Exterior extension problems for strongly elliptic operators: solvability and approximation using fundamental solutions

In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of the Sobolev spaces over a domain with relatively smooth boundaries. We consider the existence of solutions to these problems, the dense solvability and conditional well-posedness of these problems for a wide class of strongly elliptic systems. We also consider the approximation of solutions to these problems by a single layer potential and by a linear combination of "discrete" fundamental solutions in relation to a narrower class of strongly elliptic operators of the second order. The obtained results justify the applicability of the indirect method of boundary integral equations and for numerical solving the exterior extension problems

Investigation of the Influence of Nanodispersed Compositions Obtained by Plasmochemical Synthesis on the Crystallization Processes of Structural Alloys

The state of the problem of stabilizing the structure, improving the quality and properties of structural alloys is studied. To solve the problem, it is proposed to modify melts of low–alloyed alloys with nanodispersed compositions obtained by plasma–chemical synthesis. Process technological parameters are developed. Nanopowders of carbide and carbonitride class SiC and Ti (C, N) with a size of 50 ... 100 nm are obtained. The crystallographic parameters of the nanocompositions and the specific surface are determined, and the dependency curves are plotted. The macro– and microstructure of structural steels and alloys was studied before and after the modification. A significant (in 2 ... 3.5 times) grain refinement and stabilization of the alloy structure as a result of nanopowder modification of titanium carbonitride have been achieved. Thermodynamic calculations of the dimensions of crystalline seeds during the crystallization of steels and alloys are carried out. A complex criterial estimation of the efficiency of nanodispersed compositions in a steel melt is proposed. The features of crystallization and structure formation of modified structural steels are studied. The obtained results are of theoretical and practical importance for production of critical parts from structural steels and high–quality alloys

On uniqueness theorems for the inverse problem of Electrocardiography in the Sobolev spaces

We consider a mathematical model related to reconstruction of cardiac electrical activity from ECG measurements on the body surface. An application of recent developments in solving boundary value problems for elliptic and parabolic equations in Sobolev type spaces allows us to obtain uniqueness theorems for the model. The obtained results can be used as a sound basis for creating numerical methods for non-invasive mapping of the heart.Comment: arXiv admin note: substantial text overlap with arXiv:2106.0412

FPGA Implementation of Convolutional Neural Networks with Fixed-Point Calculations

Neural network-based methods for image processing are becoming widely used in practical applications. Modern neural networks are computationally expensive and require specialized hardware, such as graphics processing units. Since such hardware is not always available in real life applications, there is a compelling need for the design of neural networks for mobile devices. Mobile neural networks typically have reduced number of parameters and require a relatively small number of arithmetic operations. However, they usually still are executed at the software level and use floating-point calculations. The use of mobile networks without further optimization may not provide sufficient performance when high processing speed is required, for example, in real-time video processing (30 frames per second). In this study, we suggest optimizations to speed up computations in order to efficiently use already trained neural networks on a mobile device. Specifically, we propose an approach for speeding up neural networks by moving computation from software to hardware and by using fixed-point calculations instead of floating-point. We propose a number of methods for neural network architecture design to improve the performance with fixed-point calculations. We also show an example of how existing datasets can be modified and adapted for the recognition task in hand. Finally, we present the design and the implementation of a floating-point gate array-based device to solve the practical problem of real-time handwritten digit classification from mobile camera video feed