Fast Direct Solvers for Boundary Integral Equations

拟议的研究旨在开发快速，准确和强大的计算技术，用于解决一类被称为“线性边值问题”的数学方程。 这类方程在工程和科学中无处不在，而寻找近似解的任务往往是数值模拟中最昂贵的部分。目前存在大量求解线性边值问题的计算技术，包括一些既高精度又非常快的技术。 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是有限的，在他们的适用性，有点变化无常，在这个意义上说，软件需要特定问题的调整，以执行良好。 建议的研究的主要目标是消除这些缺点，为特定类别的高性能的技术，从而使这些算法访问到广泛的重要的计算problems.从技术上讲，建议的研究关注一类计算技术的基础上制定的问题作为一个方程的边界上的计算域。 已知所得方程在某些环境中可以非常快速地求解。 现有的这项任务的技术是基于所谓的“迭代求解器”，它构建了一个序列的近似解决方案，逐步接近精确的解决方案。 拟议的研究旨在开发“直接求解器”求解边界方程。 不严格地说，“直接求解器”操纵数学方程以产生一种算法，该算法一次性地从给定的数据中确定未知变量。 直接求解器通常比迭代求解器更受欢迎，但在许多环境中，它们似乎过于昂贵。 然而，最近的发展表明，它是可能的构造直接方法，是一样快，有时甚至比，现有的迭代法，许多好处将产生从直接方法的发展，解决与线性边值问题相关联的边界方程，其中包括：（1）解决某些问题的能力，是超出了现有的快速算法的范围。 一个例子是电磁和声学散射问题的精确解，涉及波频率接近散射体共振频率的大型物体。 (2)在需要针对不同数据多次求解相同方程的环境中，计算速度得到提高。 初步的实验，涉及建模的生化过程和大的散射问题表明，一个或两个数量级的加速是可以预期的。 (3)高性能计算技术的可用性，这些技术足够强大，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.