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The article is devoted to studying the application of convex optimization methods to solve the Cauchy problem for the Helmholtz equation, which is ill-posed since the equation belongs to the elliptic type. The Cauchy problem is formulated as an inverse problem and is reduced to a convex optimization problem in a Hilbert space. The functional to be optimized and its gradient are calculated using the solution of boundary value problems, which, in turn, are well-posed and can be approximately solved by standard numerical methods, such as finite-difference schemes and Fourier series expansions. The convergence of the applied fast gradient method and the quality of the solution obtained in this way are experimentally investigated. The experiment shows that the accelerated gradient method — the Similar Triangle Method — converges faster than the non-accelerated method. Theorems on the computational complexity of the resulting algorithms are formulated and proved. It is found that Fourier’s series expansions are better than finite-difference schemes in terms of the speed of calculations and improve the quality of the solution obtained. An attempt was made to use restarts of the Similar Triangle Method after halving the residual of the functional. In this case, the convergence does not improve, which confirms the absence of strong convexity. The experiments show that the inaccuracy of the calculations is more adequately described by the additive concept of the noise in the first-order oracle. This factor limits the achievable quality of the solution, but the error does not accumulate. According to the results obtained, the use of accelerated gradient optimization methods can be the way to solve inverse problems effectively.

Data hiding in digital images is a promising direction of cybersecurity. Digital steganography methods provide imperceptible transmission of secret data over an open communication channel. The information embedding efficiency depends on the embedding imperceptibility, capacity, and robustness. These quality criteria are mutually inverse, and the improvement of one indicator usually leads to the deterioration of the others. A balance between them can be achieved using metaheuristic optimization. Metaheuristics are a class of optimization algorithms that find an optimal, or close to an optimal solution for a variety of problems, including those that are difficult to formalize, by simulating various natural processes, for example, the evolution of species or the behavior of animals. In this study, we propose an approach to data hiding in the hybrid spatial-frequency domain of digital images based on metaheuristic optimization. Changing a block of image pixels according to some change matrix is considered as an embedding operation. We select the change matrix adaptively for each block using metaheuristic optimization algorithms. In this study, we compare the performance of three metaheuristics such as genetic algorithm, particle swarm optimization, and differential evolution to find the best change matrix. Experimental results showed that the proposed approach provides high imperceptibility of embedding, high capacity, and error-free extraction of embedded information. At the same time, storage of change matrices for each block is not required for further data extraction. This improves user experience and reduces the chance of an attacker discovering the steganographic attachment. Metaheuristics provided an increase in imperceptibility indicator, estimated by the PSNR metric, and the capacity of the previous algorithm for embedding information into the coefficients of the discrete cosine transform using the QIM method [Evsutin, Melman, Meshcheryakov, 2021] by 26.02% and 30.18%, respectively, for the genetic algorithm, 26.01% and 19.39% for particle swarm optimization, 27.30% and 28.73% for differential evolution.

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.

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.

We'd like to present a specific grid infrastructure and web application development and deployment. The purpose of infrastructure and web application is to solve particular geophysical problems that require heavy computational resources. Here we cover technology overview and connector framework internals. The connector framework links problem-specific routines with middleware in a manner that developer of application doesn't have to be aware of any particular grid software. That is, the web application built with this framework acts as an interface between the user 's web browser and Grid's (often very) own middleware.

Our distributed computing system is built around Gridway metascheduler. The metascheduler is connected to TORQUE resource managers of virtual compute nodes that are being run atop of compute cluster utilizing the virtualization technology. Such approach offers several notable features that are unavailable to bare-metal compute clusters.

The first application we've integrated with our framework is seismic anisotropic parameters determination by inversion of SKS and converted phases. We've used probabilistic approach to inverse problem solution based on a posteriory probability distribution function (APDF) formalism. To get the exact solution of the problem we have to compute the values of multidimensional function. Within our implementation we used brute-force APDF calculation on rectangular grid across parameter space.

The result of computation is stored in relational DBMS and then represented in familiar human-readable form. Application provides several instruments to allow analysis of function's shape by computational results: maximum value distribution, 2D cross-sections of APDF, 2D marginals and a few other tools. During the tests we've run the application against both synthetic and observed data.