Integral equations on fractal domains: analysis and computation

积分方程是泛函分析数学领域的基本对象。它们也是分析和求解偏微分方程描述的许多物理过程的数学模型的有力工具。特别是，它们广泛应用于声学、电磁和弹性波散射应用，如噪声控制、雷达/声纳/医疗/地震成像、移动通信和气候科学。基于积分方程的计算方法的重要例子包括边界元法和离散偶极近似。现有的用积分方程分析和计算波散射的数学工具只适用于散射体相对简单的情况，具有一定的数学“平滑性”。然而，在许多应用中，散射体可能是高度复杂和极其粗糙的，具有多个长度尺度的微观结构。例子包括树木和植被，建筑立面，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。这种散射体通常被建模为“分形”，在所有长度尺度上表现出自相似性的非光滑数学对象。本项目旨在推广积分方程理论，以便能够处理这种分形散射体。这需要在数学分析和数值近似方面取得进步。该项目将导致：(A)函数空间和积分算子理论的新的数学结果，允许对现有理论范围之外的分形散射问题进行严格的分析；(B)精确有效地求解分形域积分方程的新数值方法，得到系统数学分析的支持（与现有文献不同）。该项目的理论成果将应用于实际相关应用，包括(i)电气工程中的分形天线设计，以及（ii）气象和气候科学中分形大气粒子（雪/冰/尘埃）的光/雷达散射。

Integral equations are fundamental objects in the mathematical area of functional analysis. They are also a powerful tool for analysing and solving mathematical models of many physical processes described by partial differential equations. In particular they are widely used in acoustic, electromagnetic and elastic wave scattering applications such as noise control, radar/sonar/medical/seismic imaging, mobile communications and climate science. Important examples of computational methods based on integral equations include the boundary element method and the discrete dipole approximation.Existing mathematical tools for analysis and computation with integral equations in wave scattering apply only to situations in which the scatterer is relatively simple, possessing a certain degree of mathematical "smoothness". However, in many applications scatterers can be highly complex and extremely rough, with microstructure on multiple lengthscales. Examples include trees and vegetation, building facades, surface of the ocean, certain antenna designs in electrical engineering, and atmospheric particles such as snow/ice crystals and dust aggregates. Such scatterers are often modelled as "fractals", non-smooth mathematical objects exhibiting self-similarity on all lengthscales. This project aims to generalise the theory of integral equations to be able to handle such fractal scatterers. This requires advances in mathematical analysis and numerical approximation. The project will lead to:(A) New mathematical results in the theory of function spaces and integral operators, permitting the rigorous analysis of fractal scattering problems that are beyond the scope of existing theory;(B) New numerical methods for accurately and efficiently solving integral equations on fractal domains, supported (unlike those currently available in the literature) by a systematic mathematical analysis.The theoretical results of the project will be applied to practically relevant applications including (i) fractal antenna design in electrical engineering, and (ii) light/radar scattering by fractal atmospheric particles (snow/ice/dust) in meteorology and climate science.