Analytical and Numerical Methods For Slowly Convergent Integrals and Applications

慢收敛积分在科学和工程中起着重要的作用。当通过迭代、离散化或其他展开方法解决从最简单到最复杂的问题时，它们就会出现。缓慢收敛的积分必须在傅里叶级数中求值，以获得比标准泰勒级数收敛更快的函数近似格式。许多在应用数学和物理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。传统的求积分规则已经不能为缓慢收敛的积分提供精确的近似，因此新的求积分规则受到了人们的高度追捧。最具挑战性的应用涉及计算能量分量（如核吸引能）所需的分子多中心积分。这些积分的计算占用了分子结构计算的40%的壁时间。因此，计算时间的任何减少都将显著影响用于分子结构计算的任何软件的性能。***在这个研究项目中，我们将开发专门用于慢收敛积分的新方法，以克服现有方法面临的困难。我们将介绍计算分子多中心积分需要数百万次的一类球面贝塞尔积分的解析进展。积分的渐近级数表示也将被使用。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。在积分的分析发展将是无果的情况下，例如那些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一个正交规则。梯形法则的双指数变换的最优性清楚地表明了将其应用于这类积分的巨大潜力。梯形定则被认为是Sinc数值方法的副产品，作为一种通用积分器，它具有非常有前途的性质。对于在端点处观察到双指数衰减率的积分，导出了梯形法则的元最优性和指数收敛率。***所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，并将在准确性和效率方面与现有方法进行比较。这个研究项目的结果将对应用数学有帮助；分子物理和化学，振荡积分是很普遍的

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.**