Том 63, № 2 (2023)

We present a unified analysis of methods for such a wide class of problems as variational inequalities, which include minimization and saddle point problems as special cases. The analysis is developed relying on the extragradient method, which is a classic technique for solving variational inequalities. We consider the monotone and strongly monotone cases, which correspond to convex-concave and strongly-convex-strongly-concave saddle point problems. The theoretical analysis is based on parametric assumptions about extragradient iterations. Therefore, it can serve as a strong basis for combining existing methods of various types and for creating new algorithms. Specifically, to show this, we develop new robust methods, including methods with quantization, coordinate methods, and distributed randomized local methods. Most of these approaches have never been considered in the generality of variational inequalities and have previously been used only for minimization problems. The robustness of the new methods is confirmed by numerical experiments with GANs.

Let A and B  be Hermitian n*n  matrices with A  being nonsingular. According to a well-known theorem of matrix analysis, these matrices can be brought to diagonal form by one and the same Hermitian congruence transformation if and only if the matrix C = A-1B  has a real spectrum and can be diagonalized by a similarity. An extension of this assertion to the case where two unitoid matrices are simultaneously reduced to diagonal form is stated and proved.

We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

A linear integro-differential equation with a singular differential operator in the principal part is studied. For its approximate solution in the space of generalized functions, special generalized versions of the methods of moments and subdomains are proposed and substantiated. Optimality of the methods in order of accuracy is established.

Volume and surface potentials arising in Cauchy problems for nonlinear equations in the theory of ion acoustic and drift waves in a plasma are considered, and their properties are examined. For the volume potential, an estimate is derived, which is used to prove a Schauder-type a priori estimate and Schauder-type estimates for weighted potentials.

The gas Couette flow for a cylindrical geometry of bounding surfaces that move in the longitudinal direction relative to their symmetry axis is considered. For a monoatomic gas, the relation between the shear stress and the energy flux transferred to the longitudinal-flow surface (Reynolds analogy) is studied. In the case of continuous medium and free molecular flow regimes, simple explicit analytical expressions for the Reynolds analogy coefficient are obtained, which depend only on the Eckert number and are independent of the ratio of the cylinder radii. The transitional regime for various values of the Knudsen number is studied using the direct simulation Monte Carlo (DSMC) method. It is shown that, in this case, the Reynolds analogy coefficient at a fixed ratio of the radii and the Knudsen number depends on the relative velocity and temperatures of the surfaces mainly through the Eckert number. A relationship between the linear energy fluxes transferred to the cylindrical surfaces is found.

The Cauchy problem for a first-order evolutionary equation with memory with the time derivative of the Volterra integral term and difference kernel in the finite-dimensional Banach space is considered. The fundamental difficulties of the approximate solution of such problems are caused by nonlocality with respect to time when the solution at the current time depends on the entire history. Transformation of the first-order integrodifferential equation to a system of evolutionary local equations with the approximation of the difference kernel by a sum of exponential functions is used. For the weakly coupled system of local equations with additional ordinary differential equations, estimates of stability of solution with respect to initial data and right-hand side are obtained using the concept of logarithmic norm. Similar estimates are obtained for the approximate solution using two-level time approximations.