Mathematical Sciences: Numerical Analysis and Software for Integral Equations in Three Dimensions

小行星9403589 建议的重点是边界积分方程（BIE）的数值解的拉普拉斯方程的三维重建。 这项工作将涉及两个正在使用的数值方法和一个大型计算机包（BIEPACK）的进一步发展，以解决BIE在三维表面上的理论知识的发展。 要研究的数值分析问题包括进一步发展理论的逼近表面，数值积分的奇异和非奇异函数的表面，使用分级网格近似功能，表现不佳的边缘和角落的表面，和迭代求解的大型密集的线性方程组时出现的离散BIE。 此外，涉及使用和数值解的BIE的新课题将被考虑。 其中包括：（1）在解拉普拉斯方程的狄利克雷和诺依曼问题中使用第一类边界积分方程;（2）超奇异边界积分方程的近似和解;（3）使用“聚类方法”来减少作为大多数数值方法的一部分的数值积分和矩阵向量乘法的成本。 这些研究的许多结果将被纳入目前的包BIEPACK，既改善其目前的计划，并将其扩展到新的BIE。 还计划将为国际积分方程组开发的数值方法应用于计算机图形学中产生的积分方程的求解，特别是“绘制方程”。 许多工程问题涉及到一个数学方程的解决方案称为“拉普拉斯方程”或扰动它;和名称“位理论”的研究和应用拉普拉斯方程和密切相关的方程。 拉普拉斯方程本身出现在诸如流体力学（无旋不可压缩流体流动）、传热、重力场、静电场等应用中。 拉普拉斯方程的变体出现在流体力学（斯托克斯流）、线性弹性理论、电磁波和声波传播以及其他主题中。 拉普拉斯方程的解是一个定义在空间某个区域上的所有点的函数，而我们的目标就是找到这个函数。 在三维中，这可能是一个非常耗时的过程。 许多分析和数值技术被用来解决拉普拉斯方程和相关方程，包括有限差分法，有限元法和边界积分方程方法。 后者的优点之一是，该问题被重新定义为一个新的方程的边界上的原始区域上定义的拉普拉斯方程的兴趣，这往往可以导致节省时间在整个解决方案的过程。 这个新问题被称为“边界积分方程”或BIE。 本研究的目的是改善数值方法的数值解的BIE，以及发展一个进一步的理论理解的数值方法求解BIE。 这些结果将被用来改进一个软件包，称为BIEPACK，已开发的数值解的拉普拉斯方程的许多BIE。 还计划将为边界积分方程开发的数值方法应用于计算机制图中出现的积分方程的求解。

9403589 Atkinson The focus of the proposal is the numerical solution of boundary integral equation (BIE) reformulations of Laplace's equation in three dimensions. The work will involve both the development of further theoretical knowledge on the numerical methods being used and the further development of a large computer package (BIEPACK) to solve BIE on surfaces in three dimensions. The numerical analysis problems to be studied include the further development of theory for the approximation of surfaces, the numerical integration of singular and nonsingular functions over surfaces, the use of graded meshes to approximate functions that are ill-behaved around edges and corners of the surface, and the iterative solution of the large dense linear systems of equations which arise when discretizing the BIE. In addition, new topics involving the use and numerical solution of BIE are to be considered. These include: (1) the use of BIE of the first kind in solving the Dirichlet and Neumann problems for Laplace's equation; (2) the approximation and solution of hypersingular BIE; and (3) the use of "clustering methods" to reduce the cost of the numerical integration and matrix-vector multiplications that are a part of most numerical methods. Many of the results of these studies will be incorporated into the present package BIEPACK, to both improve its present programs and to extend it to new BIE. It is also planned to apply numerical methods that were developed for BIE to the solution of integral equations arising in computer graphics, and in particular, the "rendering equation". Many engineering problems involve the solution of a mathematical equation called "Laplace's equation" or perturbations of it; and the name "potential theory" is given to the study and application of Laplace's equation and closely related equations. The Laplace equation itself occurs in applications such as fluid mechanics (irrotational incompressible fluid flows), heat transfer, gravit ational fields, electrostatic fields, and others. Variations on Laplace's equation occur in fluid mechanics (Stokes' flows), linear elasticity theory, electromagnetic and acoustic wave propagation, and other topics. The solution of Laplace's equation is a function defined for all points of some region in space, and the objective is to find this function. In three dimensions, this can be a very time-consuming process. Many analytical and numerical techniques are used to solve Laplace's equation and related equations, including finite difference methods, finite element methods, and boundary integral equation methods. One of the advantages of the latter is that the problem is re-defined as a new equation over the boundary of the original region on which the Laplace equation of interest is defined, and this can often lead to time savings in the overall solution process. The new problem is called a "boundary integral equation" or BIE. The intention of the present study is to improve numerical methods for the numerical solution of BIE, as well as to develop a further theoretical understanding of numerical methods for solving BIE. These results will be used to improve a package, called BIEPACK, that has been developed for the numerical solution of many BIE for Laplace's equation. It is also planned to apply numerical methods that were developed for BIE to the solution of integral equations arising in computer graphics.