Fast Direct Solvers for Boundary Integral Equations

提出的研究旨在开发快速，准确和强大的计算技术来解决一类被称为“线性边值问题”的数学方程。这样的方程在工程和科学中无处不在，而找到它们的近似解的任务往往是数值模拟中最昂贵的组成部分。目前存在许多求解线性边值问题的计算技术，其中一些既高精度又非常快速。在过去的二十年里，这些方法的出现极大地提高了我们在科学、工程、医学和许多其他领域模拟复杂现象的能力。然而，现有的高性能计算技术往往在适用性上受到限制，并且在某种意义上，软件需要针对特定问题进行调优才能良好地执行。所提出的研究的主要目标是消除特定类别的高性能技术的这些缺点，从而使此类算法可用于广泛的重要计算问题。从技术上讲，提出的研究涉及一类基于将问题表述为计算域边界上的方程的计算技术。众所周知，所得到的方程在某些环境下可以解得非常快。这项任务的现有技术是基于所谓的“迭代求解器”，它构建了一个逐渐接近精确解的近似解序列。提出的研究旨在开发求解边界方程的“直接求解器”。粗略地说，“直接求解器”操纵数学方程，产生一种算法，从给定的数据中一次性确定未知变量。直接求解器通常比迭代求解器更受欢迎，但它们在许多环境中显得过于昂贵。然而，最近的发展表明，可以构建与现有迭代方法一样快，有时甚至更快的直接方法。发展解决与线性边值问题有关的边界方程的直接方法将带来许多好处；这些包括：(1)解决现有快速算法无法解决的某些问题的能力。一个例子是精确解决涉及大物体的电磁和声散射问题，其波频率接近于散射器的谐振频率。(2)在需要对不同数据多次求解同一方程的环境中，计算速度的提高。涉及生化过程建模和大散射问题的初步实验表明，预计会有一到两个数量级的加速。(3)高性能计算技术的可用性，这些技术足够健壮，可以纳入通用软件包。

The proposed research seeks to develop fast, accurate, and robust computational techniques for solving a class of mathematical equations known as "linear boundary-value problems". Such equations are ubiquitous in engineering and science, and the task of finding approximate solutions to them is frequently the most expensive component of numerical simulations.There currently exists a multitude of computational techniques for solving linear boundary-value problems, including some that are both highly accurate and very fast. The emergence of such methods over the last two decades has vastly increased our ability to simulate complex phenomena in science, engineering, medicine, and many other fields. However, existing high-performance computational techniques tend to be limited in their applicability, and somewhat fickle, in the sense that software needs problem-specific tuning to perform well. The principal goal of the proposed research is to eliminate these drawbacks for a particular class of high-performance techniques, thus making such algorithms accessible to a wide range of important computational problems.Technically speaking, the proposed research is concerned with a class of computational techniques based on formulating the problem as an equation on the boundary of the computational domain. It is known that the resulting equations can in some environments be solved extraordinarily rapidly. Existing techniques for this task are based on so-called "iterative solvers", which construct a sequence of approximate solutions that gradually approach the exact solution. The proposed research seeks to develop "direct solvers" for solving the boundary equations. Loosely speaking, a "direct solver" manipulates the mathematical equation to produce an algorithm that determines the unknown variables from the given data in one shot. Direct solvers are generally preferred to iterative ones, but they have in many environments appeared to be prohibitively expensive. However, recent developments indicate that it is possible to construct direct methods that are as fast as, and sometimes even faster than, existing iterative ones.Many benefits would accrue from the development of direct methods for solving the boundary equations associated with linear boundary-value problems; these include: (1) The ability to solve certain problems that are beyond the reach of existing fast algorithms. An example is the accurate solution of electromagnetic and acoustic scattering problems involving large objects at wave frequencies close to resonant frequencies of the scatterer. (2) An increase in computational speed in environments where the same equation needs to be solved multiple times for different data. Preliminary experiments involving the modeling of biochemical processes and large scattering problems indicate that a speed-up of one or two orders of magnitude is to be expected. (3) The availability of high-performance computational techniques that are sufficiently robust to be incorporated into general purpose software packages.