Analytical and Numerical Methods For Slowly Convergent Integrals and Applications

缓慢收敛积分的分析和数值方法及其应用

• 批准号: RGPIN-2016-04317
• 负责人: Safouhi, Hassan
• 金额: $ 1.6万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=709096
• 关键词: Analytical; Numerical; Methods; Slowly; Convergent

Slowly convergent integrals play a major role in science and engineering. They arise when solving the simplest to the most complicated problems through iterative, discretization or other expansion methods. Slowly convergent integrals must be evaluated in Fourier series to obtain a function approximation scheme that converges faster than the standard Taylor series. Many special functions, which play an extremely important role in applied mathematics and physics, have slowly convergent integral representations. Traditional quadrature rules have failed to provide accurate approximations to slowly convergent integrals, therefore new techniques are highly sought after and desired. The most challenging application concerns the computation of molecular multi-centre integrals needed for the computation of energy components, such as nuclear attraction energy. The computation of these integrals takes as much as 40% of the wall time of molecular structure calculations. Accordingly, any reduction in the calculation time will significantly impact the performance of any software used for molecular structure calculations. In this research program, we will develop new methods specifically for slowly convergent integrals that will overcome the difficulties that existing methods face. We will introduce analytic developments of a class of spherical Bessel integrals required millions of times for the computation of molecular multi-centre integrals. Asymptotic series representations for integrals will also be used. For the summation of the asymptotic series, we will use sequence transformations and convergence acceleration methods. In the case of integrals for which the analytic development will be fruitless, such as those involved in the four-centre molecular integrals, we will focus our efforts on specifically tailoring a quadrature rule to these integrals. The optimality of the double exponential transformation for the trapezoidal rule clearly indicates the tremendous potential in applying it to this kind of integrals. The trapezoidal rule, recognized as a by-product of Sinc numerical methods, has been shown to have extremely promising properties as a general-purpose integrator. The meta-optimality and exponential convergence rate of the trapezoidal rule is derived for integrands that observe a double exponential decay rate at the endpoints. The developed methods will be applied to molecular multi-centre integrals and to other challenging integrals including the incomplete Bessel functions, tail probability distributions, and the generalized hyperbolic distribution. Through these applications, we will assess the proposed methods and will perform comparisons with regard to accuracy and efficiency with the existing ones. The results of this research program will be useful in applied mathematics; molecular physics and chemistry, where oscillatory integrals are prevalent.

缓慢收敛积分在科学和工程中发挥着重要作用。当通过迭代、离散化或其他扩展方法解决最简单到最复杂的问题时，就会出现这种情况。必须在傅里叶级数中评估缓慢收敛的积分，以获得比标准泰勒级数收敛得更快的函数逼近方案。许多在应用数学和物理学中起着极其重要作用的特殊函数都具有缓慢收敛的积分表示。传统的求积规则无法为缓慢收敛的积分提供精确的近似，因此新技术受到高度追捧和渴望。最具挑战性的应用涉及计算能量成分（例如核吸引力能）所需的分子多中心积分的计算。这些积分的计算占用了分子结构计算的 40% 的时间。因此，计算时间的任何减少都会显着影响用于分子结构计算的任何软件的性能。 在本研究计划中，我们将开发专门针对缓慢收敛积分的新方法，以克服现有方法面临的困难。我们将介绍一类需要数百万次计算分子多中心积分的球面贝塞尔积分的解析发展。还将使用积分的渐近级数表示。对于渐近级数的求和，我们将使用序列变换和收敛加速方法。对于解析开发没有结果的积分，例如涉及四中心分子积分的积分，我们将集中精力专门为这些积分定制求积规则。梯形规则的双指数变换的最优性清楚地表明了将其应用于此类积分的巨大潜力。梯形法则被认为是 Sinc 数值方法的副产品，已被证明作为通用积分器具有极有前景的特性。梯形规则的元最优性和指数收敛率是针对在端点处观察到双指数衰减率的被积函数得出的。 开发的方法将应用于分子多中心积分和其他具有挑战性的积分，包括不完全贝塞尔函数、尾部概率分布和广义双曲分布。通过这些应用，我们将评估所提出的方法，并将与现有方法的准确性和效率进行比较。该研究计划的结果将有助于应用数学；分子物理和化学，其中振荡积分很普遍。