of approximation). This leads to the large value of global error. We committed a qualitative stability analysis of the explicit Euler method and were able to estimate the model’s parameters influence on the border of the region of absolute stability. The latter is used when setting the value of the timestep for simulations a priori.

A proposed approximate model of the rolling of a deforming wheel with a pneumatic tire allows one to account as well forces in tires as the effect of the dry friction on the stability of the rolling upon the shimmy phenomenon prognosis. The model os based on the theory of the dry friction with combined kinematics of relative motion of interacting bodies, i. e. under the condition of simultaneous rolling, sliding, and spinning with accounting for the real shape of a contact spot and contact pressure distribution. The resultant vector and couple of the forces generated by the contact interaction with dry friction are defined by integration over the contact area, whereas the static contact pressure under the conditions of vanishing velocity of sliding and angular velocity of spinning is computed after the finite-element solution for the statical contact of a pneumatic with a rigid road with accounting forreal internal structure and properties of a tire. The solid finite element model of a typical tire with longitudinal thread is used below as a background. Given constant boost pressure, vertical load and static friction factor 0.5 the numerical solution is constructed, as well as the appropriate solutions for lateral and torsional kinematic loading. It is shown that the contact interaction of a pneumatic tire and an absolutely rigid road could be represented without crucial loss of accuracy as two typical stages, the adhesion and the slip; the contact area shape remains nevertheless close to a circle. The approximate diagrams are constructed for both lateral force and friction torque; on the initial stage the diagrams are linear so that corresponds to the elastic deformation of a tire while on the second stage both force and torque values are constant and correspond to the dry friction force and torque. For the last stages the approximate formulae for the longitudinal and lateral friction force and the friction torque are constructed on the background of the theory of the dry friction with combined kinematics. The obtained model can be treated as a combination of the Keldysh model of elastic wheel with no slip and spin and the Klimov rigid wheel model interacting with a road by dry friction forces.

The article substantiates the need to develop and analyze mathematical models that take into account the mutual influence of long (Kondratiev) waves of economic development. The analysis of the available publications shows that at the model level, the direct and inverse relationships between intersecting long waves are still insufficiently studied. As practice shows, the production of the current long wave can receive an additional impetus for growth from the technologies of the next long wave. The technologies of the next industrial revolution often serve as improving innovations for the industries born of the previous industrial revolution. As a result, the new long wave increases the amplitude of the oscillations of the trajectory of the previous long wave. Such results of the interaction of long waves in the economy are similar to the effects of interference of physical waves. The mutual influence of the recessions and booms of the economies of different countries gives even more grounds for comparing the consequences of this mutual influence with the interference of physical waves. The article presents a model for the development of the technological base of production, taking into account the possibilities of combining old and new technologies. The model consists of several sub-models. The use of a different mathematical description for the individual stages of updating the technological base of production allows us to take into account the significant differences between the successive phases of the life cycle of general purpose technologies, considered in modern literature as the technological basis of industrial revolutions. One of these phases is the period of formation of the appropriate infrastructure necessary for the intensive diffusion of new general purpose technology, for the rapid development of industries using this technology. The model is used for illustrative calculations with the values of exogenous parameters corresponding to the logic of changing long waves. Despite all the conditionality of the illustrative calculations, the configuration of the curve representing the change in the return on capital in the simulated period is close to the configuration of the real trajectory of the return on private fixed assets of the US economy in the period 1982-2019. The factors that remained outside the scope of the presented model, but which are advisable to take into account when describing the interference of long waves of economic development, are indicated.

Within the framework of the actual problem of comet-asteroid danger, the physical processes causing the destruction and fragmentation of meteor bodies in the Earth’s atmosphere are numerically investigated. Based on the developed physicalmathematical models that determines the movements of space objects of natural origin in the atmosphere and their interaction with it, the fall of three, one of the largest and by some parameters unusual bolides in the history of meteoritics, are considered: Tunguska, Vitim and Chelyabinsk. Their singularity lies in the absence of any material meteorite remains and craters in the area of the alleged crash site for the first two bodies and the non-detection, as it is assumed, of the main mother body for the third body (due to the too small amount of mass of the fallen fragments compared to the estimated mass). The effect of aerodynamic loads and heat flows on these bodies are studied, which leads to intensive surface mass loss and possible mechanical destruction. The velocities of the studied celestial bodies and the change in their masses are determined from the modernized system of equations of the theory of meteoric physics. An important factor that is taken into account here is the variability of the meteorite mass entrainment parameter under the action of heat fluxes (radiation and convective) along the flight path. The process of fragmentation of meteoroids in this paper is considered within the framework of a progressive crushing model based on the statistical theory of strength, taking into account the influence of the scale factor on the ultimate strength of objects. The phenomena and effects arising at various kinematic and physical parameters of each of these bodies are revealed. In particular, the change in the ballistics of their flight in the denser layers of the atmosphere, consisting in the transition from the fall mode to the ascent mode. At the same time, the following scenarios of the event can be realized: 1) the return of the body back to outer space at its residual velocity greater than the second cosmic one; 2) the transition of the body to the orbit of the Earth satellite at a residual velocity greater than the first cosmic one; 3) at lower values of the residual velocity of the body, its return after some time to the fall mode and falling out at a considerable distance from the intended crash site. It is the implementation of one of these three scenarios of the event that explains, for example, the absence of material traces, including craters, in the case of the Tunguska bolide in the vicinity of the forest collapse. Assumptions about the possibility of such scenarios have been made earlier by other authors, and in this paper their implementation is confirmed by the results of numerical calculations.

The article considers an approach to modeling the impact of sanctions and import substitution on the performance of high-tech product markets based on the use of control theory methods (operational calculus, z-transform). The model under consideration assumes that an equipment manufacturer supplies unique high-tech equipment to a high-tech product (HP) manufacturer that dominates the equipment consumer market. The HP manufacturer, fearing disruption of equipment supplies due to the introduction of all kinds of restrictions and sanctions, invests in the development of import-substituting equipment production in a third company, which can also find application in the external market, at the expense of deductions from its profits. The influence of the following factors and actions on the performance of the conditional market is analyzed: 1) the degree of inertia of the development and production development processes in the company; 2) the share of equipment of the import-substituting company supplied to the HP manufacturer; 3) sanctions (general and selective) on the supply of equipment to the company-manufacturer of the import substitution, as well as blocking the import substitution process in the third company by the first company.

The calculations show that the acceleration of the equipment development and production processes leads to a faster decrease in the production volumes of the first company. At the same time, an increase in price is observed, which is associated with a change in the parameters of the inverse demand function.

An increase in the share of equipment of the import-substituting company consumed by the second company can lead to a sharp increase in production volumes in the second and third companies, stabilization of production volumes in the first company and an increase in price.

The introduction of sanctions leads to a decrease in the production volumes and income of all companies relative to the baseline version. A significant change in price also occurs. However, due to the inertia of the equipment production processes in the example under consideration, a significant change in production volumes in the aggregate of companies occurs with a significant lag. This is especially characteristic of the third company, in which a noticeable deviation from the baseline version begins after 20 years. The blocking by the first equipment manufacturing company of investments in the development of import substitution in the third company ensures a relatively small gain for the first company in production volumes and NPV although allows to raise her market share.