Экстремальная функция интерполяционных формул в пространстве W2(2,0)

One of the main problems of computational mathematics is the optimization of computational methods in functional spaces. Optimization of computational methods are well demonstrated in the problems of the theory of interpolation formulas. In this paper, we study the problem of constructing an optimal interpolation formula in a Hilbert space. Here, using the Sobolev method, the first part of the problem is solved, i.e., an explicit expression of the square of the norm of the error functional of the optimal interpolation formulas in the Hilbert space W2(2,0) is found. Одна из основных проблем вычислительной математики — оптимизация вычислительных методов в функциональных пространствах. Оптимизация вычислительных методов хорошо проявляется в задачах теории интерполяционных формул. В данной статье исследуется проблема построения оптимальной интерполяционной формулы в гильбертовом пространстве. Здесь с помощью метода Соболева решается первая часть задачи — явное выражение квадрата нормы функционала погрешности оптимальных интерполяционных формул в гильбертовом пространстве W2(2,0) .

Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂t β u = −(− Δ) α/2 u − (− Δ) γ/2 u

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called “double-scale” anomalous diffusion ∂ t β u ( t , x ) = − ( − Δ ) α / 2 u ( t , x ) − ( − Δ ) γ / 2 u ( t , x ) , t > 0 , − 1 < x < 1 , $$\begin{array}{} \displaystyle \partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x), \, \, t \gt 0, \, -1 \lt x \lt 1, \end{array}$$ where ∂ t β $\begin{array}{} \displaystyle \partial_t^\beta \end{array}$ is the Caputo fractional derivative of order β ∈ (0, 1) and 0 < α ≤ γ < 2. We consider a nonlocal inverse problem and show that the fractional exponents β, α and γ are determined uniquely by the data u(t, 0) = g(t), 0 < t ≤ T. The existence of the solution for the inverse problem is proved using the quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input-output mapping is defined and its continuity is established. The uniqueness of the solution for the inverse problem is proved by means of eigenfunction expansion of the solution of the forward problem and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, namely the steepest descent method and a least squares approach, is proposed for the solution of the inverse problem. The numerical method determines the fractional exponents simultaneously. Finally, numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

Method for the Synthesis of Adaptive Algorithms for Estimating the Parameters of Dynamic Systems Based on the Decomposition Principle and the Joint Maximum Methodology

A method of synthesis of a filter for estimating the state of dynamic systems of Kalman type with an adaptive model built on the basis of the principle of decomposition of the system using kinematic relations from the condition of constancy of motion invariants has been developed. The structure of the model is determined from the condition of the maximum function of the generalized power up to a nonlinear synthesizing function that determines the rate of dissipation and, accordingly, the degree of structural adaptation. The resulting model has an explicit relation with the gradient of the estimation error functional, which makes it possible to adapt to the intensity of regular and random influences and can be used to construct a filter for estimating the state of the Kalman structure. On the basis of the developed method, a discrete algorithm is obtained and its comparative analysis with the classical Kalman filter is carried out.

Оптимальные формулы типа Эйлера-Маклорена для численного интегрирования в пространстве Соболева

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N ≥ m-3 and for any m ≥ 4 using S. L. Sobolev method which is based on the discrete analogue of the differential operator d2m/dx2m. In particular, for m = 4 and m = 5 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=6 new optimal quadrature formulas are obtained. At the end of this work some numerical results are presented. В настоящей статье рассматривается задача построения оптимальных квадратурных формул в смысле Сарда в пространстве L2(m)(0,1). Здесь квадратурная сумма состоить из значений подынтегральной функции в узлах и значений первой и третьей производных подынтегральной функции на концах интервала интегрирования. Найдены коэффициенты оптимальных квадратурных формул и вычислена норма оптимального функционала погрешности для любого натурального числа N ≥ m-3 и для любого m ≥ 4, используя метод С. Л. Соболева который основывается на дискретный аналог дифференциального оператора d2m/dx2m. В частности, при m = 4 и m = 5 получен оптимальность классической формулы Эйлера-Маклорена. Начиная с m = 6 получены новые оптимальные квадратурные формулы. В конце работы приведаны некоторые численные результаты.

On Error Estimates in Sp for Cubature Formulas Exact for Haar Polynomials

On the spaces Sp, an upper and lower estimates for the norm of the error functional cubature formulas possessing the Haar d-property are obtained for the n-dimensional case

Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by the middle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.

The order of convergence of an optimal quadrature formula with derivative in the space W(2,1)2

The present work is devoted to extension of the trapezoidal rule in the space W(2,1)2. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the first derivative of an integrand. Using the discrete analog of the operator d2/dx2-1 the explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set F = span{1,x,ex,e-x}. Finally, in the space W(2,1) 2 the square of the norm of the error functional of the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the Euler-Maclaurin quadrature formula on the space L(2)2 .

TOP EVALUATION FOR THE RA TION FOR THE RATE OF FUNC TE OF FUNCTIONAL OF ERROR AL OF ERROR WEIGHT CUBATURE FORMUL TURE FORMULA IN SP A IN SPACE

An upper bound is obtained for the norm of the error functional of the weight cubature formula in the Sobolev space . The modern formulation of the problem of optimization of approximate integration formulas is to minimize the norm of the error functional of the formula on the selected normalized spaces. In these works, the problem of optimality with respect to some definite space is investigated. Most of the problems are considered in the Sobolev space

Some Remarks on a Variational Method for Stiff Differential Equations

We have recently proposed a variational framework for the approximation of systems of differential equations. We associated, in a natural way, with the original problem, a certain error functional. The discretization is based on standard descent schemes, and we can use a variable-step implementation. The minimization problem has a unique solution, and the approach has a global convergence. The use of our error-functional strategy was considered by other authors, but using a completely different way to derive the discretization. Their technique was based on the use of an integral form of the Euler equation for a related optimal control problem, combined with an adapted version of the shooting method, and the cyclic coordinate descent method. In this note, we illustrate and compare our strategy to theirs from a numerical point of view.

Comparison of two neural network approaches to modeling processes in a chemical reactor

In this paper, we conduct the comparative analysis of two neural network approaches to the problem of constructing approximate neural network solutions of non-linear differential equations. The first approach is connected with building a neural network with one hidden layer by minimization of an error functional with regeneration of test points. The second approach is based on a new continuous analog of the shooting method. In the first step of the second method, we apply our modification of the corrected Euler method, and in the second and subsequent steps, we apply our modification of the St?rmer method. We have tested our methods on a boundary value problem for an ODE which describes the processes in the chemical reactor. These methods allowed us to obtain simple formulas for the approximate solution of the problem, but the problem is special because it is highly non-linear and also has ambiguous solutions and vanishing solutions if we change the parameter value.