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.

Shifted Sobol points and multigrid Monte Carlo simulation

Multidimensional integrals arise in many problems of physics. For example, moments of the distribution function in the problems of transport of various particles (photons, neutrons, etc.) are 6-dimensional integrals. When calculating the coefficients of electrical conductivity and thermal conductivity, scattering integrals arise, the dimension of which is equal to 12. There are also problems with a significantly large number of variables. The Monte Carlo method is the most effective method for calculating integrals of such a high multiplicity. However, the efficiency of this method strongly depends on the choice of a sequence that simulates a set of random numbers. A large number of pseudo-random number generators are described in the literature. Their quality is checked using a battery of formal tests. However, the simplest visual analysis shows that passing such tests does not guarantee good uniformity of these sequences. The magic Sobol points are the most effective for calculating multidimensional integrals. In this paper, an improvement of these sequences is proposed: the shifted magic Sobol points that provide better uniformity of points distribution in a multidimensional cube. This significantly increases the cubature accuracy. A significant difficulty of the Monte Carlo method is a posteriori confirmation of the actual accuracy. In this paper, we propose a multigrid algorithm that allows one to find the grid value of the integral simultaneously with a statistically reliable accuracy estimate. Previously, such estimates were unknown. Calculations of representative test integrals with a high actual dimension up to 16 are carried out. The multidimensional Weierstrass function, which has no derivative at any point, is chosen as the integrand function. These calculations convincingly show the advantages of the proposed methods.

Unified error bounds for all Newton–Cotes quadrature rules

It is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction.

New error bounds for Gauss-Legendre quadrature rules

It is well-known that the remaining term of any n-point interpolatory quadrature rule such as Gauss-Legendre quadrature formula depends on at least an n-order derivative of the integrand function, which is of no use if the integrand is not smooth enough and requires a lot of differentiation for large n. In this paper, by defining a specific linear kernel, we resolve this problemand obtain new bounds for the error of Gauss-Legendre quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function. Some illustrative examples are given in this direction.

Evaluation of peak functions on ultra-coarse grids

Integration of sampled data arises in many practical applications, where the integrand function is available from experimental measurements only. One extensive field of research is the problem of pest monitoring and control where an accurate evaluation of the population size from the spatial density distribution is required for a given pest species. High aggregation population density distributions (peak functions) are an important class of data that often appear in this problem. The main difficulty associated with the integration of such functions is that the function values are usually only available at a few locations; therefore, new techniques are required to evaluate the accuracy of integration as the standard approach based on convergence analysis does not work when the data are sparse. Thus, in this paper, we introduce the new concept of ultra-coarse grids for high aggregation density distributions. Integration of the density function on ultra-coarse grids cannot provide the prescribed accuracy because of insufficient information (uncertainty) about the integrand function. Instead, the results of the integration should be treated probabilistically by considering the integration error as a random variable, and we show how the corresponding probabilities can be calculated. Handling the integration error as a random variable allows us to evaluate the accuracy of integration on very coarse grids where asymptotic error estimates cannot be applied.

Cauchy Integral Formula on the Integration Path C on Singularity in Mechanics of Materials

This paper is a combination of conditions and the knowledge of singular integrals, the integrand function analysis of the deformation, the singularity in the integral for a class on the path integral come up with a complex new formula for the solution,the formula can be used in these areas,such as mechanics of materials.