Convergence Estimates Research Articles

ПРОБЛЕМА ЗБІЖНОСТІ ПРОЦЕДУРИ ПОБУДОВИ КЛАСИФІКАТОРІВ У СХЕМАХ ЛОГІЧНИХ І АЛГОРИТМІЧНИХ ДЕРЕВ КЛАСИФІКАЦІЇ

The problem of convergence of the procedure for synthesizing classifier schemes in the methods of logical and algorithmic classification trees is considered. An upper estimate of the complexity of the algorithm tree scheme is proposed in the problem of approximating an array of real data with a set of generalized features with a fixed criterion for stopping the branching procedure at the stage of constructing a classification tree. This approach allows you to ensure the necessary accuracy of the model, assess its complexity, reduce the number of branches and achieve the necessary performance indicators. For the first time, methods for constructing structures of logical and algorithmic classification trees are given an upper estimate of the convergence of constructing classification trees. The proposed convergence estimate of the procedure for constructing classifiers for LCT/ACT structures makes it possible to build economical and efficient classification models of a given accuracy. The method of constructing an algorithmic classification tree is based on a step-by-step approximation of an initial sample of arbitrary volume and structure by a set of independent classification algorithms. When forming the current vertex of an algorithmic tree, node, or generalized feature, this method highlights the most efficient, high-quality autonomous classification algorithms from the initial set. This approach to constructing the resulting classification tree can significantly reduce the size and complexity of the tree, the total number of branches, vertices, and tiers of the structure, improve the quality of its subsequent analysis, interpretability, and ability to decompose. Methods for synthesizing logical and algorithmic classification trees were implemented in the library of algorithms of the “Orion III” software system for solving various applied problems of artificial intelligence. Practical applications have confirmed the operability of the constructed classification tree models and the developed software. The paper estimates the convergence of the procedure for constructing recognition schemes for cases of logical and algorithmic classification trees under conditions of weak and strong class separation of the initial sample. Prospects for further research and testing may consist in evaluating the convergence of the ACT synthesis procedure in a limited method of the algorithmic classification tree, which consists in maintaining a criterion for stopping the procedure for constructing a tree model by the depth of the structure, optimizing its software implementations, introducing new types of algorithmic trees, as well as experimental studies of this method for a wider range of practical problems.

Stancu type operators including generalized Brenke polynomials

Our objective in this paper is to present the sequence of Stancu type operators including generalized Brenke polynomials. We answer the problem of uniform approximation of continuous functions on closed bounded interval and the problem of the order of this convergence estimate by known tools in approximation theory. Furthermore, we apply some of the results obtained in this work to Miller-Lee polynomials and Gould-Hopper polynomials.

Approximation for Gupta type general operators based on Mihean basis functions

Recently Indian known mathematician V. Gupta introduced a general family of linear positive operators, which produces a large number of well-known linear positive operators as particular cases. The family of operators, proposed by Gupta provides a unified approach. This motivated us to extend the studies and we establish some convergence estimates which include an asymptotic formula and the rate of convergence for the function having derivatives of bounded variation.

О сходимости разностных схем метода опорных операторов для ротационных операций векторного анализа на тетраэдрических сетках

One approach to the description of the metric properties of an irregular grid is proposed for discretizing repeated rotational operations of vector analysis (rot rot) as applied to modeling the magnetic and electric fields. On the basis of the support operator method and the used grid metric, integrally consistent vector analysis operations rot, div, grad are constructed that are necessary to obtain convergence estimates for difference schemes for repeated rotational operations, as well as solving physical problems with discontinuous magnetic viscosity, dielectric permeability or thermal resistance of a medium. On smooth solutions of the model magnetostatic problem on a tetrahedral grid with first order of accuracy in the mean-square sense, the convergence of the constructed difference schemes with zero eigenvalue of the spectral problem is proved. At the same time, no restrictions are imposed on an irregular tetrahedral mesh, except for its non-degeneracy.

Optimal $W^{1,\infty}$-estimates for an isoparametric finite element discretization of elliptic boundary value problems

We consider an elliptic boundary value problem on a domain with regularboundary and discretize it with isoparametric finite elements of order$k\geq1$.We show optimal order of convergence of the isoparametric finiteelement solution in the $W^{1,\infty}$-norm. As an intermediate step,we derive stability and convergence estimates of optimal order $k$ fora (generalized) Ritz map.

Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution

<p style='text-indent:20px;'>The motivation behind the current paper is to elucidate the approximation properties of a Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. We construct quantitative-Voronovskaya and Grüss-Voronovskaya type theorems and determine the convergence estimates of the above operators. We also contrive the statistical convergence and talk about the approximation degree of a bivariate extension of these operators by exhibiting the convergence rate in terms of the complete and partial moduli of continuity. We build GBS (Generalized Boolean Sum) operators allied with the bivariate operators and estimate their convergence rate using mixed modulus of smoothness and Lipschitz class of B<inline-formula><tex-math id="M1">\begin{document}$ \ddot{o} $\end{document}</tex-math></inline-formula>gel continuous functions. We also evaluate the order of approximation of the GBS operators in the spaces of B-continuous (Bögel continuous) and B-differentiable (Bögel differentiable) functions. In addition, we depict the comparison between the rate of convergence of the proposed bivariate operators and the corresponding GBS operators for some functions by graphical illustrations using MATLAB software.</p>

Конвергенция неравенства доходов в российских регионах

In this paper we test convergence of income inequality in Russia’s regions for the period 1995–2020. To do this, conditional and unconditional beta convergence models for the regional Gini index are evaluated on cross-sectional and panel data using time and spatial effects. Estimates of the models show that both conditional and unconditional convergence of income inequality takes place in Russia’s regions. It is shown that the rate of convergence varies significantly within the considered period: the levels of income inequality in the regions converged most strongly at the beginning of the period with a gradual slowdown in the rate of convergence in subsequent periods. This result may be related to the recovery growth and redistribution policy in the 2000s, as well as the consequences of the 2014 crisis. The use of the same initial characteristics, such as GRP per capita, level of education and population, accelerates convergence. Spatial effects are statistically significant for models of unconditional, but not conditional convergence, but do not affect the estimates obtained. When considering a panel data structure with the inclusion of fixed time effects, convergence estimates increase for both unconditional and conditional convergence

A new minimizing-movements scheme for curves of maximal slope

Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on minimizing-movements schemes, most notably on the Euler scheme. We present here an alternative minimizing-movements approach, yielding more regular discretizations, serving as a-posteriori convergence estimator, and allowing for a simple convergence proof.

Diffusion bias-compensated recursive maximum correntropy criterion algorithm with noisy input

In recent years, the distributed estimation problem based on the diffusion strategy has received more and more attention, where the node filters cooperate with other in-network node filters to estimate the unknown parameter vector. In this paper, we propose a Diffusion Bias Compensated Recursive Maximum Correntropy Criterion (DBCRMCC) algorithm based on the idea of distributed estimation for adaptive filtering in network containing input noise and non-Gaussian output noise. The Recursive Maximum Correntropy Criterion (RMCC) is an adaptive filtering algorithm with Maximum Correntropy Criterion (MCC) as the cost function which is robust to large outliers. Considering that the estimates directly obtained by the RMCC algorithm are biased when the input is disturbed by noise, this paper proposes the DBCRMCC algorithm under some reasonable assumptions by using the principle of unbiased estimation and combining it with a diffusion strategy. Through bias-compensation for biased estimates and cooperation among node filters, the asymptotic unbiased estimates of the unknown parameters can be obtained. The simulation results show that the proposed DBCRMCC algorithm has acceptable convergence speed and estimation accuracy in the environment where the input signal is noisy and the output noise is non-Gaussian.

Recovering the initial distribution for a logarithmic nonlinear biparabolic equation

In this paper, we consider the problem of finding the initial distribution for a biparabolic equation with logarithmic nonlinearity source. The problem is severely ill‐posed in the sense of Hadamard. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Lq space with some q ≥ 2 and the Sobolev space Ws, ℓ for some constants s > 0, ℓ ≥ 1.

Approximation properties of a new family of Gamma operators and their applications

The present paper introduces a new modification of Gamma operators that protects polynomials in the sense of the Bohman–Korovkin theorem. In order to examine their approximation behaviours, the approximation properties of the newly introduced operators such as Voronovskaya-type theorems, rate of convergence, weighted approximation, and pointwise estimates are presented. Finally, we present some numerical examples to verify that the newly constructed operators are an approximation procedure.

Convergence estimates of finite elements for a class of quasilinear elliptic problems

This paper is concerned with conforming finite element discretizations for quasilinear elliptic problems in divergence form, of a class that generalizes the p-Laplace equation and allows to show existence and uniqueness of the continuous and discrete problems. We derive discretization error estimates under general regularity assumptions for the solution and using high order polynomial spaces, resulting in convergence rates that are then verified numerically. A key idea of this error analysis is to consider carefully the relation between the natural W1,p-seminorm and a specific quasinorm introduced in the literature. In particular, we are able to derive interpolation estimates in this quasinorm from known interpolation estimates in the W1,p-seminorm. We also give a simplified proof of known near-best approximation results in W1,p-seminorm starting from the corresponding result in the mentioned quasinorm.

Two parameter singular perturbation problems for sine-Gordon type equations

In the real Sobolev space $H_0^1(\Omega)$ we consider the Cauchy-Dirichlet problem for sine-Gordon type equation with strongly elliptic operators and two small parameters. Using some {\it a priori} estimates of solutions to the perturbed problem and a relationship between solutions in the linear case, we establish convergence estimates for the difference of solutions to the perturbed and corresponding unperturbed problems. We obtain that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of $t=0.$

Convergence estimates for abstract second order differential equations with two small parameters and Lipschitzian nonlinearities

In a real Hilbert space $H$ we consider the following singularly perturbed Cauchy problem ... We study the behavior of solutions $u_{\varepsilon\delta}$ in two different cases: $\varepsilon\to 0$ and $\delta \geq \delta_0>0;$ $\varepsilon\to 0$ and $\delta \to 0,$ relative to solution to the corresponding unperturbed problem.We obtain some {\it a priori} estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of $t=0.$

Tau and Galerkin operational matrices of derivatives for treating singular and Emden–Fowler third-order-type equations

In this paper, our target is to implement and analyze numerical algorithms for the numerical solutions of initial and boundary third-order singular-type equations, and in particular the Emden–Fowler-type equations. Both linear and nonlinear equations are investigated. Two kinds of basis functions in terms of the shifted Chebyshev polynomials of the second kind are employed. Regarding the initial value problems, a new Galerkin operational matrix of derivatives is established and utilized, while in the case of boundary value problems (BVPs), the operational matrix of derivatives of the shifted Chebyshev polynomials of the second kind is employed. Convergence and error estimates of the two proposed expansions are given. The obtained solutions are spectral and they are consequences of the application of tau and collocation methods. These methods serve to convert the problems governed by their initial/boundary conditions into systems of linear or nonlinear algebraic equations which can be solved via suitable numerical solvers. Finally, we support our theoretical study by presenting several examples to ensure the accuracy and efficiency of the proposed algorithms.

Landweber Iterative Regularization Method for Identifying the Initial Value Problem of the Rayleigh–Stokes Equation

In this paper, we study an inverse problem to identify the initial value problem of the homogeneous Rayleigh–Stokes equation for a generalized second-grade fluid with the Riemann–Liouville fractional derivative model. This problem is ill posed; that is, the solution (if it exists) does not depend continuously on the data. We use the Landweber iterative regularization method to solve the inverse problem. Based on a conditional stability result, the convergent error estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given. Some numerical experiments are performed to illustrate the effectiveness and stability of this method.

Bi-parametric operator preconditioning

We extend the operator preconditioning framework Hiptmair (2006) [10] to Petrov-Galerkin methods while accounting for parameter-dependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner, our bi-parametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and super-linear convergence estimates for iterative solvers, such as h-independent convergence bounds, when preconditioning with low-accuracy or, equivalently, with highly compressed approximations.

A novel linearized Galerkin finite element scheme with fractional Crank–Nicolson method for the nonlinear coupled delay subdiffusion system with smooth solutions

A second order accurate linearized fractional Crank–Nicolson–Galerkin finite element scheme is proposed for solving the nonlinear coupled delay subdiffusion system. The scheme presented in this paper has the advantage of making iterative processes redundant. Existence and uniqueness results for the fully discrete solution are analyzed in detail. Further, a priori bound and convergence estimate for the fully discrete solution are derived in L∞(L2) norm. Finally, we perform our numerical experiments on one‐ and two‐dimensional problems to demonstrate the accuracy and efficiency of the proposed scheme.

Two-scale methods for convex envelopes

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp. 1689–1694]). We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method provided by Oberman [Math. Models Methods Appl. Sci. 18 (2008), pp. 759–780].

Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.