Efficient and highly accurate solvers for integral equations on surfaces with edges and corners

这个项目涉及物理现象的计算机建模。特别是，PI寻求准确地模拟电磁波和声波的行为。这一课题由来已久，之前已经取得了许多重要成果，但目前我们缺乏准确模拟涉及复杂几何图形的情况的能力。这个项目试图通过将来自纯数学的几个观察结果与新的工程方法相结合来解决这些困难。该项目的最终结果将是能够对电磁波和声波进行准确建模的工具。这些工具将适用于许多问题，但PI特别感兴趣的是将它们应用于集成电路分析和生物力学模拟(例如，囊泡流动)。数学物理中的许多偏微分方程式可以有益地重新表述为积分方程组。这些方法在电动力学、流体动力学和弹性力学问题中有重要的应用。然而，大多数应用都涉及到具有奇异性的区域，在这些区域上求解积分方程组时，要达到高精度和高效率是出了名的困难。从原理上讲，在具有边角奇点的曲面上以蛮力方式求解一大类积分方程组应该不会有什么困难。数学物理中许多重要的边值问题都可以用积分算子来表示，这些积分算子在平方整函数空间上是可逆的且条件良好。这类算子的Galerkin离散化是收敛的，并且与基础算子一样具有良好的条件。根据这些观察，假设离散化的所有方面都得到了正确的处理，在足够密集的网格上简单地用多项式基函数局部表示解将导致高度精确的近似。然而，在使用这种蛮力方法的实践中，出现了几个困难的问题；其中最主要的是：(1)密集的网格导致过大的线性系统，即使是现代的O(N)快速求解器也不能解决；(2)表示边缘附近的奇点最好使用高度各向异性的网格，这给目前可用的离散化技术和快速求解器带来了困难；(3)高精度地评估系数矩阵的条目，涉及估计奇异和“接近”的奇异积分，在一般情况下是相当具有挑战性的，并且在角落和边缘区域更是如此。本项目的目标是开发高精度、快速、健壮的积分方程组求解器，以克服这些困难，实现高精度和高效率。它将在不使用先验渐近估计的情况下做到这一点(这在许多感兴趣的情况下是不可用的)。该项目包括三个阶段的方法：(1)PI将实现一个高精度和非常健壮的“蛮力”过程；(2)将部署多种工具，包括本地快速自适应网格生成器和算子压缩技术，以加速暴力求解器；(3)最后，将使用加速求解器构造高效的数值预计算求积公式，该公式表征解的奇异性并替代先验渐近估计。

This project concerns the computer modeling of physical phenomena. In particular, the PI seeks to accurately model the behavior of electromagnetic and acoustic waves. This topic has a long history, and many important results have been previously achieved, but we currently lack the ability to accurately model situations involving complex geometry. This project seeks to address these difficulties by combing several observations from pure mathematics with new engineering approaches. The end result of this project will be tools which allow for the accurate modeling of electromagnetic and acoustic waves. These tools will be applicable to many problems, but the PI is particularly interested in applying them to integrated circuit analysis and biomechanical simulations (for instance, vesicle flows).Many of the partial differential equations of mathematical physics can be profitably reformulated as integral equations. Such methods have important applications to problems in electrodynamics, fluid dynamics and elasticity. However, most applications involve domains with singularity, and it is notoriously difficult to achieve high-accuracy and efficiency when solving integral equations on such domains. In principle, it appears that there should be no difficulty in solving a large class of integral equations given on surfaces with edge and corner singularities in a brute-force fashion. Many important boundary value problems in mathematical physics can be formulated using integral operators which are invertible and well-conditioned on spaces of square integral functions. Galerkin discretizations of such operators converge and are as well-conditioned as the underlying operator. It follows from these observations that, assuming all aspects of discretization are correctly handled, simply representing solutions locally with polynomial basis functions on a sufficiently dense mesh will result in highly accurate approximations. However, several difficult problems arise in practice with this brute-force approach; chief among them are: (1) dense meshes lead to excessively large linear systems that even modern O(N) fast solvers are inadequate to address; (2) representing singularities near edges is best done with highly anisotropic meshes which cause difficulties for currently available discretization techniques and fast solvers; (3) evaluating the entries of coefficient matrices to high accuracy, which involves estimating singular and "nearly" singular integrals, is quite challenging in general and substantially more so near corner and edge regions. The goal of this project is to develop highly-accurate fast robust solvers for integral equations on surfaces with edge and corner singularities which overcome these difficulties and achieve high-accuracy and efficiency. It will do so without the use of a priori asymptotic estimates (which are not available in many cases of interest). The project consists of a three-phased approach: (1) The PI will implement a highly-accurate and very robust "brute-force" procedure; (2) several tools, including local fast adaptive mesh generators and operator compression techniques will be deployed in order to accelerate the brute-force solver; (3) finally, efficient numerically precomputed quadrature formulae, which characterize the singularities of solutions and serve as a substitutes for a priori asymptotic estimates, will be constructed using the accelerated solver.