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The paper considers the problem of transverse impact on a thin preloaded fiber. The commonly accepted theory of transverse impact on a thin fiber is based on the classical works of Rakhmatulin and Smith. The simple relations obtained from the Rakhmatulin – Smith theory are widely used in engineering practice. However, there are numerous evidences that experimental results may differ significantly from estimations based on these relations. A brief overview of the factors that cause the differences is given in this article.

This paper focuses on the shear wave velocity, as it is the only feature that can be directly observed and measured using high-speed cameras or similar methods. The influence of the fiber preload on the wave speed is considered. This factor is important, since it inevitably arises in the experimental results. The reliable fastening and precise positioning of the fiber during the experiments requires its preload. This work shows that the preload significantly affects the shear wave velocity in the impacted fiber.

Numerical calculations were performed for Kevlar 29 and Spectra 1000 yarns. Shear wave velocities are obtained for different levels of initial tension. A direct comparison of numerical results and analytical estimations with experimental data is presented. The speed of the transverse wave in free and preloaded fibers differed by a factor of two for the setup parameters considered. This fact demonstrates that measurements based on high-speed imaging and analysis of the observed shear waves should take into account the preload of the fibers.

This paper proposes a formula for a quick estimation of the shear wave velocity in preloaded fibers. The formula is obtained from the basic relations of the Rakhmatulin – Smith theory under the assumption of a large initial deformation of the fiber. The formula can give significantly better results than the classical approximation, this fact is demonstrated using the data for preloaded Kevlar 29 and Spectra 1000. The paper also shows that direct numerical calculation has better corresponding with the experimental data than any of the considered analytical estimations.

Distributed saddle point problems (SPPs) have numerous applications in optimization, matrix games and machine learning. For example, the training of generated adversarial networks is represented as a min-max optimization problem, and training regularized linear models can be reformulated as an SPP as well. This paper studies distributed nonsmooth SPPs with Lipschitz-continuous objective functions. The objective function is represented as a sum of several components that are distributed between groups of computational nodes. The nodes, or agents, exchange information through some communication network that may be centralized or decentralized. A centralized network has a universal information aggregator (a server, or master node) that directly communicates to each of the agents and therefore can coordinate the optimization process. In a decentralized network, all the nodes are equal, the server node is not present, and each agent only communicates to its immediate neighbors.

We assume that each of the nodes locally holds its objective and can compute its value at given points, i. e. has access to zero-order oracle. Zero-order information is used when the gradient of the function is costly, not possible to compute or when the function is not differentiable. For example, in reinforcement learning one needs to generate a trajectory to evaluate the current policy. This policy evaluation process can be interpreted as the computation of the function value. We propose an approach that uses a smoothing technique, i. e., applies a first-order method to the smoothed version of the initial function. It can be shown that the stochastic gradient of the smoothed function can be viewed as a random two-point gradient approximation of the initial function. Smoothing approaches have been studied for distributed zero-order minimization, and our paper generalizes the smoothing technique on SPPs.

The paper is devoted to one approach to constructing subgradient methods for strongly convex programming problems with several functional constraints. More precisely, the strongly convex minimization problem with several strongly convex (inequality-type) constraints is considered, and first-order optimization methods for this class of problems are proposed. The special feature of the proposed methods is the possibility of using the strong convexity parameters of the violated functional constraints at nonproductive iterations, in theoretical estimates of the quality of the produced solution by the methods. The main task, to solve the considered problem, is to propose a subgradient method with adaptive rules for selecting steps and stopping rule of the method. The key idea of the proposed methods in this paper is to combine two approaches: a scheme with switching on productive and nonproductive steps and recently proposed modifications of mirror descent for convex programming problems, allowing to ignore some of the functional constraints on nonproductive steps of the algorithms. In the paper, it was described a subgradient method with switching by productive and nonproductive steps for strongly convex programming problems in the case where the objective function and functional constraints satisfy the Lipschitz condition. An analog of the proposed subgradient method, a mirror descent scheme for problems with relatively Lipschitz and relatively strongly convex objective functions and constraints is also considered. For the proposed methods, it obtained theoretical estimates of the quality of the solution, they indicate the optimality of these methods from the point of view of lower oracle estimates. In addition, since in many problems, the operation of finding the exact subgradient vector is quite expensive, then for the class of problems under consideration, analogs of the mentioned above methods with the replacement of the usual subgradient of the objective function or functional constraints by the $\delta$-subgradient were investigated. The noted approach can save computational costs of the method by refusing to require the availability of the exact value of the subgradient at the current point. It is shown that the quality estimates of the solution change by $O(\delta)$. The results of numerical experiments illustrating the advantages of the proposed methods in comparison with some previously known ones are also presented.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.

The possibility of numerical 3D simulation of thrombi formation is considered.

The developed up to now detailed mathematical models describing formation of thrombi and clots include a great number of equations. Being implemented in a CFD code, the detailed mathematical models require essential computer resources for simulation of the thrombi growth in a blood flow. A reasonable alternative way is using reduced mathematical models. Two models based on the reduced mathematical model for the thrombin generation are described in the given paper.

The first model describes growth of a thrombus in a great vessel (artery). The artery flows are essentially unsteady. They are characterized by pulse waves. The blood velocity here is high compared to that in the vein tree. The reduced model for the thrombin generation and the thrombus growth in an artery is relatively simple. The processes accompanying the thrombin generation in arteries are well described by the zero-order approximation.

A vein flow is characterized lower velocity value, lower gradients, and lower shear stresses. In order to simulate the thrombin generation in veins, a more complex system of equations has to be solved. The model must allow for all the non-linear terms in the right-hand sides of the equations.

The simulation is carried out in the industrial software FlowVision.

The performed numerical investigations have shown the suitability of the reduced models for simulation of thrombin generation and thrombus growth. The calculations demonstrate formation of the recirculation zone behind a thrombus. The concentration of thrombin and the mass fraction of activated platelets are maximum here. Formation of such a zone causes slow growth of the thrombus downstream. At the upwind part of the thrombus, the concentration of activated platelets is low, and the upstream thrombus growth is negligible.

When the blood flow variation during a hart cycle is taken into account, the thrombus growth proceeds substantially slower compared to the results obtained under the assumption of constant (averaged over a hard cycle) conditions. Thrombin and activated platelets produced during diastole are quickly carried away by the blood flow during systole. Account of non-Newtonian rheology of blood noticeably affects the results.

This paper considers the problem of water consumption in the regions of Russia with low water availability. We provide a review of the existing methods to control quality and quantity of water resources at different scales — from households to worldwide. The paper itself considers regions with low “water availability” parameter which is amount of water per person per year. Special attention is paid to the regions, where this parameter is low because of natural features of the region, not because of high population. In such regions many resources are spend on water processing infrastructure to store water and transport water from other regions. In such regions the main water consumers are industry and agriculture.

We propose dynamic two-level hierarchical model which matches water consumption of a region with its gross regional product. On the top level there is a regional administration (supervisor) and on the lower level there are region enterprises (agents). The supervisor sets fees for water consumption. We study the model with Pontryagin’s maximum principle and provide agents’s optimal control in analytical form. For the supervisor’s control we provide numerical algorithm. The model has six free coefficients, which can be chosen so the model represents a particular region. We use data from Russia Federal State Statistics Service for identification process of a model. For numerical analysis we use trust region reflective algorithms. We provide calculations for a few regions with low water availability. It is shown that it is possible to reduce water consumption of a region more than by 20% while gross regional product drop is less than 10%.

In the first part of the paper we present convergence analysis of AGMsDR method on a new class of functions — in general non-convex with $M$-Lipschitz-continuous gradients that satisfy Polyak – Lojasiewicz condition. Method does not need the value of $\mu^{PL}>0$ in the condition and converges linearly with a scale factor $\left(1 - \frac{\mu^{PL}}{M}\right)$. It was previously proved that method converges as $O\left(\frac1{k^2}\right)$ if a function is convex and has $M$-Lipschitz-continuous gradient and converges linearly with a~scale factor $\left(1 - \sqrt{\frac{\mu^{SC}}{M}}\right)$ if the value of strong convexity parameter $\mu^{SC}>0$ is known. The novelty is that one can save linear convergence if $\frac{\mu^{PL}}{\mu^{SC}}$ is not known, but without square root in the scale factor.

The second part presents modification of AGMsDR method for solving problems that allow alternating minimization (Alternating AGMsDR). The similar results are proved.

As the result, we present adaptive accelerated methods that converge as $O\left(\min\left\lbrace\frac{M}{k^2},\,\left(1-{\frac{\mu^{PL}}{M}}\right)^{(k-1)}\right\rbrace\right)$ on a class of convex functions with $M$-Lipschitz-continuous gradient that satisfy Polyak – Lojasiewicz condition. Algorithms do not need values of $M$ and $\mu^{PL}$. If Polyak – Lojasiewicz condition does not hold, the convergence is $O\left(\frac1{k^2}\right)$, but no tuning needed.

We also consider the adaptive catalyst envelope of non-accelerated gradient methods. The envelope allows acceleration up to $O\left(\frac1{k^2}\right)$. We present numerical comparison of non-accelerated adaptive gradient descent which is accelerated using adaptive catalyst envelope with AGMsDR, Alternating AGMsDR, APDAGD (Adaptive Primal-Dual Accelerated Gradient Descent) and Sinkhorn's algorithm on the problem dual to the optimal transport problem.

Conducted experiments show faster convergence of alternating AGMsDR in comparison with described catalyst approach and AGMsDR, despite the same asymptotic rate $O\left(\frac1{k^2}\right)$. Such behavior can be explained by linear convergence of AGMsDR method and was tested on quadratic functions. Alternating AGMsDR demonstrated better performance in comparison with AGMsDR.

The problem of developing efficient numerical methods for non-convex (including non-smooth) problems is relevant due to their widespread use of such problems in applications. This paper is devoted to subgradient methods for minimizing Lipschitz $\mu$-weakly convex functions, which are not necessarily smooth. It is well known that subgradient methods have low convergence rates in high-dimensional spaces even for convex functions. However, if we consider a subclass of functions that satisfies sharp minimum condition and also use the Polyak step, we can guarantee a linear convergence rate of the subgradient method. In some cases, the values of the function or it’s subgradient may be available to the numerical method with some error. The accuracy of the solution provided by the numerical method depends on the magnitude of this error. In this paper, we investigate the behavior of the subgradient method with a Polyak step when inaccurate information about the objective function value or subgradient is used in iterations. We prove that with a specific choice of starting point, the subgradient method with some analogue of the Polyak step-size converges at a geometric progression rate on a class of $\mu$-weakly convex functions with a sharp minimum, provided that there is additive inaccuracy in the subgradient values. In the case when both the value of the function and the value of its subgradient at the current point are known with error, convergence to some neighborhood of the set of exact solutions is shown and the quality estimates of the output solution by the subgradient method with the corresponding analogue of the Polyak step are obtained. The article also proposes a subgradient method with a clipped step, and an assessment of the quality of the solution obtained by this method for the class of $\mu$-weakly convex functions with a sharp minimum is presented. Numerical experiments were conducted for the problem of low-rank matrix recovery. They showed that the efficiency of the studied algorithms may not depend on the accuracy of localization of the initial approximation within the required region, and the inaccuracy in the values of the function and subgradient may affect the number of iterations required to achieve an acceptable quality of the solution, but has almost no effect on the quality of the solution itself.

The data of episodic field measurements of carbon dioxide balance components (soil respiration — R_soil, ecosystem respiration — R_eco, net ecosystem exchange — NEE) of hayfields under use and abandoned one are interpreted by modelling. The field measurements were carried within five field campaigns in 2018 and 2019 on the drained part of the Dubna Peatland in Taldom District, Moscow Oblast, Russia. The territory is within humid continental climate zone. Peatland drainage was done out for milled peat extraction. After extraction was stopped, the residual peat deposit (1–1.5 m) was ploughed and grassed (Poa pratensis L.) for hay production. The current ground water level (GWL) varies from 0.3–0.5 m below the surface during wet and up to 1.0 m during dry periods. Daily dynamics of CO2 fluxes was measured using dynamic chamber method in 2018 (August) and 2019 (May, June, August) for abandoned ditch spacing only with sanitary mowing once in 5 years and the ditch spacing with annual mowing. NEE and Reco were measured on the sites with original vegetation, and Rsoil — after vegetation removal. To model a seasonal dynamics of NEE, the dependence of its components (Reco, Rsoil, and Gross ecosystematmosphere exchange of carbon dioxide — GEE) from soil and air temperature, GWL, photosynthetically active radiation, underground and aboveground plant biomass were used. The parametrization of the models has been carried out considering the stability of coefficients estimated by the bootstrap method. R2 (α = 0.05) between simulated and measured Reco was 0.44 (p < 0.0003) on abandoned and 0.59 (p < 0.04) on under use hayfield, and GEE was 0.57 (p < 0.0002) and 0.77 (p < 0.00001), respectively. Numerical experiments were carried out to assess the influence of different haymaking regime on NEE. It was found that NEE for the season (May 15 – September 30) did not differ much between the hayfield without mowing (4.5±1.0 tC·ha–1·season–1) and the abandoned one (6.2±1.4). Single mowing during the season leads to increase of NEE up to 6.5±0.9, and double mowing — up to 7.5±1.4 tC·ha–1·season–1. This means increase of carbon losses and CO2 emission into the atmosphere. Carbon loss on hayfield for both single and double mowing scenario was comparable with abandoned hayfield. The value of removed phytomass for single and double mowing was 0.8±0.1 tC·ha^–1·season^–1 and 1.4±0.1 (45% carbon content in dry phytomass) or 3.0 and 4.4 t·ha^–1·season^–1 of hay (17% moisture content). In comparison with the fallow, the removal of biomass of 0.8±0.1 at single and 1.4±0.1 tC·ha^–1·season^–1 double mowing is accompanied by an increase in carbon loss due to CO2 emissions, i.e., the growth of NEE by 0.3±0.1 and 1.3±0.6 tC·ha–1·season–1, respectively. This corresponds to the growth of NEE for each ton of withdrawn phytomass per hectare of 0.4±0.2 tС·ha–1·season–1 at single mowing, and 0.9±0.7 tС·ha^–1·season^–1 at double mowing. Therefore, single mowing is more justified in terms of carbon loss than double mowing. Extensive mowing does not increase CO2 emissions into the atmosphere and allows, in addition, to “replace” part of the carbon loss by agricultural production.

Studies of the crisis socio-demographic situation in the Russian Far East require not only the use of traditional statistical methods, but also a conceptual analysis of possible development scenarios based on the synergy principles. The article is devoted to the analysis and modeling of the number of employed, unemployed and economically inactive population using nonlinear autonomous differential equations. We studied a basic mathematical model that takes into account the principle of pair interactions, which is a special case of the model for the struggle between conditional information of D. S. Chernavsky. The point estimates for the parameters are found using least squares method adapted for this model. The average approximation error was no more than 5.17%. The calculated parameter values correspond to the unstable focus and the oscillations with increasing amplitude of population number in the asymptotic case, which indicates a gradual increase in disparities between the employed, unemployed and economically inactive population and a collapse of their dynamics. We found that in the parametric space, not far from the inertial scenario, there are domains of blow-up and chaotic regimes complicating the ability to effectively manage. The numerical study showed that a change in only one model parameter (e.g. migration) without complex structural socio-economic changes can only delay the collapse of the dynamics in the long term or leads to the emergence of unpredictable chaotic regimes. We found an additional set of the model parameters corresponding to sustainable dynamics (stable focus) which approximates well the time series of the considered population groups. In the mathematical model, the bifurcation parameters are the outflow rate of the able-bodied population, the fertility (“rejuvenation of the population”), as well as the migration inflow rate of the unemployed. We found that the transition to stable regimes is possible with the simultaneous impact on several parameters which requires a comprehensive set of measures to consolidate the population in the Russian Far East and increase the level of income in terms of compensation for infrastructure sparseness. Further economic and sociological research is required to develop specific state policy measures.