The paper considers two weakly singular Fredholm boundary integral equations of the first kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on the spectra, where they are ill posed. For the first equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the transmission problem. The second equation in this case always has infinitely many solutions, with only one of them giving a solution of the transmission problem. The interpolation method for finding approximate solutions of the integral equations and the transmission problem in question is discussed.

A direct method (self-regularization method) for the numerical solution of a weakly singular integral equation of the first kind on a closed surface is considered. This equation is an integral formulation of the internal and external three-dimensional Dirichlet problems for the Laplace equation if their solutions are sought in the form of a single-layer potential. It is approximated by a system of linear algebraic equations, which is solved numerically. In this case, a new method of averaging the kernel of the integral operator is used. It preserves the conditional correctness of the discretized problem and significantly increases the rate of convergence of its solution to the exact solution of the integral equation.

Fredholm boundary integral equations of the first kind with a single unknown function are considered. Each equation is conditionally equivalent to a scalar diffraction (transmission) problem on a three-dimensional homogeneous inclusion and is solved numerically. A modified numerical method for solving the diffraction problem on the spectrum of an integral operator is proposed and tested in the case where the conditions for the correct solvability of the integral equation and its equivalence to the original problem are violated.

The pipe wall thickness was estimated based on three-dimensional images of the pipe recovered from several X-ray projections, which were made in a limited angle of view. Since the effects of scattered radiation and beam hardening are up to 50 % of the main radiation, ignoring them leads to blur of the image and inaccuracy in determining dimensions. To restore pipe images from projections, a volume and/or shell representation of the pipe is used, as well as iterative Bayesian methods. Using these methods, the error in estimating the pipe wall thickness from the projection data can be equal to or less than 300 μm. It has been shown that standard X-ray projections on the film or imaging plates used to obtain data can be used to restore pipe wall thickness profiles in factory conditions.

A three-dimensional scalar stationary scattering problem is considered. It is formulated in the form of a weakly singular Fredholm boundary integral equation of the first kind with a single unknown function. The equation is approximated by a system of linear algebraic equations, which is then solved numerically by an iterative method. The mosaic-skeleton method is used at the stage of the approximate solution of this system in order to reduce the computational complexity of the approach.

Рассматриваются пространственные задачи Дирихле для уравнения Гельмгольца в обобщенных постановках. При помощи потенциалов простого слоя они сводятся к граничным интегральным уравнениям Фредгольма I рода. Для дискретизации этих уравнений используется специальный метод осреднения интегральных операторов со слабыми особенностями в ядрах. В результате интегральные уравнения аппроксимируются системами линейных алгебраических уравнений с легко вычисляемыми коэффициентами, которые затем решаются численно обобщенным методом минимальных невязок.

Описывается метод численного решения трехмерной стационарной задачи дифракции акустических волн на однородном включении. Исходная задача сводится к граничным слабо сингулярным интегральным уравнениям Фредгольма первого рода с одной неизвестной функцией, имеющим дискретный спектр. Эти уравнения решаются численно. Для поиска решения исходной задачи на спектре предложен метод интерполяции решения. Приводятся результаты численных экспериментов.

Исследуются вопросы организации многопользовательской работы гибридных вычислительных систем. На примере кластера Центра коллективного пользования Центр данных ДВО РАН, построенного на архитектуре OpenPOWER, рассмотрены особенности функционирования систем подобного класса и предложены решения для организации их работы. С использованием механизма виртуальных узлов проведена адаптация системы диспетчеризации заданий PBS Professional, позволяющая организовать эффективное распределение аппаратных ресурсов кластера между пользовательскими задачами. Реализованное программное окружение кластера с системой комплексного планирования заданий рассчитано на работу с широким перечнем компьютерных приложений, включая программы, построенные с использованием различных технологий параллельного программирования. Для эффективного исполнения в данной среде решений на основе машинного обучения, глубокого обучения и искусственного интеллекта применены технологии виртуализации. С использованием возможностей среды контейнеризации Singularity сформирован специализированный стек программного обеспечения и реализован особый режим его работы в формате единой вычислительной цифровой платформы. Purpose. Improving the technology of machine learning, deep learning and artificial intelligence plays an important role in acquiring new knowledge, technological modernization and the digital economy development.An important factor of the development in these areas is the availability of an appropriate highperformance computing infrastructure capable of providing the processing of large amounts of data. The creation of coprocessorbased hybrid computing systems, as well as new parallel programming technologies and application development tools allows partial solving this problem. However, many issues of organizing the effective multiuser operation of this class of systems require a separate study. The current paper addresses research in this area. Methodology. Using the OpenPOWER architecturebased cluster in the Shared Services Center The Data Center of the Far Eastern Branch of the Russian Academy of Sciences, the features of the functioning of hybrid computing systems are considered and solutions are proposed for organizing their work in a multiuser mode. Based on the virtual nodes concept, an adaptation of the PBS Professional job scheduling system was carried out, which provides an efficient allocation of cluster hardware resources among user tasks. Application virtualization technology was used for effective execution of machine learning and deep learning problems. Findings. The implemented cluster software environment with the integrated task scheduling system is designed to work with a wide range of computer applications, including programs built using parallel programming technologies. The virtualization technologies were used in this environment for effective execution of the software, based on machine learning, deep learning and artificial intelligence. Having the capabilities of the container Singularity, a specialized software stack and its operation mode was implemented for execution machine learning, deep learning and artificial intelligence tasks on a unified computing digital platform. Originality. The features of hybrid computing platforms functioning are considered, and the approach for their effective multiuser work mode is proposed. An effective resource manage model is developed, based on the virtualization technology usage.

Одни из наиболее важных вычислительных задач разработка и адаптация итерационных методов для решения сверхбольших разреженных систем алгебраических уравнений. К таким вычислительным задачам приводит задача итерационной параллельной реконструкции трехмерных изображений промышленных изделий. Важно, что итерационные методы решения вычислительных задач большой размерности реализуются на параллельных структурах намного эффективнее, чем прямые методы их решения. В этой работе описан синхронный параллельный алгоритм, основанный на использовании системы MPI для решения задачи реконструкции трехмерных изображений промышленных изделий. Purpose. Currently, one of the most important tasks is the development and adaptation of iterative methods for solving ultra-large sparse systems of algebraic equations. Such computational problems are caused by the iterative parallel reconstruction of three-dimensional images of industrial products. It is important that iterative methods for solving computational problems of large size are implemented on parallel structures much more efficiently than direct methods for solving them. This paper describes a synchronous parallel algorithm based on the MPI system for solving the problem of reconstruction of three-dimensional images of industrial products. Methodology. It is important that iterative methods for solving computational problems of larger dimensionality on parallel structures are implemented more efficiently than the direct ones. Most algorithms based on direct solving methods have a significant hereditary sequential structure and require a large number of processors interactions which cannot be executed in parallel mode. Iterative methods, for the most part, require a significantly smaller number of interactions of this type and are relatively easily mapped onto parallel computational structures. Equally important, in most cases parallel implementations of classical iterative methods are more effective in terms of computational speed. The parallel execution of the algorithm is based on the distribution of the process in some way between various groups of processors. Depending on the interaction method between local processors, two different types of parallel iterative algorithms execution are distinguished: synchronous and asynchronous. In the former case, it is assumed that the processors complete the calculations and exchange all the necessary results before the start of a new iteration. The main disadvantage of synchronous parallel algorithms is that they require synchronization of iterations. This is a very difficult task, especially with large number of processors. In addition, the overall calculation speed is limited by the speed of the slowest processor. At the same time, faster processors spend most of their time in the waiting mode. But, nevertheless, the implementation of these parallel algorithms can be effectively achieved using the MPI standard. Findings. Synchronous parallel computing algorithms and program codes for threedimensional tomographic reconstruction in a conical beam were developed. Program code debugging and numerical calculations were performed on a hybrid cluster based on the OpenPOWER architecture using the MPI system. For designing a parallel threedimensional tomographic reconstruction, a voxel form of parallelism was used. Originality. Implemented parallel iterative technology of three-dimensional images reconstruction has undeniable advantages over traditional sequential iterative tomographic reconstruction. It allows reducing the time of tomographic reconstruction as many as tens of times, provides the ability to reconstruct products with sizes of 5123 to 10243 voxels with simultaneous storage of the submatrix node of the projection matrix in RAM, which eliminates the need for recalculation of matrix coefficients for each new iteration.

The three-dimensional diffraction problem of stationary acoustic waves on a homogeneous inclusion is considered. It is reduced to the weakly singular boundary Fredholm integral equations of the first kind with one unknown function, each of which is conditionally equivalent to the original problem. By using the original method of averaging the integral operators kernels, these equations are approximated by systems of linear algebraic equations. The resulting systems are solved numerically by the generalized minimal residual method (GMRES). Then the solution of the initial problem is calculated. To find the solution on the spectrum of integral operators, where the condition of equivalence of integral equations to the original problem is violated, the interpolation solution method is proposed. It does not require knowledge of the spectrum and allows us to find the approximated solutions with high accuracy. The proposed algorithms have been implemented in the computing cluster of Computing Center FEB RAS. The results of the calculations that allow us to assess the possibilities of this approach are presented.

Решается задача сшивки фотографических изображений при построении карт морского дна на основе известных положений и ориентаций камеры автономного необитаемого подводного аппарата и расположения характерных (особых) точек на дне. Рассматриваются два алгоритма: простая сшивка, когда участок дна, видимый на каждом изображении, аппроксимируется плоскостью, и сшивка на основе трехмерной модели дна. Purpose. Seabed mosaics are built of tens of thousands of images obtained by the AUV at a small distance from the bottom. The height of the survey is limited by the power of the AUV lighting equipment and the transparency of the water and is often comparable with the differences in the heights of the bottom. This leads to strong distortions caused by parallax, which makes standard stitching methods inapplicable to the construction of photographic maps of the bottom with a complex relief. Methodology. The article proposes to consider the problem of constructing seabed mosaics as a problem of 3D reconstruction. Two approaches to stitching images are described: simple stitching and based on a 3D bottom model. With simple stitching, the relief represented by each image is approximated by a plane that is then projected onto the common plane of the seabed mosaic. When stitching based on a 3D model, the bottom section model is first constructed using the Delaunay triangulation, and then each triangle of the model is projected onto the plane of the map using a graphic accelerator GPU. To mix colors, a simple method of weighting the pixels of images is used, depending on their distance from the edges of the image. Findings. Stitching algorithms proposed in the paper were tested on images obtained by both real AUV and synthetic images. This allowed us to verify efficiency of stitching algorithms for conditions of a highly complex relief. In combination with simple color blending techniques, proposed algorithms have shown their practical efficiency. The stitching algorithm, based on the 3D model demonstrated its robustness to distortions caused by parallax. Originality/value. The main advantage of described approach is an absence of necessity to use computationally consuming, nontrivial color blending techniques while constructing seabed mosaics in the case of complex bottom relief.

Interior and exterior three-dimensional Dirichlet problems for the Helmholtz equation are solved numerically. They are formulated as equivalent boundary Fredholm integral equations of the first kind and are approximated by systems of linear algebraic equations, which are then solved numerically by applying an iteration method. The mosaic-skeleton method is used to speed up the solution procedure.

Рассматриваются пространственные задачи Дирихле для уравнения Гельмгольца в обобщенных постановках. При помощи потенциалов простого слоя они сводятся к граничным интегральным уравнениям Фредгольма I рода. Для дискретизации этих уравнений используется специальный метод осреднения интегральных операторов со слабыми особенностями в ядрах. В результате интегральные уравнения аппроксимируются системами линейных алгебраических уравнений с легко вычисляемыми коэффициентами, которые затем решаются численно обобщенным методом минимальных невязок.

Three-dimensional Dirichlet problems for the Helmholtz equation are considered in generalized formulations. By applying single-layer potentials, they are reduced to Fredholm boundary integral equations of the first kind. The equations are discretized using a special averaging method for integral operators with weak singularities in the kernels. As a result, the integral equations are approximated by systems of linear algebraic equations with easy-to-compute coefficients, which are solved numerically by applying the generalized minimal residual method. A modification of the method is proposed that yields solutions in the spectra of interior Dirichlet problems and integral operators when the integral equations are not equivalent to the original differential problems and are not well-posed. Numerical results are presented for assessing the capabilities of the approach.

Questions of solution of three-dimensional diffraction problems are considered. The problems are formulated as weakly singular integral equations of 1 kind with alone unknown density. Discretization of these equations is realized by means of special smoothing method of fit integral operators. Numerical solutions of systems of linear algebraic equations, approximating integral equations of diffraction problems, were found by using of the variational iterative method and parallel computing technology. We gave the numerical experiment results.

The Far Eastern Branch of RAS (FEB RAS) establishedcorporate network connects all the regional RASresearch centers and provides support for basic networksystems and corporate application services(videoconferencing system, storage and computing).