All issues

Two algorithms of evaluation of the continuous wavelet transform with the Morlet wavelet are presented. The first one is the solution of PDE with transformed signal, which plays a role of the initial value. The second allows to explore the influence of central frequency variation via the diffusion smoothing of the data modulated by the harmonic functions. These approaches are illustrated by the analysis of the chaotic oscillations of the coupled Roessler systems.

Small practical value of many numerical methods for solving single-ended systems of linear equations with ill-conditioned matrices due to the fact that these methods in the practice behave quite differently than in the case of precise calculations. Historically, sustainability is not enough attention was given, unlike in numerical algebra ‘medium-sized’, and emphasis is given to solving the problems of maximal order in data capabilities of the computer, including the expense of some loss of accuracy. Therefore, the main objects of study is the most appropriate storage of information contained in the sparse matrix; maintaining the highest degree of rarefaction at all stages of the computational process. Thus, the development of efficient numerical methods for solving unstable systems refers to the actual problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving systems of linear equations, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach consists in minimization of filling the main lines of the multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats. The storage format of sparse matrices has been studied and the advantage of this format consists in possibility of parallel execution any matrix operations without unboxing, which significantly reduces the execution time and memory footprint.

Direct multiplier methods for solving systems of linear equations are best suited for solving problems of large size on a computer — sparse matrix systems allow you to get multipliers, the main row of which is also sparse, and the operation of multiplication of a vector-row of the multiplier according to the complexity proportional to the number of nonzero elements of this multiplier.

As a direct continuation of this work is proposed in the basis for constructing a direct multiplier algorithm of linear programming to put a modification of the direct multiplier algorithm for solving systems of linear equations based on integration of technique of linear programming for methods to select the host item. Direct multiplicative methods of linear programming are best suited for the construction of a direct multiplicative algorithm set the direction of descent Newton methods in unconstrained optimization by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

Multiplicative methods for sparse matrices are best suited to reduce the complexity of operations solving systems of linear equations performed on each iteration of the simplex method. The matrix of constraints in these problems of sparsely populated nonzero elements, which allows to obtain the multipliers, the main columns which are also sparse, and the operation of multiplication of a vector by a multiplier according to the complexity proportional to the number of nonzero elements of this multiplier. In addition, the transition to the adjacent basis multiplier representation quite easily corrected. To improve the efficiency of such methods requires a decrease in occupancy multiplicative representation of the nonzero elements. However, at each iteration of the algorithm to the sequence of multipliers added another. As the complexity of multiplication grows and linearly depends on the length of the sequence. So you want to run from time to time the recalculation of inverse matrix, getting it from the unit. Overall, however, the problem is not solved. In addition, the set of multipliers is a sequence of structures, and the size of this sequence is inconvenient is large and not precisely known. Multiplicative methods do not take into account the factors of the high degree of sparseness of the original matrices and constraints of equality, require the determination of initial basic feasible solution of the problem and, consequently, do not allow to reduce the dimensionality of a linear programming problem and the regular procedure of compression — dimensionality reduction of multipliers and exceptions of the nonzero elements from all the main columns of multipliers obtained in previous iterations. Thus, the development of numerical methods for the solution of linear programming problems, which allows to overcome or substantially reduce the shortcomings of the schemes implementation of the simplex method, refers to the current problems of computational mathematics.

In this paper, the approach to the construction of numerically stable direct multiplier methods for solving problems in linear programming, taking into account sparseness of matrices, presented in packaged form. The advantage of the approach is to reduce dimensionality and minimize filling of the main rows of multipliers without compromising accuracy of the results and changes in the position of the next processed row of the matrix are made that allows you to use static data storage formats.

As a direct continuation of this work is the basis for constructing a direct multiplicative algorithm set the direction of descent in the Newton methods for unconstrained optimization is proposed to put a modification of the direct multiplier method, linear programming by integrating one of the existing design techniques significantly positive definite matrix of the second derivatives.

The second part of paper is devoted to final study of three classic partial differential equations (Laplace, Diffusion and Wave) solution using simple numerical methods in terms of Cellular Automata. Specificity of this solution has been shown by different examples, which are related to the hexagonal grid. Also the next statements that are mentioned in the first part have been proved: the matter conservation law and the offensive effect of excessive hexagonal symmetry.

From the point of CA view diffusion equation is the most important. While solving of diffusion equation at the infinite time interval we can find solution of boundary value problem of Laplace equation and if we introduce vector-variable we will solve wave equation (at least, for scalar). The critical requirement for the sampling of the boundary conditions for CA-cells has been shown during the solving of problem of circular membrane vibrations with Neumann boundary conditions. CA-calculations using the simple scheme and Margolus rotary-block mechanism were compared for the quasione-dimensional problem “diffusion in the half-space”. During the solving of mixed task of circular membrane vibration with the fixed ends in a classical case it has been shown that the simultaneous application of the Crank–Nicholson method and taking into account of the second-order terms is allowed to avoid the effect of excessive hexagonal symmetry that was studied for a simple scheme.

By the example of the centrally symmetric Neumann problem a new method of spatial derivatives introducing into the postfix CA procedure, which is reflecting the time derivatives (on the base of the continuity equation) was demonstrated. The value of the constant that is related to these derivatives has been empirically found in the case of central symmetry. The low rate of convergence and accuracy that limited within the boundaries of the sample, in contrary to the formal precision of the method (4-th order), prevents the using of the CAmethods for such problems. We recommend using multigrid method. During the solving of the quasi-diffusion equations (two-dimensional CA) it was showing that the rotary-block mechanism of CA (Margolus mechanism) is more effective than simple CA.

The paper deals with the problems of approximation of periodic functions of high smoothness by arithmetic means of Fourier sums. The simplest and natural example of a linear process of approximation of continuous periodic functions of a real variable is the approximation of these functions by partial sums of the Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the entire class of continuous $2\pi$-periodic functions. In connection with this, a significant number of papers is devoted to the study of the approximative properties of other approximation methods, which are generated by certain transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for each function $f \in C$. In particular, over the past decades, de la Vallee Poussin sums and Fejer sums have been widely studied. One of the most important directions in this field is the study of the asymptotic behavior of upper bounds of deviations of arithmetic means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol’sky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk, etc.

The aim of the work systematizes known results related to the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of $r$-repeated de la Vallee Poussin sums on the classes of periodic functions that can be regularly extended into the fixed strip of the complex plane. We obtain asymptotic formulas for upper bounds of the deviations of repeated de la Vallee Poussin sums taken over classes of periodic analytic functions. In certain cases, these formulas give a solution of the corresponding Kolmogorov–Nikolsky problem. We indicate conditions under which the repeated de la Vallee Poussin sums guarantee a better order of approximation than ordinary de la Vallee Poussin sums.

Padé approximation is a useful tool for extracting singularity information from a power series. A linear Padé approximant is a rational function and can provide estimates of pole and zero locations in the complex plane. A quadratic Padé approximant has square root singularities and can, therefore, provide additional information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing quadratic Padé approximants as well as some applications. Two algorithms for computing the coefficients in the approximant are discussed: a direct method involving the solution of a linear system (well-known in the mathematics community) and a recursive method (well-known in the physics community). We compare the accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons. In addition, we extend Luke’s perturbation analysis of linear Padé approximation to the quadratic case and identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be identified by the recursive method mentioned above. Another complication with the quadratic approximant arises in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by means of an example. Two concluding applications show the superiority of the quadratic approximant over its linear counterpart: one involving a special function (the Lambert $W$-function) and the other a nonlinear PDE (the continuation of a solution of the inviscid Burgers equation into the complex plane).

In this work we consider a possibility to use the conception of $(\delta, L)$-model of a function for optimization tasks, whereby solving a primal problem there is a necessity to recover a solution of a dual problem. The conception of $(\delta, L)$-model is based on the conception of $(\delta, L)$-oracle which was proposed by Devolder–Glineur–Nesterov, herewith the authors proposed approximate a function with an upper bound using a convex quadratic function with some additive noise $\delta$. They managed to get convex quadratic upper bounds with noise even for nonsmooth functions. The conception of $(\delta, L)$-model continues this idea by using instead of a convex quadratic function a more complex convex function in an upper bound. Possibility to recover the solution of a dual problem gives great benefits in different problems, for instance, in some cases, it is faster to find a solution in a primal problem than in a dual problem. Note that primal-dual methods are well studied, but usually each class of optimization problems has its own primal-dual method. Our goal is to develop a method which can find solutions in different classes of optimization problems. This is realized through the use of the conception of $(\delta, L)$-model and adaptive structure of our methods. Thereby, we developed primal-dual adaptive gradient method and fast gradient method with $(\delta, L)$-model and proved convergence rates of the methods, moreover, for some classes of optimization problems the rates are optimal. The main idea is the following: we find a dual solution to an approximation of a primal problem using the conception of $(\delta, L)$-model. It is much easier to find a solution to an approximated problem, however, we have to do it in each step of our method, thereby the principle of “divide and conquer” is realized.

Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.

Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for solving loaded partial differential equations of integer and fractional orders are widely used, since analytical solving methods for solving are impossible. A fairly effective method for solving this kind of problem is the finite difference method, or the grid method.

We studied the initial-boundary value problem in the rectangle $\overline{D}=\{(x,\,t)\colon 0\leqslant x\leqslant l,\;0\leqslant t\leqslant T\}$ for the loaded differential heat equation with composition fractional derivative of Riemann – Liouville and Caputo – Gerasimov and with boundary conditions of the first and third kind. We have gotten an a priori assessment in differential and difference interpretations. The obtained inequalities mean the uniqueness of the solution and the continuous dependence of the solution on the input data of the problem. A difference analogue of the composition fractional derivative of Riemann – Liouville and Caputo –Gerasimov order $(2-\beta )$ is obtained and a difference scheme is constructed that approximates the original problem with the order $O\left(\tau +h^{2-\beta } \right)$. The convergence of the approximate solution to the exact one is proven at a rate equal to the order of approximation of the difference scheme.

The process of heat conduction, accompanied by the first order phase transitions is discussed in this article. Using cellular automates simulation was investigated class of problems that have broad application in practice. In this paper we calculate the temperature distribution in the depth of the soil at different times for a problem of freezing of moist soil. Another task — zone growing — has been modeled by cellular automates too. The coincidence of real and modeling parameters of the system confirms the feasibility of using the selected method of modeling of physical processes.