High-precision numerical integration: Progress and challenges

One of the most fruitful advances in the field of experimental mathematics has been the development of practical methods for very high-precision numerical integration, a quest initiated by Keith Geddes and other researchers in the 1980s and 1990s. These techniques, when coupled with equally powerful integer relation detection methods, have resulted in the analytic evaluation of many integrals that previously were beyond the realm of symbolic techniques. This paper presents a survey of the current state-of-the-art in this area (including results by the present authors and others), mentions some new results, and then sketches what challenges lie ahead.     

Effect of numerical integration on meshless methods

In this paper, we present the effect of numerical integration on meshless methods with shape functions that reproduce polynomials of degree k1. The meshless method was used on a second order Neumann problem and we derived an estimate for the energy norm of the error between the exact solution and the approximate solution from the meshless method under the presence of numerical integration. This estimate was obtained under the assumption that the numerical integration scheme satisfied a form of Green’s formula. We also indicated how to obtain numerical integration schemes satisfying this property.     

Simpson's rule of discretized Feynman path integration

We suggest a Simpson's rule for discretized Feynman path integral approximation of density matrix element. For a class of bounded below potential functions, we rigorously establish the error boundO(1/N ²) for itsN-step discretized representation. As a model, we use harmonic oscillator to compare the Simpson's rule with the conventional trapezoidal rule.     

A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

We develop a numerical method to price discrete barrier options on an underlying described by the constant elasticity of variance model with jump-diffusion (CEVJD). In particular, the partial integro differential equation associated to this model is discretized in time using an operator splitting scheme whose accuracy is enhanced by repeated Richardson extrapolation. Such an approach allows us to approximate the differential terms and the jump integral by means of two different numerical techniques. Precisely, the spatial derivatives, which exist only in the weak sense, are discretized using a finite element method based on piecewise quadratic polynomials, whereas the jump integral is directly collocated at the mesh points, so that it can be easily evaluated by Simpson numerical quadrature. As shown by extensive numerical simulation, the proposed approach is very efficient from the computational standpoint, and performs significantly better than the finite difference scheme developed in Wade et al. [On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158].     

Assessment and improvement of precise time step integration method

In this paper, the numerical stability and accuracy of Precise Time Step Integration Method are discussed in detail. It is shown that the method is conditionally stable and it has inherent algorithmic damping, algorithmic period error and algorithmic amplitude decay. However for discretized structural models, it is relatively easy for this time integration scheme to satisfy the stability conditions and required accuracy. Based on the above results, the optimum values of the truncation order L and bisection order N are presented. The Gauss quadrature method is used to improve the accuracy of the Precise Time Step Integration Method. Finally, two numerical examples are presented to show the feasibility of this improvement method.     

NICE—An explicit numerical scheme for efficient integration of nonlinear constitutive equations

The paper presents a simple but efficient new numerical scheme for the integration of nonlinear constitutive equations. Although it can be used for the integration of a system of algebraic and differential equations in general, the scheme is primarily developed for use with the direct solution methods for solving boundary value problems, e.g. explicit dynamic analysis in ABAQUS/Explicit. In the developed explicit scheme, where no iteration is required, the implementation simplicity of the forward-Euler scheme and the accuracy of the backward-Euler scheme are successfully combined. The properties of the proposed NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, are compared with the properties of the classical forward-Euler scheme and backward-Euler scheme. For this purpose two highly nonlinear examples, with the von Mises and GTN material model considered, have been studied. The accuracy of the new scheme is demonstrated to be at least of the same level as experienced by the backward-Euler scheme, if we compare them on the condition of the same CPU time consumption. Besides, the simplicity of the NICE scheme, which is due to implementation similarity with the classical forward-Euler scheme, is its great Advantage.     

粒子群和蚁群融合算法的自主清洁机器人路径

为了克服粒子群算法和蚁群算法的缺陷,将改进的粒子群算法和蚁群算法进行融合,形成了PAAA算法,并将此算法应用于自主清洁机器人行为路径的仿真实验。结果表明：PAAA在求解性能上优于粒子群算法,在时间效率上优于蚁群算法。     

数控加工中路径规划与速度插补综述

数控技术在现代制造工业中被广泛使用,相关研究一直为学界和业界共同关注。数控技术的传统流程主要包含刀具路径规划和进给速度插补。为实现高速高精加工,人们通常将路径规划与速度插补中的若干问题转换成数理优化模型,针对工程应用问题的复杂性,采用分步迭代优化的思路进行求解,但所得的结果往往只是局部最优解。其次,路径规划与速度插补都是为了加工一个工件曲面,分两步进行处理虽然简化了计算,但也导致不能进行整体优化。因此,为了更好地开展路径规划与速度插补一体化设计与全局最优求解的研究,系统性地了解并学习已有的代表性工作是十分有必要的。所以将逐次介绍数控加工中刀具路径规划与速度插补的相关方法与技术进展,包括基于端铣的加工路径规划;刀轴方向优化;G代码加工以及拐角过渡;参数曲线路径的进给速度规划等国内外相关研究以及最新提出的一些新型加工优化方法。     

Gaussian integration with rescaling of abscissas and weights

An algorithm for integration of the polynomial functions with a variable weight is considered. It provides an extension of the Gaussian integration, with appropriate scaling of the abscissas and weights. In a first step, orthogonal polynomials are computed for a fixed a=1. Then, using approximate scaling, the initial guess is constructed for a≠1. Finally, numerical values of the abscissas and weights are refined, solving polynomial system using Newton–Raphson method. The final form of the algorithm provides good alternative to usually adopted interval splitting, automatically avoiding problems with limiting values of parameter present in the weight function. Construction of the method requires arbitrary precision arithmetic and special functions, polylogarithms in particular. The final form of the algorithm can be coded using machine precision floating point numbers and standard mathematical library.     

室内外一体化最优路径分析算法实现

为了弥补传统路径导航服务在室内立体空间方面的不足,提出了一种室内外一体化的网络数据模型和最优路径分析解决方案。以几何网络模型为基础,设计了一种楼层数据偏移策略,实现室内三维空间路径拓扑模型快速构建和二维可视化表达。对开源pgRouting库内置的高效Dijkstra路径查询函数进行扩展,实现了基于PostgreSQL库的任意两点之间最优路径和转弯方向语义信息查询。最后,利用GeoServer和OpenLayers等开源软件开发了室内外一体化路径查询原型系统,并采用大规模室内外一体化路径网络模型数据进行测试,定性与定量分析对比结果验证了该方法的正确性和高效性。该方法能够最大化兼容城市交通网络数据和成熟的最短路径分析算法,具有普适性与实用性。     

The path and location planning of workpieces by genetic algorithms

This paper presents the planning of a near-optimum path and location of a workpiece by genetic algorithms. The purpose of this planning is to minimize the processing time required for a robot to complete its work on a workpiece. The location of the workpiece can be anywhere by translating it along any direction and by rotating it about the fixedz-axis of the robot coordinate system. Owing to the changeable location of the workpiece and the alterable motion time required for a robot to move between two workpoints, the path and location planning problem is much more complicated than the travelling salesman problem. It is definitely impossible to obtain an optimum path and location within an acceptable time. In this paper, genetic algorithms are applied to solve this problem. The location of the workpiece is defined by three position parameters and one angular parameter, and the path is determined based on the values of the parameters for all workpoints. All the path and location parameters are encoded into a binary string. They are modified simultaneously by genetic algorithms to search for a global solution. As the workpiece can be anywhere, a penalty function is used to prevent the selection of illegal paths. Two experiments are given to show the performance of genetic algorithms: one has 30 workpoints and the other has 50 workpoints. Compared with four human-generated plannings, planning by genetic algorithms has much better performance in minimizing the processing time.     

Fast fitting of joint models for longitudinal and event time data using a pseudo-adaptive Gaussian quadrature rule

Joint models for longitudinal and time-to-event data have recently attracted a lot of attention in statistics and biostatistics. Even though these models enjoy a wide range of applications in many different statistical fields, they have not yet found their rightful place in the toolbox of modern applied statisticians mainly due to the fact that they are rather computationally intensive to fit. The main difficulty arises from the requirement for numerical integration with respect to the random effects. This integration is typically performed using Gaussian quadrature rules whose computational complexity increases exponentially with the dimension of the random-effects vector. A solution to overcome this problem is proposed using a pseudo-adaptive Gauss-Hermite quadrature rule. The idea behind this rule is to use information for the shape of the integrand by separately fitting a mixed model for the longitudinal outcome. Simulation studies show that the pseudo-adaptive rule performs excellently in practice, and is considerably faster than the standard Gauss-Hermite rule.     

Optimal and collision free tool posture in five-axis machining through the tight integration of tool path generation and machine simulation

The generation of collision free NC-programs for multi-axis milling operations is a critical task, which leads to multi-axis milling machines being exploited below their full capacities. Today, CAM systems, generating the tool path, do not take the multi-axis machine movements into account. They generate a multi-axis tool path, described by a sequence of tool postures (tool tip+tool orientation), which is then converted by a NC-postprocessor to a machine specific NC-program. As the postprocessing is normally done in batch mode, the NC-programmer does not know how the machine will move and the chance for having collisions between (moving) machine components is often very high. The execution of a machine test run or the application of a machine simulation system (NC-simulation) is the only solution to inform the NC-programmer about possible machine collisions during operation.This paper describes a multi-axis tool path generation algorithm where the tool orientation is optimised to avoid machine collisions and at the same time to maximise the material removal rate along the tool track. To perform efficient collision avoidance, the tool path generation module (traditional CAM), the postprocessing (axes transformation) and machine simulation has been integrated into one system. Cutting tests have been carried out to define the allowable tool orientation changes for optimisation and collision avoidance without disturbing the surface quality.The developed multi-axis tool path generation algorithm is applicable for the machining of several part surfaces within one operation. This, together with tool path generation functionality to adapt the tool orientation for both, maximal material removal and avoidance of collisions between (moving) machine components, are the innovative aspects of the presented research work.     

Theory and implementation of path planning by negotiation for decentralized agents

This paper presents a cooperative decentralized path-planning algorithm for a group of autonomous agents that provides guaranteed collision-free trajectories in real-time. The algorithm is robust with respect to arbitrary delays in the wireless traffic, possible sources being transmission time and error correction. Agents move on reserved areas which are guaranteed not to intersect, therefore ensuring safety. A handshaking procedure guarantees recent information states for the agents. Conflicts between agents are resolved by a cost-based negotiation process. The basic algorithm is augmented by the introduction of waypoints, which increase performance at the cost of additional wireless traffic. An implementation of the algorithm is tested in simulation and successfully applied to a real system of autonomous robots. The results are presented and discussed.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.