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The probabilistic topic model of a text document collection finds two matrices: a matrix of conditional probabilities of topics in documents and a matrix of conditional probabilities of words in topics. Each document is represented by a multiset of words also called the “bag of words”, thus assuming that the order of words is not important for revealing the latent topics of the document. Under this assumption, the problem is reduced to a low-rank non-negative matrix factorization governed by likelihood maximization. In general, this problem is ill-posed having an infinite set of solutions. In order to regularize the solution, a weighted sum of optimization criteria is added to the log-likelihood. When modeling large text collections, storing the first matrix seems to be impractical, since its size is proportional to the number of documents in the collection. At the same time, the topical vector representation (embedding) of documents is necessary for solving many text analysis tasks, such as information retrieval, clustering, classification, and summarization of texts. In practice, the topical embedding is calculated for a document “on-the-fly”, which may require dozens of iterations over all the words of the document. In this paper, we propose a way to calculate a topical embedding quickly, by one pass over document words. For this, an additional constraint is introduced into the model in the form of an equation, which calculates the first matrix from the second one in linear time. Although formally this constraint is not an optimization criterion, in fact it plays the role of a regularizer and can be used in combination with other regularizers within the additive regularization framework ARTM. Experiments on three text collections have shown that the proposed method improves the model in terms of sparseness, difference, logLift and coherence measures of topic quality. The open source libraries BigARTM and TopicNet were used for the experiments.

Most biomechanical tasks of interest to clinicians can be solved only using personalized mathematical models. Such models allow to formalize and relate key pathophysiological processes, basing on clinically available data evaluate non-measurable parameters that are important for the diagnosis of diseases, predict the result of a therapeutic or surgical intervention. The use of models in clinical practice imposes additional restrictions: clinicians require model validation on clinical cases, the speed and automation of the entire calculated technological chain, from processing input data to obtaining a result. Limitations on the simulation time, determined by the time of making a medical decision (of the order of several minutes), imply the use of reduction methods that correctly describe the processes under study within the framework of reduced models or machine learning tools.

Personalization of models requires patient-oriented parameters, personalized geometry of a computational domain and generation of a computational mesh. Model parameters are estimated by direct measurements, or methods of solving inverse problems, or methods of machine learning. The requirement of personalization imposes severe restrictions on the number of fitted parameters that can be measured under standard clinical conditions. In addition to parameters, the model operates with boundary conditions that must take into account the patient’s characteristics. Methods for setting personalized boundary conditions significantly depend on the clinical setting of the problem and clinical data. Building a personalized computational domain through segmentation of medical images and generation of the computational grid, as a rule, takes a lot of time and effort due to manual or semi-automatic operations. Development of automated methods for setting personalized boundary conditions and segmentation of medical images with the subsequent construction of a computational grid is the key to the widespread use of mathematical modeling in clinical practice.

The aim of this work is to review our solutions for personalization of mathematical models within the framework of three tasks of clinical cardiology: virtual assessment of hemodynamic significance of coronary artery stenosis, calculation of global blood flow after hemodynamic correction of complex heart defects, calculating characteristics of coaptation of reconstructed aortic valve.

Laminar-turbulent transition is the subject of an active research related to improvement of economic efficiency of air vehicles, because in the turbulent boundary layer drag increases, which leads to higher fuel consumption. One of the directions of such research is the search for efficient methods, that can be used to find the position of the transition in space. Using this information about laminar-turbulent transition location when designing an aircraft, engineers can predict its performance and profitability at the initial stages of the project. Traditionally, $e^N$ method is applied to find the coordinates of a laminar-turbulent transition. It is a well known approach in industry. However, despite its widespread use, this method has a number of significant drawbacks, since it relies on parallel flow assumption, which limits the scenarios for its application, and also requires computationally expensive calculations in a wide range of frequencies and wave numbers. Alternatively, flow analysis can be done by using Dynamic Mode Decomposition, which allows one to analyze flow disturbances using flow data directly. Since Dynamic Mode Decomposition is a dimensionality reduction method, the number of computations can be dramatically reduced. Furthermore, usage of Dynamic Mode Decomposition expands the applicability of the whole method, due to the absence of assumptions about the parallel flow in its derivation.

The presented study proposes an approach to finding the location of a laminar-turbulent transition using the Dynamic Mode Decomposition method. The essence of this approach is to divide the boundary layer region into sets of subregions, for each of which the transition point is independently calculated, using Dynamic Mode Decomposition for flow analysis, after which the results are averaged to produce the final result. This approach is validated by laminar-turbulent transition predictions of subsonic and supersonic flows over a 2D flat plate with zero pressure gradient. The results demonstrate the fundamental applicability and high accuracy of the described method in a wide range of conditions. The study focuses on comparison with the $e^N$ method and proves the advantages of the proposed approach. It is shown that usage of Dynamic Mode Decomposition leads to significantly faster execution due to less intensive computations, while the accuracy is comparable to the such of the solution obtained with the $e^N$ method. This indicates the prospects for using the described approach in a real world applications.

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.

Many properties of ordinary differential equations systems solutions are determined by the properties of the equations in variations. An ODE system, which includes both the original nonlinear system and the equations in variations, will be called an extended system further. When studying the properties of the Cauchy problem for the systems of ordinary differential equations, the transition to extended systems allows one to study many subtle properties of solutions. For example, the transition to the extended system allows one to increase the order of approximation for numerical methods, gives the approaches to constructing a sensitivity function without using numerical differentiation procedures, allows to use methods of increased convergence order for the inverse problem solution. Authors used the Broyden method belonging to the class of quasi-Newtonian methods. The Rosenbroke method with complex coefficients was used to solve the stiff systems of the ordinary differential equations. In our case, it is equivalent to the second order approximation method for the extended system.

As an example of the proposed approach, several related mathematical models of the blood coagulation process were considered. Based on the analysis of the numerical calculations results, the conclusion was drawn that it is necessary to include a description of the factor XI positive feedback loop in the model equations system. Estimates of some reaction constants based on the numerical inverse problem solution were given.

Effect of factor V release on platelet activation was considered. The modification of the mathematical model allowed to achieve quantitative correspondence in the dynamics of the thrombin production with experimental data for an artificial system. Based on the sensitivity analysis, the hypothesis tested that there is no influence of the lipid membrane composition (the number of sites for various factors of the clotting system, except for thrombin sites) on the dynamics of the process.

The paper is devoted to the study of relationships between discrete and continuous models financial processes and their probabilistic characteristics. First, a connection is established between the price processes of stocks, hedging portfolio and options in the models conditioned by binomial perturbations and their limit perturbations of the Brownian motion type. Secondly, analogues in the coefficients of stochastic equations with various random processes, continuous and jumpwise, and in the coefficients corresponding deterministic equations for their probabilistic characteristics. Statement of the results on the connections and finding analogies, obtained in this paper, led to the need for an adequate presentation of preliminary information and results from financial mathematics, as well as descriptions of related objects of stochastic analysis. In this paper, partially new and known results are presented in an accessible form for those who are not specialists in financial mathematics and stochastic analysis, and for whom these results are important from the point of view of applications. Specifically, the following sections are presented.

• In one- and n-period binomial models, it is proposed a unified approach to determining on the probability space a risk-neutral measure with which the discounted option price becomes a martingale. The resulting martingale formula for the option price is suitable for numerical simulation. In the following sections, the risk-neutral measures approach is applied to study financial processes in continuous-time models.

• In continuous time, models of the price of shares, hedging portfolios and options are considered in the form of stochastic equations with the Ito integral over Brownian motion and over a compensated Poisson process. The study of the properties of these processes in this section is based on one of the central objects of stochastic analysis — the Ito formula. Special attention is given to the methods of its application.

• The famous Black – Scholes formula is presented, which gives a solution to the partial differential equation for the function $v(t, x)$, which, when $x = S (t)$ is substituted, where $S(t)$ is the stock price at the moment time $t$, gives the price of the option in the model with continuous perturbation by Brownian motion.

• The analogue of the Black – Scholes formula for the case of the model with a jump-like perturbation by the Poisson process is suggested. The derivation of this formula is based on the technique of risk-neutral measures and the independence lemma.

The article describes a model of a human in the informational economy and demonstrates the multicriteria optimizational approach to the metric analysis of model-generated data. The traditional approach using the identification and study involves the model’s identification by time series and its further prediction. However, this is not possible when some variables are not explicitly observed and only some typical borders or population features are known, which is often the case in the social sciences, making some models pure theoretical. To avoid this problem, we propose a method of metric data analysis (MMDA) for identification and study of such models, based on the construction and analysis of the Kolmogorov – Shannon metric nets of the general population in a multidimensional space of social characteristics. Using this method, the coefficients of the model are identified and the features of its phase trajectories are studied. In this paper, we are describing human according to his role in information processing, considering his awareness and cognitive abilities. We construct two lifetime indices of human capital: creative individual (generalizing cognitive abilities) and productive (generalizing the amount of information mastered by a person) and formulate the problem of their multi-criteria (two-criteria) optimization taking into account life expectancy. This approach allows us to identify and economically justify the new requirements for the education system and the information environment of human existence. It is shown that the Pareto-frontier exists in the optimization problem, and its type depends on the mortality rates: at high life expectancy there is one dominant solution, while for lower life expectancy there are different types of Paretofrontier. In particular, the Pareto-principle applies to Russia: a significant increase in the creative human capital of an individual (summarizing his cognitive abilities) is possible due to a small decrease in the creative human capital (summarizing awareness). It is shown that the increase in life expectancy makes competence approach (focused on the development of cognitive abilities) being optimal, while for low life expectancy the knowledge approach is preferable.

The article discusses the problem of the influence of the research goals on the structure of the multivariate model of regression analysis (in particular, on the implementation of the procedure for reducing the dimension of the model). It is shown how bringing the specification of the multiple regression model in line with the research objectives affects the choice of modeling methods. Two schemes for constructing a model are compared: the first does not allow taking into account the typology of primary predictors and the nature of their influence on the performance characteristics, the second scheme implies a stage of preliminary division of the initial predictors into groups, in accordance with the objectives of the study. Using the example of solving the problem of analyzing the causes of burnout of creative workers, the importance of the stage of qualitative analysis and systematization of a priori selected factors is shown, which is implemented not by computing means, but by attracting the knowledge and experience of specialists in the studied subject area. The presented example of the implementation of the approach to determining the specification of the regression model combines formalized mathematical and statistical procedures and the preceding stage of the classification of primary factors. The presence of this stage makes it possible to explain the scheme of managing (corrective) actions (softening the leadership style and increasing approval lead to a decrease in the manifestations of anxiety and stress, which, in turn, reduces the severity of the emotional exhaustion of the team members). Preclassification also allows avoiding the combination in one main component of controlled and uncontrolled, regulatory and controlled feature factors, which could worsen the interpretability of the synthesized predictors. On the example of a specific problem, it is shown that the selection of factors-regressors is a process that requires an individual solution. In the case under consideration, the following were consistently used: systematization of features, correlation analysis, principal component analysis, regression analysis. The first three methods made it possible to significantly reduce the dimension of the problem, which did not affect the achievement of the goal for which this task was posed: significant measures of controlling influence on the team were shown. allowing to reduce the degree of emotional burnout of its participants.

The paper presents the results of applying a scheme of very high accuracy and resolution to obtain numerical solutions of the Navier – Stokes equations of a compressible gas describing the occurrence and development of instability of a two-dimensional laminar boundary layer on a flat plate. The peculiarity of the conducted studies is the absence of commonly used artificial exciters of instability in the implementation of direct numerical modeling. The multioperator scheme used made it possible to observe the subtle effects of the birth of unstable modes and the complex nature of their development caused presumably by its small approximation errors. A brief description of the scheme design and its main properties is given. The formulation of the problem and the method of obtaining initial data are described, which makes it possible to observe the established non-stationary regime fairly quickly. A technique is given that allows detecting flow fluctuations with amplitudes many orders of magnitude smaller than its average values. A time-dependent picture of the appearance of packets of Tollmien – Schlichting waves with varying intensity in the vicinity of the leading edge of the plate and their downstream propagation is presented. The presented amplitude spectra with expanding peak values in the downstream regions indicate the excitation of new unstable modes other than those occurring in the vicinity of the leading edge. The analysis of the evolution of instability waves in time and space showed agreement with the main conclusions of the linear theory. The numerical solutions obtained seem to describe for the first time the complete scenario of the possible development of Tollmien – Schlichting instability, which often plays an essential role at the initial stage of the laminar-turbulent transition. They open up the possibilities of full-scale numerical modeling of this process, which is extremely important for practice, with a similar study of the spatial boundary layer.

Technological development determines the emergence of highly detailed data in time and space, which expands the possibilities of analysis, allowing us to consider consumer decisions and the competitive behavior of enterprises in all their diversity, taking into account the context of the territory and the characteristics of time periods. Despite the promise of such studies, they are currently limited in the scientific literature. This is due to the range of problems, the solution of which is considered in this paper. The article draws attention to the complexity of the analysis of depersonalized high-frequency data and the possibility of modeling consumption changes in time and space based on them. The features of the new type of data are considered on the example of real depersonalized data received from the fiscal data operator “First OFD” (JSC “Energy Systems and Communications”). It is shown that along with the spectrum of problems inherent in high-frequency data, there are disadvantages associated with the process of generating data on the side of the sellers, which requires a wider use of data mining tools. A series of statistical tests were carried out on the data under consideration, including a Unit-Root Test, test for unobserved individual effects, test for serial correlation and for cross-sectional dependence in panels, etc. The presence of spatial autocorrelation of the data was tested using modified tests of Lagrange multipliers. The tests carried out showed the presence of a consistent correlation and spatial dependence of the data, which determine the expediency of applying the methods of panel and spatial analysis in relation to high-frequency data accumulated by fiscal operators. The constructed models made it possible to substantiate the spatial relationship of sales growth and its dependence on the day of the week. The limitation for increasing the predictive ability of the constructed models and their subsequent complication, due to the inclusion of explanatory factors, was the lack of open access statistics grouped in the required detail in time and space, which determines the relevance of the formation of high-frequency geographically structured data bases.