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Siberian Journal of Numerical Mathematics

2019 year, number 4

1.
Numerical solution to a three-dimensional coefficient inverse problem for the wave equation with integral data in a cylindrical domain

A.B. Bakushinsky1, A.S. Leonov2
1Institute for Systems Analysis, Moscow, Russia
2National Nuclear Research University В«MEPHI», Moscow, Russia
Keywords: трехмерное волновое уравнение, обратная коэффициентная задача, регуляризующий алгоритм, быстрое преобразование Фурье, three-dimensional wave equation, wave field, inverse coefficient problem, regularizing algorithm, fast Fourier transform

Abstract >>
A three-dimensional coefficient inverse problem for the wave equation (with losses) in a cylindrical domain is under consideration. The data for its solution are special time integrals of the wave field measured in a cylindrical layer. We present and substantiate an efficient algorithm for solving such a three-dimensional problem based on the fast Fourier transform. The algorithm proposed makes possible to obtain a solution on grids of 512x512x512 size in a time of about 1.4 hours on a typical PC without parallelizing the calculations. The results of the numerical experiments for solving the corresponding model inverse problems are presented.



2.
A numerical method for predicting hemodynamic effects in vascular prostheses

V.G. Borisov1, Y.N. Zakharov1, Y.I. Shokin2, E.A. Ovcharenko3, K.Y. Klyshnikov3, I.N. Sizova3, A.V. Batranin4, Y.A. Kudryavtseva3, P.S. Onishchenko2,3
1Institute of Computational Technologies SB RAS, Kemerovo, Russia
2Institute of Computational Technologies SB RAS, Novosibirsk, Russia
3Research Institute for Complex Issues of Cardiovascular Diseases under the Siberian Branch of the Russian Academy of Medical Sciences, Kemerovo, Russia
4Tomsk Polytechnic University, Tomsk, Russia
Keywords: компьютерное моделирование, течение крови, биопротезы, пристеночное напряжение сдвига, computer modeling, blood flow, bioprostheses, wall shear stress

Abstract >>
The three-dimensional unsteady-state periodic flow of blood in xenogenic vascular bioprostheses is simulated using computational fluid dynamics methods. The geometry of the computational domain is based on microtomographic scanning of bioprostheses. To set a variable pressure gradient causing a non-stationary flow in the prostheses, personal-specific data of the Doppler-echography of the blood flow of a particular patient are used. A comparative analysis of the velocity fields in the flow areas corresponding to three real samples of bioprostheses with multiple stenoses is carried out. In the zones of stenosis and outside of them, the distribution of the near-wall shear stress, which influences the risk factors for thrombosis in the prostheses, is analyzed. An algorithm for predicting the hemodynamic effects arising in vascular bioprostheses, based on the numerical modeling of a blood flow in them, is proposed.



3.
The Walrasian equilibrium and centralized distributed optimization in terms of modern convex optimization methods on the example of resource allocation problem

E.A. Vorontsova1,2, A.V. Gasnikov3,4,5, A.C. Ivanova3, E.A. Nurminsky1
1Far Eastern Federal University, Vladivostok, Russia
2Universite de Grenoble-Alpes, Saint-Martin-d'Heres, France
3Moscow Institute of Physics and Technology, Dolgoprudny, Russia
4Institute for Information Transmission Problems RAS, Moscow, Russia
5Adyghe State University, Maikop, Russia
Keywords: вальрасов механизм, децентрализация цен, прямо-двойственный метод, субградиентный метод, условие Слейтера, Walrasian equilibrium, decentralized pricing, primal-dual method, subgradient method, Slater condition

Abstract >>
We consider the resource allocation problem and its numerical solution. The following is demonstrated: 1) the Walrasian price-adjustment mechanism for determining the equilibrium; 2) the decentralized role of the prices; 3) Slater's method for price restrictions (dual Lagrange multipliers); 4) a new mechanism for determining equilibrium prices, in which prices are fully controlled not by Center (Government), but by economic agents -- nodes (factories). In the economic literature, only the convergence of the methods considered is proved. In contrast, this paper provides an accurate analysis of the convergence rate of the described procedures for determining the equilibrium. The analysis is based on the primal-dual nature of the algorithms proposed. More precisely, in this paper, we propose the economic interpretation of the following numerical primal-dual methods of the convex optimization: dichotomy and subgradient projection method.



4.
A solution of the degenerate Neumann problem by the finite element method

M.I. Ivanov1, I.A. Kremer1,2, M.V. Urev1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: вырожденная задача Неймана, условия согласования, ортогонализация правой части, конечные элементы, degenerate Neumann problem, matching conditions, orthogonalization of the right-hand side, finite elements

Abstract >>
This paper deals with the solution of the degenerate Neumann problem for the diffusion equation by the finite element method. First, an extended generalized formulation of the Neumann problem in the Sobolev space H1(Ω) is derived and investigated. Then a discrete analogue of this problem is formulated using standard finite element approximations of the space H1(Ω). An iterative method for solving the corresponding SLAE is proposed. Some examples of solving the model problems are used to discuss the numerical peculiarities of the algorithm proposed.



5.
On computation of the Bessel function by summing up the series

E.A. Karatsuba
Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow, Russia
Keywords: функции Бесселя, быстрые алгоритмы, сложность вычисления, метод БВЕ, большой аргумент, эффективное вычисление, Bessel functions, fast algorithms, computational complexity, FEE method, large argument, efficient calculation

Abstract >>
Two algorithms for an effective calculation of the Bessel function are presented: a fast algorithm with an increasing accuracy of computation and a computational algorithm for the case of a large argument of the Bessel function.



6.
Increasing the stability of triangular decomposition of ill-conditioned matrices

V.N. Lutay
Southern Federal University, Rostov-on-Don, Russia
Keywords: плохо обусловленные матрицы, треугольное разложение, повышение устойчивости, отсечение младших разрядов, неполное скалярное произведение, ill-conditioned matrix, triangular decomposition, improving resilience, cutting off the least significant bits of partial scalar product

Abstract >>
An approach to increasing the stability of triangular decomposition of a dense positive definite matrix with a large condition number with the use of the Gauss and the Cholesky methods is considered. It is proposed to introduce additions to standard computational schemes, which consist in the use of an incomplete scalar product of two vectors, which is formed by cutting off the lower digits of the sum of the products of two numbers. Cutting off being performed in the process of factorization leads to an increase in the diagonal elements of triangular matrices to a random number and prevents the appearance of very small numbers during the decomposition according to Gauss and a negative radical expression in the Cholesky method. The number of additional operations required to obtain an accurate solution is estimated. The results of computational experiments are presented.



7.
Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation

A.G. Megrabov1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Novosibirsk State Technical University, Novosibirsk, Russia
Keywords: кинематическая сейсмика, геометрическая оптика, уравнение эйконала, семейство лучей, семейство фронтов, законы сохранения, дифференциальная геометрия, геометрия векторных полей, kinematic seismic, geometric optics, eikonal equation, family of rays, family of wavefronts, conservation laws, differential geometry, geometry of vector fields

Abstract >>
In the previous studies, the author has obtained the conservation laws for the 2D eikonal equation in an inhomogeneous isotropic medium. These laws represent the divergent identities of the form div F =0. The vector field F is expressed in terms of the solution to the eikonal equation (the time field), the refractive index (the equation parameter) and their partial derivatives. Also, there were found equivalent conservation laws (divergent identities) for the families of rays and the families of wavefronts in terms of their geometric characteristics. Thus, the geometric essence (interpretation) of the above-mentioned conservation laws for the 2D eikonal equation was discovered. In this paper, the 3D analogs to the results obtained are presented: differential conservation laws for the 3D eikonal equation and the conservation laws (divergent identities of the form div F =0) for the family of rays and the family of wavefronts, the vector field F is expressed in terms of classical geometric characteristics of the ray curves: their Frenet basis (unit tangent vector, a principal normal and a binormal), the first curvature and the second curvature, or in terms of the classical geometric characteristics of the wavefront surfaces, i. e. their normal, principal directions, principal curvatures, the Gaussian curvature and the mean curvature. All the results have been obtained based on the vector and geometric formulas (differential conservation laws and some formulas) obtained for the families of arbitrary smooth curves, the families of arbitrary smooth surfaces and arbitrary smooth vector fields.



8.
The simulation of nonlinear oscillations in a micro clock generator

S.I. Fadeev1,2, V.V. Kogai1,2
1Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
2Novosibirsk State University, Novosibirsk, Russia
Keywords: математическая модель, микрогенератор, задача Коши, краевая задача, периодические колебания, предельный цикл, устойчивость, фазовая плоскость, продолжение решения по параметру, mathematical model, microgenerator, Cauchy problem, boundary value problem, periodic oscillations, limit cycle, phase plane, continuation of the solution with respect to the parameter

Abstract >>
In this paper, we consider a mathematical model of a new type of a microgenerator, based on generating the oscillations of a mobile electrode in a micro-gap due to electrostatic forces. The principle of operation of the generator is analogous to the well-known theory of the clock-escapement mechanism, with the difference that in the equation of motion the form of the right-hand side corresponds to the electrostatic nature of the impulse action. The numerical analysis shows that the bounded oscillations with an increase in time tend towards a stable limit cycle in the phase plane and, thereby, the emerging oscillations are stable with respect to external perturbations. In studying periodic oscillations, depending on the parameters of a model, we use the solution of the boundary value problem for the equation with a discontinuous right-hand side, transformed to a form allowing the application of the numerical continuation method. In this way, the area in the plane of the model parameters is defined, in which stable limit cycles exist.