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Vol 59, No 5 (2019)

Article

On Implementation of Non-Polynomial Spline Approximation

Belyakova O.V.

Abstract

In this paper, different variants of processing of number flows using Lagrange and Hermite non-polynomial splines are studied. The splines are constructed from approximate relations including a generating vector function with components of different character, including non-polynomial. Approximations by first-order Lagrange and third-order Hermite splines are considered. The efficiency of the approximations constructed is demonstrated on the examples of flows of the values of a function and flows of the values of a function and its derivative. The advantages of the splines considered are the simplicity of construction, maximum smoothness, interpolation and approximation properties, and the accuracy on a priori given functions (on the components of the generating vector function).

Computational Mathematics and Mathematical Physics. 2019;59(5):689-695
pages 689-695 views

Approach to the Numerical Solution of Optimal Control Problems for Loaded Differential Equations with Nonlocal Conditions

Abdullayev V.M., Aida-zade K.R.

Abstract

An approach to the numerical solution of an optimal control problem described by systems of ordinary differential equations with point loading is proposed. The problem involves nonlocal conditions in which the values of the phase state at certain points and its integral values on various preset intervals are contained in coupled form. For the cost functional gradient of the problem, formulas are derived, which are used to solve the problem by applying numerical optimization methods of the first order. Results of numerical experiments are presented.

Computational Mathematics and Mathematical Physics. 2019;59(5):696-707
pages 696-707 views

Input Reconstruction in a Dynamic System from Measurements of a Part of Phase Coordinates

Maksimov V.I.

Abstract

The unknown input disturbance in a system of nonlinear ordinary differential equations is reconstructed from measurements of some of the state coordinates. A solution algorithm is proposed that is robust to information noises and computational errors. The algorithm is constructed using guaranteed control theory.

Computational Mathematics and Mathematical Physics. 2019;59(5):708-717
pages 708-717 views

On a Nonlinear Spectral Problem for a Dielectric Waveguide with Kerr Nonlinearity

Delitsyn A.L., Delitsyn L.L.

Abstract

The frequency dependence of the propagation constants of plane layered dielectric waveguides with the Kerr nonlinearity is considered. An explanation to the possible difference of their behavior from the linear case, related exclusively to a fixed value of an eigenfunction at the boundary of the layer, is given. Explicit formulas for calculating the dispersion curves are obtained. Their behavior for different ways of defining the eigenfunction of the nonlinear problem is analyzed.

Computational Mathematics and Mathematical Physics. 2019;59(5):718-730
pages 718-730 views

Computation of Optimal Disturbances for Delay Systems

Nechepurenko Y.M., Khristichenko M.Y.

Abstract

Novel fast algorithms for computing the maximum amplification of the norm of solution and optimal disturbances for delay systems are proposed and justified. The proposed algorithms are tested on a system of four nonlinear delay differential equations providing a model for the experimental infection caused by the lymphocytic choriomeningitis virus (LCMV). Numerical results are discussed.

Computational Mathematics and Mathematical Physics. 2019;59(5):731-746
pages 731-746 views

Analytical Solutions of the Internal Gravity Wave Equation for a Semi-Infinite Stratified Layer of Variable Buoyancy

Bulatov V.V., Vladimirov Y.V.

Abstract

The problem of constructing asymptotics describing far-field internal gravity waves generated by an oscillating point source of perturbations moving in a vertically semi-infinite stratified layer of variable buoyancy is considered. For a model distribution of the buoyancy frequency, analytical solutions of the main boundary value problem are obtained, which are expressed in terms of Whittaker functions. An integral representation for the Green’s function is obtained, and asymptotic solutions are constructed that describe the amplitude-phase characteristics of internal gravity wave fields in a semi-infinite stratified medium with a variable buoyancy frequency far away from the perturbation source.

Computational Mathematics and Mathematical Physics. 2019;59(5):747-750
pages 747-750 views

A KP1 Scheme for Acceleration of Inner Iterations for the Transport Equation in 3D Geometry Consistent with Nodal Schemes: 2. Splitting Method for Solving the P1 System for Acceleration Corrections

Voloshchenko A.M.

Abstract

An algorithm is proposed for solving the \({{P}_{1}}\) system for acceleration corrections that arises in constructing a \(K{{P}_{1}}\) scheme for accelerating the convergence of inner iterations consistent with the nodal LD (Linear Discontinues) and LB (Linear Best) schemes of third and fourth-order accuracy in space for the transport equation in three-dimensional \(r,\;\vartheta ,\;z\) geometry. The algorithm is based on a cyclic splitting method combined with the through-computation algorithm for solving auxiliary two-point equations system. A modification of the algorithm is considered for three-dimensional \(x,\;y,\;z\) geometry.

Computational Mathematics and Mathematical Physics. 2019;59(5):751-774
pages 751-774 views

Improvement of Multidimensional Randomized Monte Carlo Algorithms with “Splitting”

Mikhailov G.A.

Abstract

Randomized Monte Carlo algorithms are constructed by jointly realizing a baseline probabilistic model of the problem and its random parameters (random medium) in order to study a parametric distribution of linear functionals. This work relies on statistical kernel estimation of the multidimensional distribution density with a “homogeneous” kernel and on a splitting method, according to which a certain number \(n\) of baseline trajectories are modeled for each medium realization. The optimal value of \(n\) is estimated using a criterion for computational complexity formulated in this work. Analytical estimates of the corresponding computational efficiency are obtained with the help of rather complicated calculations.

Computational Mathematics and Mathematical Physics. 2019;59(5):775-781
pages 775-781 views

Fundamental and Generalized Solutions of the Equations of Motion of a Thermoelastic Half-Plane with a Free Boundary

Alexeyeva L.A., Alipova B.N.

Abstract

A coupled thermoelasticity model is used to study the dynamics of a thermoelastic half-plane influenced by nonstationary body forces and heat sources. In space of Laplace transforms with respect to time, Green’s tensor of the boundary value problem for the half-plane with a boundary free of stresses and heat fluxes is constructed. The displacements and the temperature of the medium are determined for arbitrary body forces and heat sources.

Computational Mathematics and Mathematical Physics. 2019;59(5):782-790
pages 782-790 views

Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition

Ismailov M.I., Erkovan S.

Abstract

We consider an inverse problem of determining the time-dependent lowest order coefficient of two-dimensional (2D) heat equation with Ionkin boundary and total energy integral overdetermination condition. The global well-posedness of the problem is obtained by generalized Fourier method combined with the unique solvability of the second kind Volterra integral equation. For obtaining a numerical solution of the inverse problem, we propose the discretization method from a new combination. On the one hand, it is known the traditional method of uniform finite difference combined with numerical integration on a uniform grid (trapezoidal and Simpson’s), on the other hand, we give the method of non-uniform finite difference is combined by a numerical integration on a non-uniform grid (with Gauss–Lobatto nodes). Numerical examples illustrate how to implement the method.

Computational Mathematics and Mathematical Physics. 2019;59(5):791-808
pages 791-808 views

Application of a Hybrid Method to Microflow Simulation

Rovenskaya O.I.

Abstract

The gas flow in a microchannel driven by an arbitrary pressure difference is investigated numerically in a wide range of Knudsen numbers. The simulation is based on a hybrid method that dynamically couples the solutions of the S-model kinetic equation and the Navier–Stokes equations. A complete solution is obtained by matching the half-fluxes of macroscopic variables on the boundary between the domains, thus ensuring the fulfillment of the mass, momentum, and energy conservation laws through the boundary. The efficiency of the hybrid method is improved via MPI parallelization. The accuracy and robustness of the hybrid method as applied to microflows is estimated by comparing the results with the full solution of the kinetic equation.

Computational Mathematics and Mathematical Physics. 2019;59(5):809-814
pages 809-814 views

Spectral Analysis of Model Couette Flows in Application to the Ocean

Skorokhodov S.L., Kuzmina N.P.

Abstract

A method for analysis of the evolution equation of potential vorticity in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum for analyzing the stability of small perturbations of ocean currents with a linear vertical profile of the main flow is developed. The problem depends on several dimensionless parameters and reduces to solving a spectral non-self-adjoint problem containing a small parameter multiplying the highest derivative. A specific feature of this problem is that the spectral parameter enters into both the equation and the boundary conditions. Depending on the types of the boundary conditions, problems I and II, differing in specifying either a perturbations of pressure or its second derivative, are studied. Asymptotic expansions of the eigenfunctions and eigenvalues for small wavenumbers \(k\) are found. It is found that, in problem I, as \(k \to + 0\), there are two finite eigenvalues and a countable set of unlimitedly increasing eigenvalues lying on the line \(\operatorname{Re} (c) = \tfrac{1}{2}\). In problem II, as \(k \to + 0\), there are only unlimitedly increasing eigenvalues. A high-precision analytical-numerical method for calculating the eigenfunctions and eigenvalues of both problems for a wide range of physical parameters and wavenumbers k is developed. It is shown that, with variation in the wavenumber \(k\), some pairs of eigenvalues form double eigenvalues, which, with increasing \(k\), split into simple eigenvalues, symmetric with respect to the line \(\operatorname{Re} (c) = \tfrac{1}{2}\). A large number of simple and double eigenvalues are calculated with high accuracy, and the trajectories of eigenvalues with variation in k, as well as the dependence of the flow instability on the problem parameters, are analyzed.

Computational Mathematics and Mathematical Physics. 2019;59(5):815-835
pages 815-835 views

An Adaptive Proximal Method for Variational Inequalities

Gasnikov A.V., Dvurechensky P.E., Stonyakin F.S., Titov A.A.

Abstract

A novel analog of Nemirovski’s proximal mirror method with an adaptive choice of constants in the minimized prox-mappings at each iteration for variational inequalities with a Lipschitz continuous field is proposed. Estimates of the number of iterations needed to attain the desired quality of solution of the variational inequality are obtained. It is shown how the proposed approach can be extended for the case of Hölder continuous field. A modification of the proposed algorithm for the case of an inexact oracle for the field operator is also considered.

Computational Mathematics and Mathematical Physics. 2019;59(5):836-841
pages 836-841 views

Randomized Algorithms for Some Hard-to-Solve Problems of Clustering a Finite Set of Points in Euclidean Space

Kel’manov A.V., Panasenko A.V., Khandeev V.I.

Abstract

Two strongly NP-hard problems of clustering a finite set of points in Euclidean space are considered. In the first problem, given an input set, we need to find a cluster (i.e., a subset) of given size that minimizes the sum of the squared distances between the elements of this cluster and its centroid (geometric center). Every point outside this cluster is considered a singleton cluster. In the second problem, we need to partition a finite set into two clusters minimizing the sum, over both clusters, of weighted intracluster sums of the squared distances between the elements of the clusters and their centers. The center of one of the clusters is unknown and is determined as its centroid, while the center of the other cluster is set at some point of space (without loss of generality, at the origin). The weighting factors for both intracluster sums are the given cluster sizes. Parameterized randomized algorithms are presented for both problems. For given upper bounds on the relative error and the failure probability, the parameter value is defined for which both algorithms find approximation solutions in polynomial time. This running time is linear in the space dimension and the size of the input set. The conditions are found under which the algorithms are asymptotically exact and their time complexity is linear in the space dimension and quadratic in the input set size.

Computational Mathematics and Mathematical Physics. 2019;59(5):842-850
pages 842-850 views

A Fast Algorithm for Solving a Simple Search Problem

Malozemov V.N., Tamasyan G.S.

Abstract

A fast algorithm for solving the Danskin problem is proposed. The dependence of its solution on parameters is analyzed.

Computational Mathematics and Mathematical Physics. 2019;59(5):851-856
pages 851-856 views