This paper proposes a modification of Fisher’s approximate nuclear discriminant analysis that provides a computationally efficient projection of initially linearly inseparable data onto a lower-dimensional space with the formation of linear class separability. The key feature of this modification is to obtain an approximation of the projection matrix using a projection submatrix on the basis of a weighted set of reference points. The main approximation rule is the Nyström method with automatic determination of the weights of reference points. For the proposed modification, a theoretically justified formalization of the criterion is given and the computational scheme for solving the formulated projection problem is refined. The main algorithmic steps are specified and practical techniques for the efficient implementation of the numerical scheme are presented. To compare the efficiency of the proposed modification and the numerical solution, we have conducted experimental studies using both synthetic datasets that model different class topologies and a real binary classification task in computer vision: detecting unmanned aerial vehicles in images. The experiments included a series of runs with different ratios of the number of reference points to the total sample size and nonuniform data density, which allowed us to study the posterior convergence of the solution and the dependence of quality on the number of reference points. The results of computational experiments showed a steady improvement in classification quality and favorable computational characteristics of the proposed modification compared to classical and modern counterparts. Specifically, the modification of the method makes it possible to reduce the number of reference points while maintaining (or improving) the classification quality and consistently increases the final F1-measure on real datasets with a poorly separable class topology. Also, this modification demonstrates better resistance to background noise and nonuniform data density due to the automatic weighting of reference points that specify the most significant representations of the feature space. Promising fields for further research have been identified.
Results of a two-stage numerical experiment are presented. At its first stage, the system of equations of meteor physics is used to analyze the intrusion of a celestial body of complex structure into the atmosphere and its motion, taking into account the loss of mass—the outer protective crust to a low altitude above the Earth’s surface. At the second stage, the complete system of unsteady Euler equations for the motion of an ideal compressible gas is used to investigate the impact of shock waves initiated by a meteoroid explosion on a taiga forest. Here, the problem is solved numerically by approximating the initial equations with an explicit difference scheme and implementing it with a through-computation method. This process is analyzed for the test case of the Vitim bolide. The numerical experiment makes it possible to determine the main parameters of the bolide explosion: its energy and height, which characterizes the size and degree of destruction of the forest areas as coinciding with the estimates obtained by the expeditions that detected them.
The possibility of applying a completely conservative difference scheme with adaptively selected artificial viscosity to a class of problems in the dynamics of gas-dispersed media is considered. Calculations are performed for the astrophysical test problems DUSTYBOX (the so-called “dusty box” problem), with constant gas and dust densities, and DUSTYWAVE (“dusty wave”), which describes sound-wave propagation in a two-phase gas-dust medium with monodisperse particles. The numerical results agree with the analytical solutions to high accuracy, demonstrating the effectiveness of the developed algorithms and methods.
A numerical algorithm for modeling spatial flows of a mixture of heterogeneous fluids has been constructed and verified using the example of the interaction between gas bubbles and a shockwave. The algorithm is based on a heterogeneous PVT model and makes it possible to describe gas–liquid mixture flows. A system of four conservation equations closed by a bimodal state equation is used. The finite-difference algorithm is based on a finite volume method using quasi-gasdynamic regularized equations. The numerical calculations for the interaction of air bubble and a system of bubbles with a weak shockwave of low intensity in water indicate the efficiency of the proposed algorithm. The numerical algorithm has been implemented in the framework of the open-source AMReX platform.
Boundary value problems for the Black–Scholes partial differential equation, which describes the value of a financial instrument, may contain a free boundary condition if the financial instrument allows for early exercise. This paper considers free boundary value problems for the Black‒Scholes equation and the convection-diffusion equation. For the convection-diffusion equation, we present a finite difference scheme that is a generalization of a well-known second-order accurate scheme on fixed grids and propose an algorithm for determining the position of the free boundary. For the Black–Scholes equation, which is similar to, but more complex than, the convection-diffusion equation, a new numerical solution method is proposed. This method involves, on the one hand, a modification of the finite-difference scheme used for the convection-diffusion equation, and, on the other hand, elements of the smoothed particle hydrodynamics (SPH) method. The proposed method makes it possible to find both the solution and the free boundary, ensuring high accuracy. Numerical experiments confirm the efficiency of the proposed algorithm.
