Vol 64, No 9 (2024)

Finding nontrivial solutions to the trilinear Brent equations corresponds to the construction of asymptotically fast matrix multiplication algorithms is an important, but in general a very difficult computational task. Methods of parameterization of the Brent equations based on the use of symmetries of the matrix product tensor are proposed, which make it possible to repeatedly reduce the dimension of the problem. The numerical solution of the obtained trilinear or cubic systems of nonlinear equations is carried out by reducing to a nonlinear least squares problem and applying to it a specially developed iterative method that does not require calculation of derivatives. The found solutions of the parameterized Brent equations, as a rule, have a rank no higher (and sometimes even lower) than the known results. Thus, an algorithm for multiplying two 4th-order matrices in 48 active multiplications is obtained.

In Hilbert space, we consider a linear-quadratic optimal control problem with a fixed left end and a movable right end, for a fixed period of time. The target functional is the sum of the integral and terminal components of the quadratic form. Each of the components searches for its minimum on its permissible set independently of each other. At the right end of the time interval, we have a linear programming problem. The solution to this problem implicitly defines a terminal condition for controlled dynamics. A saddle approach is proposed to solve the problem, which boils down to calculating the saddle point of the Lagrange function. The approach is based on saddle inequalities in both groups of variables: direct and dual. These inequalities represent sufficient conditions for optimality. A method for calculating the saddle point of the Lagrange function is formulated. Convergence in direct and dual variables is proved, namely: weak convergence in controls, strong convergence in phase and conjugate trajectories, as well as in terminal variables of the boundary value problem. On the basis of the saddle approach, control synthesis is built, i.e. feedback in the presence of control constraints in the form of a convex closed set. This is a new result, since in the classical case, in the theory of a linear regulator, a similar statement is proved in the absence of control constraints, which makes it possible to use the Riccati matrix equation. If there are restrictions on management, these arguments no longer pass. Therefore, the basis of the obtained result is the concept of a reference plane to a set of controls.

The roots of the indicial polynomial constructed for a given linear ordinary differential operator provide information about the features of solutions of the corresponding homogeneous differential equation. Operators and equations whose coefficients are formal Laurent series are discussed. Solutions of the same kind are considered. These assumptions describe the structure of the indicial polynomial of the product of differential operators. This structural (multiplicative) property is preserved in the case of converging series.

A mathematical model based on the Baer–Nunziato multiphase model is presented. The effectiveness of the model is demonstrated by the numerical solution of shock wave problems in condensed media in the presence of an explicit contact boundary with vacuum. The results of numerical simulation of problems of interaction of femtosecond laser radiation with an aluminum target are considered. The advantage of using the Baer–Nunziato model in comparison with the single-phase hydrodynamic model in calculating the dynamics of the contact boundary is shown. The simplicity of implementation and the possibility of easy introduction of additional submodels, such as ignition, makes this approach attractive for modeling high-energy processes in multiphase media.

The problem of two-dimensional flow of a viscous slightly compressible liquid in a square cell under excitation by a static external force periodic in space (Kolmogorov flow) is considered. A new method for determining the flow structure based on the analysis of the vorticity field at various points in time is presented. This method is used to classify the types of flows whose characteristics are obtained by numerical modeling. The main flow modes are distinguished depending on the values of the bottom friction coefficient and the pumping force: laminar, chaotic and vortex modes. Transitional flow types are studied separately: the quasi-periodic regime, which arises through a sequence of bifurcations during the change of laminar and chaotic flow modes, and the regime of alternation, which occurs during the transition from chaotic to vortex flow. Phase diagrams in the space of the amplitude of the external force — the bottom friction coefficient are constructed, making it possible to classify the type of flow according to the values of the bottom friction coefficient and the pumping force.

For a long time, intense methane emissions into the atmosphere from the subsurface of permafrost have been observed in the shelf and coastal zones of the Arctic region. Due to the potential danger of this phenomenon to the environment and infrastructure, there is a need for periodic monitoring of gas pockets, including through ground-based seismic exploration. This paper contains the results of a study of this process using numerical modeling. A model of layered frozen sandy soil with curviliinear boundaries between the sheets is constructed, reflecting the main specifics of the region. The process of gas migration in vertical and horizontal directions is reconstructed by increasing the number of methane reservoirs. The defining system of hyperbolic equations of the linear theory of elasticity is used. The problem is solved by the grid-characteristic method in a two-dimensional formulation on a rectangular grid, in each cell of which the elastic characteristics of the layers are set. The resulting wave structures are studied in detail on the obtained synthetic wave patterns and seismograms, which make it possible to determine the direction of gas propagation. The results obtained can be used to interpret similar field experiments.

An explicit-implicit scheme is constructed for the numerical solution of the defining system of equations of the elastic-viscoplastic continuum model, taking into account the softening effect. The scheme includes an explicit approximation of the equations of motion and an implicit approximation of the defining relations containing a small relaxation time parameter in the denominator of the free terms. Precise correction formulas for stress deviators are obtained after the elastic calculation step in the case of the linear viscosity function and the nonlinear law of softening. The obtained solutions of the implicit approximation for the stress deviators of the elastic-viscoplastic system under consideration allow for a limiting transition when the relaxation time tends to zero. Correction formulas, obtained by such a limiting transition, can be interpreted as regularizers of numerical solutions of incorrect elastoplastic systems with a softening effect.