Mathematical Sciences: Exploitation of Symmetry in Discretization Methods

曲面和物体上积分方程组和偏微分方程组的边界元和有限元离散通常会导致非常大的线性方程组。如果区域具有任何对称性，通过有效地使用基于群表示理论的技术，可以显著减少数值求解这些系统所涉及的计算量。这种代数方法产生了将原始的大问题分解成许多较小的子问题，每个子问题对应于区域的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，该因子等于组的顺序除以表示的维度。以前关于利用对称性的工作包括将偏微分方程组简化到子域和在对称轴上调整边界条件。这种技术不适用于边界积分法。在这个项目中，建议开发灵活的代码，系统地实现各种对称结构的简化。本课程将进行尊重对称性的离散化的构造以及与稀疏性和并行实现相关的理论研究。科学和工程中的许多问题都有一个在某种程度上对称的表面或物体作为它们的自然域。例如，任何维度的圆、圆盘、球体、区间、正方形和立方体都具有明显的对称性。除了问题域的对称性之外，通常方程本身还反映了问题物理上的对称性。例如，控制热流的微分方程式是对称的，在没有任何外部刺激的情况下，热将不受任何偏好地向各个方向流动。当方程和问题的区域都具有某种对称性时，可以使用代数技术来减少在给定域上数值求解给定问题所涉及的计算工作量。本项目将专注于进行必要的理论研究，并开发在数值分析领域中利用对称性的算法。

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.