Singular and Oscillatory Quadrature on Non-Smooth Domains

数值正交在科学计算、工程和应用科学中有着大量的应用，包括特殊函数的求值，傅里叶变换和拉普拉斯变换的计算，以及ode和偏微分方程求解的数值方法的实现。尽管正交是数值分析中的一个经典课题，但它仍然是一个非常活跃的研究领域。在这个项目中，我们的目标是解决一些关于非光滑域上奇异积分和振荡积分的有效正交规则的发展、分析和实现的基本开放问题。我们希望解决的开放问题是由我们以前和正在进行的声学和电磁波散射计算研究所激发的，例如由亥姆霍兹方程和时谐麦克斯韦方程建模。这是一个非常活跃的研究领域，因为在整个科学技术的许多应用中，包括医学成像，射频和微波通信以及天气/气候预测，都存在重要的散射问题，目前没有令人满意的数值方法。主要的挑战是准确和有效地处理高频问题（波长比散射体小）和非光滑散射体（散射体有多个角、边缘和其他表面不规则）。流行的模拟方法都以数值正交为核心，包括底层偏微分方程的变分公式（导致有限元方法），以及边界和体积积分方程公式（分别导致边界元方法和离散偶极子近似等方法）。现有的这些方法的正交规则只适用于被积分函数（“被积函数”）的情况，以及进行积分的域相对简单，具有一定程度的数学“平滑性”。然而，在上面列出的每一种应用中，人们都会遇到奇异积分（在某些点膨胀到无穷大）和/或高度振荡的积分域，以及高度非光滑的积分域。当散射体比入射波长大时，就会出现前一种情况，而当散射体特别“粗糙”或形状不规则时，就会出现后一种情况，例如树木和植被、建筑物立面、海洋表面、电气工程中的某些天线设计以及大气颗粒（如雪/冰晶和尘埃聚落）的散射。该项目旨在推广数值正交理论，以便能够处理在现实应用中出现的非光滑域上的复杂奇异积分和振荡积分，特别关注大气物理。在这方面，计算大气冰晶辐射散射的工具得到改进，将大大提高目前遥感（从而预报天气）和计算辐射平衡（从而预报气候）的能力。该项目将提供新的正交规则和相关算法，以及保证其准确性和稳定性的新的理论结果。我们将开发用户友好的开源软件，用于正交规则和散射模拟，专为非专业从业者在一系列应用领域设计。

Numerical quadrature has a huge number of applications in scientific computing, engineering and applied sciences, including the evaluation of special functions, the calculation of Fourier and Laplace transforms, and the implementation of numerical methods for the solution of ODEs and PDEs. Despite being a classical topic in numerical analysis, quadrature remains a highly active research area. In this project we aim to address some fundamental open problems concerning the development, analysis and implementation of efficient quadrature rules for singular and oscillatory integrands on non-smooth domains. The open problems we wish to address are motivated by our previous and ongoing research into computational acoustic and electromagnetic wave scattering, as modelled for example by the Helmholtz equation and the time-harmonic Maxwell equations. This is a highly active research area, due to the fact that in many applications throughout science and technology including medical imaging, RF and microwave communications and weather/climate prediction, there are important scattering problems for which no satisfactory numerical method currently exists. The main challenges relate to the accurate and efficient treatment of high frequency problems (where the wavelength is small compared to the scatterer) and non-smooth scatterers (where the scatterer has multiple corners, edges and other surface irregularities). Popular simulation methods, all of which have numerical quadrature at their core, include variational formulations of the underlying PDEs (leading to finite element methods), and boundary and volume integral equation formulations (leading to boundary element method and methods such as the Discrete Dipole Approximation, respectively). Existing quadrature rules available for these methods apply only to situations where the function being integrated (the "integrand''), and the domain over which the integration is carried out, are relatively simple, possessing a certain degree of mathematical "smoothness". However, in each of the applications listed above, one encounters integrands that are singular (blow up to infinity at certain points) and/or highly oscillatory, and integration domains that are highly non-smooth. The former situation arises when the scatterer is large compared to the incident wavelength, and the latter when the scatterer is particularly "rough" or irregular in shape, as is the case e.g. for scattering by trees and vegetation, building facades, the surface of the ocean, certain antenna designs in electrical engineering, and atmospheric particles such as snow/ice crystals and dust aggregates. This project aims to generalise the theory of numerical quadrature to be able to handle the complicated singular and oscillatory integrals over non-smooth domains that arise in real-world applications, with a particular focus on atmospheric physics, where improved tools for computing scattering of radiation by atmospheric ice crystals would significantly improve current capabilities for remote sensing (and hence weather prediction) and the calculation of radiation balances (and hence climate prediction). The project will deliver new quadrature rules and associated algorithms, and new theoretical results guaranteeing their accuracy and stability. We will develop user-friendly open-source software for quadrature rules and scattering simulations, designed for non-expert practitioners in a range of application areas.