Fast Direct Solvers for Boundary Integral Equations

边界积分方程的快速直接求解器

• 批准号: 0610097
• 负责人: Per-Gunnar Martinsson
• 金额: $ 15.12万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0610097&HistoricalAwards=false
• 关键词: Fast; Direct; Solvers; Boundary; Integral

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.

这项拟议的研究旨在开发快速、准确和健壮的计算技术来解决一类被称为“线性边值问题”的数学方程。这类方程在工程和科学中普遍存在，而求其近似解的任务往往是数值模拟中最昂贵的部分。目前存在多种求解线性边值问题的计算方法，包括一些既高精度又非常快速的计算方法。在过去的二十年里，这些方法的出现极大地提高了我们模拟科学、工程、医学和许多其他领域中复杂现象的能力。然而，现有的高性能计算技术往往在其适用性方面受到限制，而且在某种意义上有些反复无常，因为软件需要针对特定问题进行调整才能很好地执行。这项研究的主要目标是消除某一类高性能技术的这些缺陷，从而使这类算法可用于广泛的重要计算问题。从技术上讲，该研究涉及一类基于将问题表示为计算域边界上的方程的计算技术。众所周知，在某些情况下，所得到的方程可以非常快速地求解。这项任务的现有技术是基于所谓的“迭代求解器”，即构造一系列逐渐接近精确解的近似解。这项拟议的研究旨在开发用于求解边界方程的“直接解算器”。不严格地说，“直接求解者”操纵数学方程，产生一种算法，一下子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，它们似乎昂贵得令人望而却步。然而，最近的发展表明，构造与现有迭代方法一样快的直接方法是可能的，有时甚至比现有的迭代方法更快。开发直接方法来求解与线性边值问题相关的边界方程将带来许多好处，其中包括：(1)能够解决现有快速算法无法解决的某些问题。一个例子是在接近散射体共振频率的波频率下精确解涉及大型物体的电磁和声散射问题。(2)在同一方程需要针对不同数据多次求解的环境中，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，速度有望提高一到两个数量级。(3)高性能计算技术的可用性，这些技术足够健壮，可以合并到通用软件包中。