Problem Of Numerical Integration Research Articles

МЕТОДИКА КРИТЕРІЇВ СУМ У ЗАДАЧАХ ТЕСТУВАННЯ НЕЗАЛЕЖНОСТІ ПОСЛІДОВНОСТЕЙ ВИПАДКОВИХ ЧИСЕЛ

Random and pseudo-random number generators (RNGs) were initially used to solve numerical integration problems (the Monte Carlo method). Currently, the RNGs are used in cryptography and simulation modeling. The latter one typically uses RNGs based on computer algorithms and programs. This article presents a method aimed at testing the independence of random numbers sequences (RNSs). The method is based on the sums properties of independent random variables. Algorithms based on this method operate fast. Here not only the instant statistics including correlation coefficients are analyzed, but also the properties of empirical functions of RNSs distributed sums. In this article, the analysis is limited only to the case of uniformly distributed RNSs. The calculations performed prove the high selective efficiency of the proposed criteria, which allows to reliably distinguish between dependent and independent RNSs. Due to the high operation speed, the proposed algorithms and criteria can be used for testing very long RNSs (especially in Big Data tasks).

Patch area and uniform sampling on the surface of any ellipsoid

Algorithms for generating uniform random points on a triaxial ellipsoid are non-trivial to verify because of the non-analytical form of the surface area. In this paper, a formula for the surface area of an ellipsoidal patch is derived in the form of a one-dimensional numerical integration problem, where the integrand is expressed using elliptic integrals. In addition, analytical formulae were obtained for the special case of a spheroid. The triaxial ellipsoid formula was used to calculate patch areas to investigate a set of surface sampling algorithms. Particular attention was paid to the efficiency of these methods. The results of this investigation show that the most efficient algorithm depends on the required coordinate system. For Cartesian coordinates, the gradient rejection sampling algorithm of Chen and Glotzer is best suited to this task, when paired with Marsaglia’s method for generating points on a unit sphere. For outputs in polar coordinates, it was found that a surface area rejection sampler is preferable.

EXPLICIT FIFTH-ORDER HYBRID METHODS FOR DYNAMICAL SYSTEMS WITH DELAY

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained.
 This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.

New approaches to constructing quadrature formulas for functions with large gradients

The problem of numerical integration of functions with large gradients in the boundary layer is studied. The use of the Newton-Cotes formulas on a uniform grid to integrate such functions can lead to significant errors. The article investigates approaches to the construction of quadrature formulas, the error of which does not increase due to large gradients of the function in the boundary layer. The following approaches are considered: the construction of quadrature formulas that are exact on the singular component responsible for the growth of the function in the boundary layer, the application of the Newton-Cotes formulas on the Shishkin and Bakhvalov meshes, minimization the error of the quadrature formula due to the construction of nodes. The investigated approaches are new, partially reflected in the publications. The results of computational experiments are presented.

APPLICATION OF THE THEORY OF NEW INFORMATION OPERATORS IN CONDUCTING RESEARCH IN THE FIELD OF INFORMATION TECHNOLOGIES

Due to the rapid development and implementation of the latest information technologies in many fields of science and technology significant changes have taken place. In particular, new mathematical theories have emerged; they can be effectively used in building and improving existing mathematical models of various phenomena and objects. Mathematical modeling is currently one of the main methods in scientific research of technological, social, economic processes and natural phenomena. No research thesis is complete without the use of mathematical methods and algorithms. In view of this it is important to acquaint young scientists with modern trends in mathematical modeling during their training for PhD degree, in particular. The development of information technology has contributed to the emergence of new approaches to obtaining, processing and analyzing information in scientific research. In this way, the theory of new information operators, which is widely used in mathematical modeling, was created. The analysis of the existing research reveals the need to consider the theoretical basics of new information operators and their application to the numerical integration of several variables functions. The example of the problem of numerical integration of two variables functions shows the way how to create cubature formulas depending on the type of the input information about function. The paper presents new cubature formulas for the approximate calculation of double integrals, in the case when the information is given by the set of traces on lines and values of the function at points. The proposed formulas have high accuracy but use much less input information compared to classical formulas for numerical integration of double integrals. All the results are supported by numerous examples made in the computer mathematics system Mathcad. The implementation of the developed methodology of using new information operators in scientific research has shown its efficacy during seminars with Master and PhD students.

Asymptotic variance of Newton–Cotes quadratures based on randomized sampling points

AbstractWe consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton–Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton–Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator.

Multiple rotations of Gaussian quadratures: An efficient method for uncertainty analyses in large-scale simulation models

Concerns regarding the impact of climate change, food price volatility, and weather uncertainty have motivated users of simulation models to consider uncertainty in their simulations. One way to do this is to integrate uncertainty components in the model equations, thus turning the model into a problem of numerical integration. Most of these problems do not have analytical solutions, and researchers, therefore, apply numerical approximation methods. This article presents a novel approach to conducting an uncertainty analysis as an alternative to the computationally burdensome Monte Carlo-based (MC) methods. The developed method is based on the degree three Gaussian quadrature (GQ) formulae and is tested using three large-scale simulation models. While the standard single GQ method often produces low-quality approximations, the results of this study demonstrate that the proposed approach reduces the approximation errors by a factor of nine using only 3.4% of the computational effort required by the MC-based methods in the most computationally demanding model.

Quantum Coin Method for Numerical Integration

AbstractLight transport simulation in rendering is formulated as a numerical integration problem in each pixel, which is commonly estimated by Monte Carlo integration. Monte Carlo integration approximates an integral of a black‐box function by taking the average of many evaluations (i.e. samples) of the function (integrand). For N queries of the integrand, Monte Carlo integration achieves the estimation error of . Recently, Johnston [Joh16] introduced quantum super‐sampling (QSS) into rendering as a numerical integration method that can run on quantum computers. QSS breaks the fundamental limitation of the convergence rate of Monte Carlo integration and achieves the faster convergence rate of approximately which is the best possible bound of any quantum algorithms we know today [NW99]. We introduce yet another quantum numerical integration algorithm, quantum coin (QCoin) [AW99], and provide numerical experiments that are unprecedented in the fields of both quantum computing and rendering. We show that QCoin's convergence rate is equivalent to QSS's. We additionally show that QCoin is fundamentally more robust under the presence of noise in actual quantum computers due to its simpler quantum circuit and the use of fewer qubits. Considering various aspects of quantum computers, we discuss how QCoin can be a more practical alternative to QSS if we were to run light transport simulation in quantum computers in the future.

Optimization of nodes of Newton-Cotes formulas in the presence of an exponential boundary layer

The problem of numerical integration of a function of one variable with large gradients in the region of the exponential boundary layer it is studied. The problem is that the use of composite Newton-Cotes formulas on a uniform grid leads to significant errors when decreasing the small parameter ε, regardless of the number of nodes of the basic quadrature formula. In the paper it is proposed to choose nodes based on minimizing the error of the composite Newton-Cotes formula. It is proved that the minimum error is achieved on the Bakhvalov mesh, while the error of the quadrature formula becomes uniform in the small parameter ε.

A New Analysis of Approximate Solutions for Numerical Integration Problems with Quadrature-based Methods

In this paper, we mainly propose the approximate solutions to solve the integration problems numerically using the quadrature method including the Trapezoidal method, Simpson’s 1/3 method, and Simpson’s 3/8 method. The three proposed methods are quite workable and practically well suitable for solving integration problems. Through the MATLAB program, our numerical solutions are determined as well as compared with the exact values to verify the higher accuracy of the proposed methods. Some numerical examples have been utilized to give the accuracy rate and simple implementation of our methods. In this study, we have compared the performance of our solutions and the computational attempt of our proposed methods. Moreover, we explore and calculate the errors of the three proposed methods for the sake of showing our approximate solution’s superiority. Then, among these three methods, we analyzed the approximate errors to prove which method shows more appropriate results. We also demonstrated the approximate results and observed errors to give clear idea graphically. Therefore, from the analysis, we can point out that only the minimum error is in Simpson’s 1/3 method which will beneficial for the readers to understand the effectiveness in solving the several numerical integration problems.

Using Neural Networks for Fast Numerical Integration and Optimization

We present a novel numerical integration technique, Neural Network Integration, or NNI, where shallow neural network design is used to approximate an integrand function within a bounded set. This function approximation is such that a closed-form solution exists to its definite integral across any generalized polyhedron within the network's domain. This closed-form solution allows for fast integral evaluation of the function across different bounds, following the initial training of the network. In other words, it becomes possible to “pre-compute” the numerical integration problem, allowing for rapid evaluation later. Experimental tests are performed using the Genz integration test functions. These experiments show NNI to be a viable integration method, working best on predictable integrand functions, but worse results on singular and non-smooth functions. NNI is proposed as a solution to problems where numerical integrations of higher dimension must be performed over different domains frequently or rapidly and with low memory requirements, such as in real-time or embedded engineering applications. The application of this method to the optimization of integral functions is also discussed.

A Globally Stable Algorithm for the Integration of High-Index Differential-Algebraic Systems

The problem of constraint stabilization and numerical integration for differential-algebraic systems is addressed using Lyapunov theory. It is observed that the application of stabilization methods which rely on a linear feedback mechanism to nonlinear systems may result in trajectories with finite escape time. To overcome this problem, we propose a method based on a nonlinear stabilization mechanism that guarantees the global existence and convergence of the solutions. Discretization schemes, which preserve the properties of the method, are also presented. The results are illustrated by means of the numerical integration of a slider-crank mechanism.

Decomposed optimization time integrator for large-step elastodynamics

Simulation methods are rapidly advancing the accuracy, consistency and controllability of elastodynamic modeling and animation. Critical to these advances, we require efficient time step solvers that reliably solve all implicit time integration problems for elastica. While available time step solvers succeed admirably in some regimes, they become impractically slow, inaccurate, unstable, or even divergent in others --- as we show here. Towards addressing these needs we present the Decomposed Optimization Time Integrator (DOT), a new domain-decomposed optimization method for solving the per time step, nonlinear problems of implicit numerical time integration. DOT is especially suitable for large time step simulations of deformable bodies with nonlinear materials and high-speed dynamics. It is efficient, automated, and robust at large, fixed-size time steps, thus ensuring stable, continued progress of high-quality simulation output. Across a broad range of extreme and mild deformation dynamics, using frame-rate size time steps with widely varying object shapes and mesh resolutions, we show that DOT always converges to user-set tolerances, generally well-exceeding and always close to the best wall-clock times across all previous nonlinear time step solvers, irrespective of the deformation applied.

Efficient temperature field evaluation in wet surface grinding for arbitrary heat flux profile

We consider the heat transfer in surface wet grinding, assuming a constant heat-transfer coefficient over the workpiece surface, as well as the usual heat flux profiles entering into the workpiece given in the literature, i.e. constant, linear, parabolic and triangular. On the one hand, we calculate in the stationary regime, the temperature distribution on the workpiece surface in series form. These series converge only for Biot numbers less than unity. By using convergence acceleration, these series can be computed more rapidly than its equivalent integral form without any appreciable loss of accuracy. Also, we avoid the numerical integration problems found in the expressions given in the literature. Moreover, the expressions found can be used to compute the maximum temperature of the workpiece very rapidly. On the other hand, we have refined some approximations for the relaxation time in wet grinding, and we have derived some new expressions for dry grinding. The relaxation time has been applied to compute the temperature field inside the workpiece in the stationary regime, obtaining a more rapid numerical evaluation without any appreciable loss of precision.

Doubly quasi-consistent fixed-stepsize numerical integration of stiff ordinary differential equations with implicit two-step peer methods

The problem of accurate and efficient numerical integration of ordinary differential equations (ODEs) is a key issue of computation science, which has a valued practical impact in various topics of research. In this respect, effective global error estimation and control strategies are of particular theoretical and practical significance. They allow many applied dynamic systems to be treated numerically for user-supplied accuracy conditions in automatic mode. The property of double quasi-consistency plays an outstanding role in the mentioned topic because the global (or true) error of a doubly quasi-consistent method is asymptotically equal to its local error. Therefore, this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run. The latter was considered to be impossible for decades. The property of double quasi-consistency goes back to 2009, when Kulikov studied it in the first time but failed to find doubly quasi-consistent numerical methods among Nordsieck formulas. Later on, Kulikov and Weiner proved the existence of doubly quasi-consistent numerical schemes among superconvergent explicit two-step parallel peer methods and examined their practical efficiency in comparison to explicit Matlab ODE solvers in 2010. The focus of the present research is on accurate numerical integration formulas for treating stiff ODEs, which often arise in practice and for which explicit methods are shown to be ineffective. Here, we make the first step towards an accurate and efficient numerical solution of stiff ODEs and prove existence of implicit doubly quasi-consistent formulas. We fulfill our investigation of double quasi-consistency within the family of fixed-stepsize implicit two-step peer schemes and construct two methods of convergence orders 3 and 4, which possess excellent stability properties. Our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper.

Pedagogical aspects of using new informative operators for the problems of numerical integration while training Masters of engineeringpedagogical specialties

The experience of many countries indicates that faster proceeding to the new level of development isimpossible without local specialists who have knowledge in the area of advanced scientific achievements.The use of the newest scientific accomplishments complies with the features and nature of futureprofessional activity of teachers-engineers who tend to study all through their professional life. It results inthe necessity to improve and expand theoretical and practical maintenance of academic courses, taking intoaccount general didactic higher education principles, in particular the principle of scientific nature.The principle of scientific nature requires the teacher to know new scientific information, tosystematize it in the curriculum, and to acquaint students with it. Many Ukrainian educational establishmentscarry out training of specialists in engineering; however, the theory of new informative operators is presentedonly in some programs. One of the topical and most often used engineering issues is a problem of numeralintegration. Therefore, it is important to develop theoretical bases for teaching in these fields, in particularfor Masters’ training in such courses as "Theoretical, physical and informative bases of branch knowledge","Mathematical methods of description of processes", "Innovative technologies and resource management inthe industry", "Technology of numeral modeling", that are based on new informative operators.

High regularity partition of unity for structural physically non-linear analysis

Meshfree techniques, such as hp-Clouds and Element Free Galerkin Methods, have been used as attractive alternatives to finite element method, due to the flexibility in constructing conforming approximations. These approximations can present high regularity, improving the description of the state variables used in physically non-linear problems. On the other hand, some drawbacks can be highlighted, as the lack of the Kronecker-delta property and numerical integration problems. These drawbacks can be overcome by using a Ck, k arbitrarily large, partition of unity (PoU) function, built over a finite element mesh, but with the approximate characteristic of the meshfree methods. Here, this procedure is for the first time investigated to simulate the non-linear behavior of structures with quasi-brittle materials. The smeared crack model is adopted and numerical results, obtained with different kinds of polynomial enrichments, are compared with the experimental results.

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 35.

Study on overflow problem of numerical integration for layered elastic half-space systems

Through a theoretical analysis for coefficients in the general expressions of stresses and displacements for a two-layer layered elastic half-space system (LEHS), the paper develops a modified method for these coefficients and new modified general expressions of stresses and displacements for LEHSs. The modified method is then applied to Love’s displacement function. The derived results show that the modified expressions of stresses and displacements for LEHSs no longer contain any positive exponential function. Theoretical derivation and numerical calculation suggest that the modified coefficients tend to zeroth, linear, or quadratic polynomial functions and that the positive exponential functions in the original general expressions of stresses and displacements for LEHSs, which result in numerical overflow, are just a balance to the quick vanishing of some coefficients. The modified method can reasonably avoid the overflow problem in the numerical integration for LEHSs. The numerical verification shows that the modified method is effective and reliable.

Very low truncation dimension for high dimensional integration under modest error demand

We consider the problem of numerical integration for weighted anchored and ANOVA Sobolev spaces of s-variate functions. Here s is large including s=∞. Under the assumption of sufficiently fast decaying weights, we prove in a constructive way that such integrals can be approximated by quadratures for functions fk with only k variables, where k=k(ε) depends solely on the error demand ε and is surprisingly small when s is sufficiently large relative to ε. This holds, in particular, for s=∞ and arbitrary ε since then k(ε)<∞ for all ε. Moreover k(ε) does not depend on the function being integrated, i.e., is the same for all functions from the unit ball of the space.