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In this paper, we consider convex stochastic optimization problems arising in machine learning applications (e. g., risk minimization) and mathematical statistics (e. g., maximum likelihood estimation). There are two main approaches to solve such kinds of problems, namely the Stochastic Approximation approach (online approach) and the Sample Average Approximation approach, also known as the Monte Carlo approach, (offline approach). In the offline approach, the problem is replaced by its empirical counterpart (the empirical risk minimization problem). The natural question is how to define the problem sample size, i. e., how many realizations should be sampled so that the quite accurate solution of the empirical problem be the solution of the original problem with the desired precision. This issue is one of the main issues in modern machine learning and optimization. In the last decade, a lot of significant advances were made in these areas to solve convex stochastic optimization problems on the Euclidean balls (or the whole space). In this work, we are based on these advances and study the case of arbitrary balls in the $p$-norms. We also explore the question of how the parameter $p$ affects the estimates of the required number of terms as a function of empirical risk.

In this paper, both convex and saddle point optimization problems are considered. For strongly convex problems, the existing results on the same sample sizes in both approaches (online and offline) were generalized to arbitrary norms. Moreover, it was shown that the strong convexity condition can be weakened: the obtained results are valid for functions satisfying the quadratic growth condition. In the case when this condition is not met, it is proposed to use the regularization of the original problem in an arbitrary norm. In contradistinction to convex problems, saddle point problems are much less studied. For saddle point problems, the sample size was obtained under the condition of $\gamma$-growth of the objective function. When $\gamma = 1$, this condition is the condition of sharp minimum in convex problems. In this article, it was shown that the sample size in the case of a sharp minimum is almost independent of the desired accuracy of the solution of the original problem.

The development of a high-temperature gas-cooled reactor (HTGR) constituting a part of nuclear power-and-process station and intended for large-scale hydrogen production is now in progress in the Russian Federation. One of the key objectives in development of the high-temperature gas-cooled reactor is the computational justification of the accepted design.

The article gives the procedure for the computational analysis of thermal and physical characteristics of the high-temperature gas-cooled reactor. The procedure is based on the use of the state-of-the-art codes for personal computer (PC).

The objective of thermal and physical analysis of the reactor as a whole and of the core in particular was achieved in three stages. The idea of the first stage is to justify the neutron physical characteristics of the block-type core during burn-up with the use of the MCU-HTR code based on the Monte Carlo method. The second and the third stages are intended to study the coolant flow and the temperature condition of the reactor and the core in 3D with the required degree of detailing using the FlowVision and the ANSYS codes.

For the purpose of carrying out the analytical studies the computational models of the reactor flow path and the fuel assembly column were developed.

As per the results of the computational modeling the design of the support columns and the neutron physical characteristics of the fuel assembly were optimized. This results in the reduction of the total hydraulic resistance of the reactor and decrease of the maximum temperature of the fuel elements.

The dependency of the maximum fuel temperature on the value of the power peaking factors determined by the arrangement of the absorber rods and of the compacts of burnable absorber in the fuel assembly is demonstrated.

The study of the structural characteristics of composites and nanostructures is of fundamental importance in materials science. Theoretical and numerical modeling and simulation of the mechanical properties of nanostructures is the main tool that allows for complex studies that are difficult to conduct only experimentally. One example of nanostructures considered in this work are carbon nanotubes (CNTs), which have good thermal and electrical properties, as well as low density and high Young’s modulus, making them the most suitable reinforcement element for composites, for potential applications in aerospace, automotive, metallurgical and biomedical industries. In this review, we reviewed the modeling methods, mechanical properties, and applications of CNT-reinforced metal matrix composites. Some modeling methods applicable in the study of composites with polymer and metal matrices are also considered. Methods such as the gradient descent method, the Monte Carlo method, methods of molecular statics and molecular dynamics are considered. Molecular dynamics simulations have been shown to be excellent for creating various composite material systems and studying the properties of metal matrix composites reinforced with carbon nanomaterials under various conditions. This paper briefly presents the most commonly used potentials that describe the interactions of composite modeling systems. The correct choice of interaction potentials between parts of composites directly affects the description of the phenomenon being studied. The dependence of the mechanical properties of composites on the volume fraction of the diameter, orientation, and number of CNTs is detailed and discussed. It has been shown that the volume fraction of carbon nanotubes has a significant effect on the tensile strength and Young’s modulus. The CNT diameter has a greater impact on the tensile strength than on the elastic modulus. An example of works is also given in which the effect of CNT length on the mechanical properties of composites is studied. In conclusion, we offer perspectives on the direction of development of molecular dynamics modeling in relation to metal matrix composites reinforced with carbon nanomaterials.

Langevin Dynamics, Monte Carlo, and all-atom Molecular Dynamics simulations in implicit solvent require a reliable source of pseudorandom numbers generated at each step of calculation. We present the two main approaches for implementation of pseudorandom number generators on a GPU. In the first approach, inherent in CPU-based calculations, one PRNG produces a stream of pseudorandom numbers in each thread of execution, whereas the second approach builds on the ability of different threads to communicate, thus, sharing random seeds across the entire device. We exemplify the use of these approaches through the development of Ran2, Hybrid Taus, and Lagged Fibonacci algorithms. As an application-based test of randomness, we carry out LD simulations of N independent harmonic oscillators coupled to a stochastic thermostat. This model allows us to assess statistical quality of pseudorandom numbers. We also profile performance of these generators in terms of the computational time, memory usage, and the speedup factor (CPU/GPU time).

The paper describes the results of computer simulation of time-dependent temperature fields arising in polar dielectrics irradiated by focused electron bunches with average electron energy when analyzing with electron microscopy techniques. The mathematical model was based on solving several-dimensional nonstationary heat conduction equation with use of numerical finite element method. The approximation of thermal source was performed taking into account the estimation of initial electron distribution determined by Monte-Carlo simulation of electron trajectories. The simulation program was designed in Matlab. The geometrical modeling and calculation results demonstrated the main features of model sample heating by electron beam were presented at the given experimental parameters as well as source approximation.

An algorithm of applied problem solving was described to calculate electrical characteristics of electrical field effects in ferroelectrics electron-beam charged. The algorithm was based on implementation of the deterministic model using finite element method as well as taking into account Monte-Carlo simulation results of electron transport. The program application was developed to perform computing experiments.

The article describes a mathematical model that simulates performance of a hierarchical organization during an early stage of a crisis. A distinguished feature of this stage of crisis is presence of so called early warning signals containing information on the approaching event. Employees are capable of catching the early warnings and of preparing the organization for the crisis based on the signals’ meaning. The efficiency of the preparation depends on both parameters of the organization and parameters of the crisis. The proposed simulation agentbased model is implemented on Java programming language and is used for conducting experiments via Monte- Carlo method. The goal of the experiments is to compare how centralized and decentralized organizational structures perform during sudden and smoldering crises. By centralized organizations we assume structures with high number of hierarchy levels and low number of direct reports of every manager, while decentralized organizations mean structures with low number of hierarchy levels and high number of direct reports of every manager. Sudden crises are distinguished by short early stage and low number of warning signals, while smoldering crises are defined as crises with long lasting early stage and high number of warning signals not necessary containing important information. Efficiency of the organizational performance during early stage of a crisis is measured by two parameters: percentage of early warnings which have been acted upon in order to prepare organization for the crisis, and time spent by top-manager on working with early warnings. As a result, we show that during early stage of smoldering crises centralized organizations process signals more efficiently than decentralized organizations, while decentralized organizations handle early warning signals more efficiently during early stage of sudden crises. However, occupation of top-managers during sudden crises is higher in decentralized organizations and it is higher in centralized organizations during smoldering crises. Thus, neither of the two classes of organizational structures is more efficient by the two parameters simultaneously. Finally, we conduct sensitivity analysis to verify the obtained results.

The well-known “Sea battle” game is in the focus of the current job. The main goal of the article is to provide modified version of “Sea battle” game and to find optimal players’ strategies in the new rules. Changes were applied to attacking strategies (new option to attack hitting four cells in one shot was added) as well as to the size of the field (sizes of 10 × 10, 20 × 20, 30 × 30 were used) and to the rules of disposal algorithms during the game (new possibility to move the ship off the attacking zone). The game was solved with the use of game theory capabilities: payoff matrices were found for each version of altered rules, for which optimal pure and mixed strategies were discovered. For solving payoff matrices iterative method was used. The simulation was in applying five attacking algorithms and six disposal ones with parameters variation due to the game of players with each other. Attacking algorithms were varied in 100 sets of parameters, disposal algorithms — in 150 sets. Major result is that using such algorithms the modified “Sea battle” game can be solved — that implies the possibility of finding stable pure and mixed strategies of behaviour, which guarantee the sides gaining optimal results in game theory terms. Moreover, influence of modifying the rules of “Sea battle” game is estimated. Comparison with prior authors’ results on this topic was made. Based on matching the payoff matrices with the statistical analysis, completed earlier, it was found out that standard “Sea battle” game could be represented as a special case of game modifications, observed in this article. The job is important not only because of its applications in war area, but in civil areas as well. Use of article’s results could save resources in exploration, provide an advantage in war conflicts, defend devices under devastating impact.

Theory of anomalous diffusion, which describe a vast number of transport processes with power law mean squared displacement, is actively advancing in recent years. Diffusion of liquids in porous media, carrier transport in amorphous semiconductors and molecular transport in viscous environments are widely known examples of anomalous deceleration of transport processes compared to the standard model.

Direct Monte Carlo simulation is a convenient tool for studying such processes. An efficient stochastic simulation algorithm is developed in the present paper. It is based on simple renewal process with interarrival times that have power law asymptotics. Analytical derivations show a deep connection between this class of random process and equations with fractional derivatives. The algorithm is further generalized by coupling it with chemical reaction simulation. It makes stochastic approach especially useful, because the exact form of integrodifferential evolution equations for reaction — subdiffusion systems is still a matter of debates.

Proposed algorithm relies on non-markovian random processes, hence one should carefully account for qualitatively new effects. The main question is how molecules leave the system during chemical reactions. An exact scheme which tracks all possible molecule combinations for every reaction channel is computationally infeasible because of the huge number of such combinations. It necessitates application of some simple heuristic procedures. Choosing one of these heuristics greatly affects obtained results, as illustrated by a series of numerical experiments.

Investigation of influence of non-equilibrium initial states and structural disorder on characteristics of anomalous slow non-equilibrium critical behavior of three-dimensional Ising model is carried out. The unique ageing properties and violations of the equilibrium fluctuation-dissipation theorem are observed for considered pure and disordered systems which were prepared in high-temperature initial state and then quenched in their critical points. The heat-bath algorithm description of ageing properties in non-equilibrium critical behavior of three-dimensional Ising model with spin concentrations p = 1.0, p = 0.8, and 0.6 is realized. On the base of analysis of such two-time quantities as autocorrelation function and dynamical susceptibility were demonstrated the ageing effects and were calculated asymptotic values of universal fluctuation-dissipation ratio in these systems. It was shown that the presence of defects leads to aging gain.