Singular and Oscillatory Quadrature on Non-Smooth Domains

数值求积在科学计算、工程和应用科学中有着广泛的应用，包括特殊函数的求值、傅立叶变换和拉普拉斯变换的计算，以及求解常微分方程组和偏微分方程组的数值方法的实现。尽管求积是数值分析中的一个经典话题，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基本的公开问题，涉及到非光滑区域上奇异和振荡被积的有效求积规则的开发、分析和实现。我们希望解决的未决问题是由我们以前和正在进行的关于计算声波和电磁波散射的研究推动的，例如，由Helmholtz方程和时间调和Maxwell方程建模。这是一个非常活跃的研究领域，因为在整个科学和技术的许多应用中，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，目前还没有令人满意的数值方法。主要挑战涉及准确和有效地处理高频问题(与散射体相比波长较小)和非光滑散射体(散射体具有多个角、边缘和其他表面不规则性)。流行的模拟方法都以数值求积为核心，包括基本偏微分方程组的变分公式(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诸如离散偶极子近似的方法)。可用于这些方法的现有求积规则仅适用于被积分的函数(被积函数‘’)和执行积分的域相对简单的情况，具有一定程度的数学“光滑性”。然而，在上面列出的每一个应用中，人们都会遇到奇异的(在某些点处膨胀到无穷大)和/或高度振荡的积分域，以及高度不光滑的积分域。前一种情况发生在散射体与入射波长相比较大时，而后一种情况发生在散射体特别“粗糙”或形状不规则时，例如树木和植被、建筑立面、海洋表面、电子工程中的某些天线设计以及雪/冰晶和尘埃集合体等大气颗粒的散射。该项目旨在推广数值求积理论，使其能够处理现实世界应用中出现的非光滑区域上的复杂奇异积分和振荡积分，特别关注大气物理学，其中计算大气冰晶辐射散射的改进工具将显著提高目前遥感(因此是天气预报)和计算辐射平衡(因此是气候预报)的能力。该项目将提供新的求积规则和相关算法，以及保证其准确性和稳定性的新理论结果。我们将开发用户友好的用于求积规则和散射模拟的开源软件，专为一系列应用领域的非专家从业者设计。

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.