Home – Home – Jornals – Numerical Analysis and Applications 2018 number 1

The parallel implementation peculiarities of a discrete stochastic model that simulates water permeation through a porous substance (soil) with a complex morphology are studied. The simulation should reveal the fluid flowing along pore curves and filling wets and cavities. The discrete stochastic model of the process, proposed earlier, is a stochastic cellular automaton (SCA), whose functioning is represented by a set of elementary local operators acting in a cellular space and imitating displacement (diffusion. convection, adsorption) and transformations (reaction, phase transition) of abstract or real particles. The microlevel of the process representation requires the cellular space of a huge size, and hence, the computations should be implemented on supercomputers. With this, the main problem is in the fact that obtaining an acceptable parallelization efficiency is possible only by inserting some determinism into the computation algorithm, i.e., by decreasing the model stochasticity. Although stochastic models are under intensive investigation, the parallel implementation methods for them are poorly studied. This gap is partially covered by the results of computational experiments, given in this paper, which allow one to assess the advantages and drawbacks of methods for the discrete stochastic mode of water permeation though implementing a porous medium on a multicore cluster.

In the second part of this paper, we consider the Buckley-Osthus model for the formation of a web-graph. For the networks generated according to this model, we numerically calculate the PageRank vector. We show that the components of this vector are distributed according to the power law. We also discuss the computational aspects of this model with respect to different numerical methods for the calculation of the PageRank vector, presented in the first part of the paper. Finally, we describe a general model for the web-page ranking and some approaches to solve the optimization problem arising when learning this model.

In this paper, convex stochastic optimization problems under different assumptions on the properties of the available stochastic subgradients are considered. It is known that if a value of the objective function is available, one can obtain, in parallel, several independent approximate solutions in terms of the objective residual expectation. Then, choosing a solution with the minimum function value, one can control the probability of large deviations of the objective residual. On the contrary, in this short paper we address the situation when the objective function value is unavailable or is too expensive to calculate. Under the «light-tail» assumption for stochastic subgradients and in the general case with a moderate probability of large deviations, it is shown that parallelization combined with averaging gives bounds for the probability of large deviations similar to those of a serial method. Thus, in these cases one can benefit from parallel computations and reduce the computational time without any loss in the solution quality.

The inverse problem of recovering the leading time-dependent coefficient by the known non-local additional information is investigated. For an approximate solution of the nonlinear inverse problems we propose the gradient method of minimizing the target functional. The comparative analysis with the method based on the linearized approximation scheme with respect to time is made. The results of the numerical calculations are presented.

The present paper uses a new two level implicit difference formula for the numerical study of one dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second order accurate in time and third order accurate in space on a non-uniform grid and in case of a uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three point spatial discretization which yields a block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for a model linear problem for a uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including complex fourth-order nonlinear equations like the Kuramoto-Sivashinsky equation and the extended Fisher-Kolmogorov equation, where comparison is made with some earlier work. It is clear from the numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competitive with the solutions derived in earlier research studies.

In this paper, we study various difference schemes on oblique stencils, i.e., the schemes using different space grids on different time levels. Such schemes can be useful when solving boundary value problems with moving boundaries and when using the regular grids of a non-standard structure (for example, triangular or cellular) and, also, when applying the adaptive methods. To study the stability, we use the analysis of First Differential Approximation of finite difference schemes and the dispersion analysis. We study the meaning of the stability conditions as constraints on the geometric location of stencil elements with respect to the characteristics of the equation. In addition, we compare our results with the geometric interpretation of the stability of classical schemes. The paper also presents the generalization of oblique schemes in the case of the quasi-linear equation of transport and numerical experiments for these schemes.

The algorithms of solving the inverse source problem for systems of the production-destruction equations are considered. Consistent in the sense of the Lagrangian identity numerical schemes for solving direct and conjugate problems have been built. With the adjoint equations, the sensitivity operator and its discrete analogue have been constructed. It links the measured values perturbations with the perturbations of the model parameters. This operator transforms the inverse problem to a quasilinear form and allows applying the Newton-Kantorovich methods to it. The paper provides a numerical comparison of the gradient algorithms based on the consistent and inconsistent numerical schemes and the Newton-Kantorovich algorithm applied to solving the inverse source problem for the nonlinear Lorenz model.

In this paper, we consider a class of second order systems of the Volterra nonlinear integral equations. This class is related to the automatic control problem of a dynamic object with vector inputs and outputs. A numerical solution technique based on the Newton-Kantorovich method is considered. To verify the efficiency of the algorithms developed, a series of test calculations were carried out.