Mathematical Sciences: Exploitation of Symmetry in Discretization Methods

数学科学：离散化方法中对称性的利用

• 批准号: 9104058
• 负责人: Eugene Allgower
• 金额: $ 28.4万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-15 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9104058&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Exploitation; Symmetry; Discretization

Boundary element and finite element discretizations of integral equations and partial differential equations over surfaces and bodies often lead to very large linear systems of equations. If the domain enjoys any symmetry, the computational effort involved in numerically solving these systems can be significantly reduced via the effective use of techniques based upon group representation theory which have recently been given by the investigators. This algebraic approach yields a decomposition of the original large problem into a number of smaller subproblems, each corresponding to an irreducible representation of the symmetry group of the domain. The size of each reduced problem is that of the large problem divided by a factor equal to the order of the group divided by the dimension of the representation. Previous work on exploiting symmetry has involved reducing partial differential equations to subdomains and adapting boundary conditions over the symmetry axes. Such techniques do not apply to boundary integral methods. It is proposed in this project to develop flexible codes which systematically implement reductions for a variety of symmetry structures. Constructions of symmetry-respecting discretizations and theoretical investigations relating to sparseness and parallel implementations will be carried out. Many problems in science and engineering have as their natural domain a surface or body which is symmetric in some way. For example, circles, circular disks, spheres, intervals, squares, and cubes, of any dimension, all have obvious symmetry. In addition to the symmetry of the domain of the problem, often the equations themselves reflect a symmetry in the physics of the problem. For example, the differential equations governing heat flow are symmetric, in the sense that heat will flow in all directions without any preferences in the absence of any outside stimulus. When both the equations and the domain of the problem enjoy some symmetry, algebraic techniques may be used to reduce the computational effort involved in numerically solving the given problem on the given domain. This project will focus on making the necessary theoretical investigations and developing algorithms for exploiting symmetry in the field of numerical analysis.

曲面和物体上的积分方程和偏微分方程的边界元和有限元离散化常常导致非常大的线性方程组。如果域具有任何对称性，则通过有效使用基于研究人员最近给出的群表示理论的技术，可以显著减少数值求解这些系统所涉及的计算工作量。这种代数方法将原来的大问题分解为许多较小的子问题，每个子问题对应于域对称群的不可约表示。每个简化问题的大小是大问题的大小除以一个因子，这个因子等于组的顺序除以表示的维数。以前利用对称性的工作包括将偏微分方程简化为子域和在对称轴上调整边界条件。这种方法不适用于边界积分方法。在这个项目中提出了开发灵活的代码，系统地实现各种对称结构的约简。关于对称的离散化的构造和关于稀疏和并行实现的理论研究将被进行。科学和工程中的许多问题都有一个表面或物体在某种程度上是对称的。例如，圆、圆盘、球体、间隔、正方形和立方体，任何维度，都具有明显的对称性。除了问题域的对称性外，通常方程本身也反映了问题物理中的对称性。例如，控制热流的微分方程是对称的，也就是说，在没有任何外界刺激的情况下，热量会向各个方向流动，没有任何偏好。当方程和问题的域都具有一定的对称性时，可以使用代数技术来减少在给定域上数值解决给定问题所涉及的计算工作量。该项目将集中于进行必要的理论研究和开发算法，以利用数值分析领域的对称性。