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The article is devoted to first-order adaptive methods for optimization problems with relatively strongly convex functionals. The concept of relatively strong convexity significantly extends the classical concept of convexity by replacing the Euclidean norm in the definition by the distance in a more general sense (more precisely, by Bregman’s divergence). An important feature of the considered classes of problems is the reduced requirements concerting the level of smoothness of objective functionals. More precisely, we consider relatively smooth and relatively Lipschitz-continuous objective functionals, which allows us to apply the proposed techniques for solving many applied problems, such as the intersection of the ellipsoids problem (IEP), the Support Vector Machine (SVM) for a binary classification problem, etc. If the objective functional is convex, the condition of relatively strong convexity can be satisfied using the problem regularization. In this work, we propose adaptive gradient-type methods for optimization problems with relatively strongly convex and relatively Lipschitzcontinuous functionals for the first time. Further, we propose universal methods for relatively strongly convex optimization problems. This technique is based on introducing an artificial inaccuracy into the optimization model, so the proposed methods can be applied both to the case of relatively smooth and relatively Lipschitz-continuous functionals. Additionally, we demonstrate the optimality of the proposed universal gradient-type methods up to the multiplication by a constant for both classes of relatively strongly convex problems. Also, we show how to apply the technique of restarts of the mirror descent algorithm to solve relatively Lipschitz-continuous optimization problems. Moreover, we prove the optimal estimate of the rate of convergence of such a technique. Also, we present the results of numerical experiments to compare the performance of the proposed methods.

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 proposes a set of fairly simple analysis algorithms that can be used to analyze a wide range of protein-protein interactions. In this work, we jointly use the methods of Brownian and molecular dynamics to describe the process of formation of a complex of plastocyanin and cytochrome f proteins in higher plants. In the diffusion-collision complex, two clusters of structures were revealed, the transition between which is possible with the preservation of the position of the center of mass of the molecules and is accompanied only by a rotation of plastocyanin by 134 degrees. The first and second clusters of structures of collisional complexes differ in that in the first cluster with a positively charged region near the small domain of cytochrome f, only the “lower” plastocyanin region contacts, while in the second cluster, both negatively charged regions. The “upper” negatively charged region of plastocyanin in the first cluster is in contact with the amino acid residue of lysine K122. When the final complex is formed, the plastocyanin molecule rotates by 69 degrees around an axis passing through both areas of electrostatic contact. With this rotation, water is displaced from the regions located near the cofactors of the molecules and formed by hydrophobic amino acid residues. This leads to the appearance of hydrophobic contacts, a decrease in the distance between the cofactors to a distance of less than 1.5 nm, and further stabilization of the complex in a position suitable for electron transfer. Characteristics such as contact matrices, rotation axes during the transition between states, and graphs of changes in the number of contacts during the modeling process make it possible to determine the key amino acid residues involved in the formation of the complex and to reveal the physicochemical mechanisms underlying this process.

Despite the widespread use of cancer chemotherapy drugs, the molecular mechanisms of action of many of them remain unclear. Some of these drugs, such as taxol, are known to affect the dynamics of microtubule assembly and stop the process of cell division in prophase-prometaphase. Recently, new spatial structures of microtubules and individual tubulin oligomers have emerged associated with various regulatory proteins and cancer chemotherapy drugs. However, knowledge of the spatial structure in itself does not provide information about the mechanism of action of drugs.

In this work, we applied the molecular dynamics method to study the behavior of taxol-bound tubulin oligomers and used our previously developed method for analyzing the conformation of tubulin protofilaments, based on the calculation of modified Euler angles. Recent structures of microtubule fragments have demonstrated that tubulin protofilaments bend not in the radial direction, as many researchers assume, but at an angle of approximately 45◦ from the radial direction. However, in the presence of taxol, the bending direction shifts closer to the radial direction. There was no significant difference between the mean bending and torsion angles of the studied tubulin structures when bound to the various natural regulatory ligands, guanosine triphosphate and guanosine diphosphate. The intra-dimer bending angle was found to be greater than the interdimer bending angle in all analyzed trajectories. This indicates that the bulk of the deformation energy is stored within the dimeric tubulin subunits and not between them. Analysis of the structures of the latest generation of tubulins indicated that the presence of taxol in the tubulin beta subunit pocket allosterically reduces the torsional rigidity of the tubulin oligomer, which could explain the underlying mechanism of taxol’s effect on microtubule dynamics. Indeed, a decrease in torsional rigidity makes it possible to maintain lateral connections between protofilaments, and therefore should lead to the stabilization of microtubules, which is what is observed in experiments. The results of the work shed light on the phenomenon of dynamic instability of microtubules and allow to come closer to understanding the molecular mechanisms of cell division.

Images of surface topography of ultrathin magnetic films have been used for Monte Carlo simulations in the framework of the ferromagnetic Ising model to study the hysteresis and thermal properties of nanomaterials. For high performance calculations was used super-scalable parallel algorithm for the finding of the equilibrium configuration. The changing of a distribution of spins on the surface during the reversal of the magnetization and the dynamics of nanodomain structure of thin magnetic films under the influence of changing external magnetic field was investigated.

The article examines the impact of non-market actions of participants in oligopolistic markets on the market structure. The following actions of one of the market participants aimed at increasing its market share are analyzed: 1) price manipulation; 2) blocking investments of stronger oligopolists; 3) destruction of produced products and capacities of competitors. Linear dynamic games with a quadratic criterion are used to model the strategies of oligopolists. The expediency of their use is due to the possibility of both an adequate description of the evolution of markets and the implementation of two mutually complementary approaches to determining the strategies of oligopolists: 1) based on the representation of models in the state space and the solution of generalized Riccati equations; 2) based on the application of operational calculus methods (in the frequency domain) which owns the visibility necessary for economic analysis.

The article shows the equivalence of approaches to solving the problem with maximin criteria of oligopolists in the state space and in the frequency domain. The results of calculations are considered in relation to a duopoly, with indicators close to one of the duopolies in the microelectronic industry of the world. The second duopolist is less effective from the standpoint of costs, though more mobile. Its goal is to increase its market share by implementing the non-market methods listed above.

Calculations carried out with help of the game model, made it possible to construct dependencies that characterize the relationship between the relative increase in production volumes over a 25-year period of weak and strong duopolists under price manipulation. Constructed dependencies show that an increase in the price for the accepted linear demand function leads to a very small increase in the production of a strong duopolist, but, simultaneously, to a significant increase in this indicator for a weak one.

Calculations carried out with use of the other variants of the model, show that blocking investments, as well as destroying the products of a strong duopolist, leads to more significant increase in the production of marketable products for a weak duopolist than to a decrease in this indicator for a strong one.

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 method for mapping of intermediate representations (IR) set of C, C++ programs to vector embedding space is considered to create an empirical estimation framework for static performance prediction using LLVM compiler infrastructure. The usage of embeddings makes programs easier to compare due to avoiding Control Flow Graphs (CFG) and Data Flow Graphs (DFG) direct comparison. This method is based on transformation series of the initial IR such as: instrumentation — injection of artificial instructions in an instrumentation compiler’s pass depending on load offset delta in the current instruction compared to the previous one, mapping of instrumented IR into multidimensional vector with IR2Vec and dimension reduction with t-SNE (t-distributed stochastic neighbor embedding) method. The D1 cache miss ratio measured with perf stat tool is considered as performance metric. A heuristic criterion of programs having more or less cache miss ratio is given. This criterion is based on embeddings of programs in 2D-space. The instrumentation compiler’s pass developed in this work is described: how it generates and injects artificial instructions into IR within the used memory model. The software pipeline that implements the performance estimation based on LLVM compiler infrastructure is given. Computational experiments are performed on synthetic tests which are the sets of programs with the same CFGs but with different sequences of offsets used when accessing the one-dimensional array of a given size. The correlation coefficient between performance metric and distance to the worst program’s embedding is measured and proved to be negative regardless of t-SNE initialization. This fact proves the heuristic criterion to be true. The process of such synthetic tests generation is also considered. Moreover, the variety of performance metric in programs set in such a test is proposed as a metric to be improved with exploration of more tests generators.

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.