卷 58, 编号 11 (2018)

The paper presents an algorithm for calculating a two-dimensional discrete convolution with an exponential kernel by solving a boundary value problem for an equation with a second-order finite-difference operator. To solve the boundary value problem, a one-step iterative process converging with a rate of a geometric progression is developed.

Extension of the mirror descent method developed for convex stochastic optimization problems to constrained convex stochastic optimization problems (subject to functional inequality constraints) is studied. A method that performs an ordinary mirror descent step if the constraints are insignificantly violated and performs a mirror descent step with respect to the violated constraint if this constraint is significantly violated is proposed. If the method parameters are chosen appropriately, a bound on the convergence rate (that is optimal for the given class of problems) is obtained and sharp bounds on the probability of large deviations are proved. For the deterministic case, the primal–dual property of the proposed method is proved. In other words, it is proved that, given the sequence of points (vectors) generated by the method, the solution of the dual method can be reconstructed up to the same accuracy with which the primal problem is solved. The efficiency of the method as applied for problems subject to a huge number of constraints is discussed. Note that the bound on the duality gap obtained in this paper does not include the unknown size of the solution to the dual problem.

The ill-posed problem of minimizing an approximately specified smooth nonconvex functional on a convex closed subset of a Hilbert space is considered. For the class of problems characterized by a feasible set with a nonempty interior and a smooth boundary, regularizing procedures are constructed that ensure an accuracy estimate proportional or close to the error in the input data. The procedures are generated by the classical Tikhonov scheme and a gradient projection technique. A necessary condition for the existence of procedures regularizing the class of optimization problems with a uniform accuracy estimate in the class is established.

A quasi-linear differential algebraic system of partial differential equations with a special structure of the pencil of Jacobian matrices of small index is considered. A nonlinear spline collocation difference scheme of high approximation order is constructed for the system by approximating the required solution by a spline of arbitrary in each independent variable. It is proved by the simple iteration method that the nonlinear difference scheme has a solution that is uniformly bounded in the grid space. Numerical results are demonstrated using a test example.

The motion of a flagellated microorganism in a free space based on specifying the space-time shape of its centreline is studied. To solve the governing Stokes equations subject to non-slip boundary conditions on the microorganism body, a computational algorithm based on the finite element method is proposed. Results of computations on meshes of various density, domains of various sizes, and the solution of the benchmark problem of flow around the Stokes sphere used for verification are presented. Using the Lighthill–Gueron–Liron theory, a semi-analytical solution of the same problem of motion of a flagellated microorganism in which the corresponding coefficients of viscous drag are found by additional test computations is obtained. It is shown that the theory and the results of direct numerical simulation are in good agreement.

Parallelizable Monte Carlo algorithms are developed for estimating the probability moments of criticality parameters for transport of particles with multiplication in a random medium. For this purpose, new iterative estimates of the multiplication factor and recurrence representations of statistical estimates of moments are constructed by applying the double randomization method and the randomized projection method. The practical efficiency of the proposed approaches is confirmed by test results obtained using special randomized homogenization with an improved diffusion approximation for a multilayered ball.

In this paper, we investigate an inhomogeneous fourth-order nonlinear Schrödinger (NLS) equation, generated by deforming the inhomogeneous Heisenberg ferromagnetic spin system through the space curve formalism and using the prolongation structure theory. Via the introduction of the auxiliary function, the bilinear form, one-soliton and two-soliton solutions for the inhomogeneous fourth-order NLS equation are obtained. Infinitely many conservation laws for the inhomogeneous fourth-order NLS equation are derived on the basis of the Ablowitz–Kaup–Newell–Segur system. Propagation and interactions of solitons are investigated analytically and graphically. The effect of the parameters \({{\mu }_{1}}\), \({{\mu }_{2}}\), \({{\nu }_{1}}\) and \({{\nu }_{2}}\) on the soliton velocity are presented. Through the asymptotic analysis, we have proved that the interaction of two solitons is not elastic.

An eigenvalue problem for the normal waves of an inhomogeneous regular waveguide is considered. The problem reduces to the boundary value problem for the tangential components of the electromagnetic field in the Sobolev spaces. The inhomogeneity of the dielectric filler and the presence of the spectral parameter in the field-matching conditions necessitate giving a special definition of the solution to the problem. To define the solution, the variational formulation of the problem is used. The variational problem reduces to the study of an operator function nonlinearly depending on the spectral parameter. The properties of the operator function, necessary for the analysis of its spectral properties, are investigated. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved. Real propagation constants are calculated. Numerical results are obtained using the Galerkin method. The numerical method proposed is implemented in a computer code. Calculations for a number of specific waveguiding structures are performed.