卷 56, 编号 4 (2016)

Surrogate modeling is widely used in many engineering problems. Data sets often have Cartesian product structure (for instance factorial design of experiments with missing points). In such case the size of the data set can be very large. Therefore, one of the most popular algorithms for approximation–Gaussian Process regression–can be hardly applied due to its computational complexity. In this paper a computationally efficient approach for constructing Gaussian Process regression in case of data sets with Cartesian product structure is presented. Efficiency is achieved by using a special structure of the data set and operations with tensors. Proposed algorithm has low computational as well as memory complexity compared to existing algorithms. In this work we also introduce a regularization procedure allowing to take into account anisotropy of the data set and avoid degeneracy of regression model.

The finite-dimensional problem of the best approximation (in the Hausdorff metric) of a convex body by a ball of arbitrary norm with a fixed radius is considered. The stability and sensitivity of the solution to errors in specifying the convex body to be approximated and the unit ball of the used norm are analyzed. It is shown that the solution of the problem is stable with respect to the functional and, if the solution is unique, the center of the best approximation ball is stable as well. The sensitivity of the solution to the error with respect to the functional is estimated (regardless of the radius of the ball). A sensitivity estimate for the center of the best approximation ball is obtained under the additional assumption that the estimated body and the ball of the used norm are strongly convex. This estimate is related to the range of radii of the approximating ball.

A family of difference schemes for the fractional-order diffusion equation with variable coefficients is considered. By the method of energetic inequalities, a priori estimates are obtained for solutions of finite-difference problems, which imply the stability and convergence of the difference schemes considered. The validity of the results is confirmed by numerical calculations for test examples.

For a singularly perturbed parabolic equation, asymptotics of the solution to an initial boundary value problem in the case of a triple root of the degenerate equation is constructed and justified. Essential distinctions from the case of a simple root are the scale of the boundary layer variables and the three-zone structure of the boundary layer.

An asymptotic expansion of the solution to the Cauchy problem for a class of hyperbolic weakly nonlinear systems with many spatial variables is constructed. A parabolic quasilinear equation describing the behavior of the solution at asymptotically large values of the independent variables is obtained. The pseudo-diffusion processes that depend on the relationship between the number of equations and the number of spatial variables are analyzed. The structure of the subspace in which there are pseudo-diffusion evolution processes of the solution in the far field is described.

In this paper we study a generalized coupled variable-coefficient modified Korteweg–de Vries (CVCmKdV) system that models a two-layer fluid, which is applied to investigate the atmospheric and oceanic phenomena such as the atmospheric blockings, interactions between the atmosphere and ocean, oceanic circulations and hurricanes. The conservation laws of the CVCmKdV system are derived using the multiplier approach and a new conservation theorem. In addition to this, a similarity reduction and exact solutions with the aid of symbolic computation are computed.

Nonlinear wave processes described by a fifth-order generalized KdV equation derived from the Fermi–Pasta–Ulam (FPU) model are considered. It is shown that, in contrast to the KdV equation, which demonstrates the recurrence of initial states and explains the FPU paradox, the fifthorder equation fails to pass the Painlevé test, is not integrable, and does not exhibit the recurrence of the initial state. The results of this paper show that the FPU paradox occurs only at an initial stage of a numerical experiment, which is explained by the existence of KdV solitons only on a bounded initial time interval.