Mathematical Sciences: Exploitation of Symmetry in Discretization Methods

9403392所有人科学中的许多问题都表现出关于一组变换的对称性。描述这类问题的方程就被称为关于这个对称群的“等变”。本项目的重点是继续研究在此类等变方程的数值解中利用对称性的系统技术。这种利用导致了更稳定的求解方法，并极大地提高了数值效率。研究人员研究的方法之一(对称性约化方法)可以看作是傅立叶变换方法(利用循环群下的对称性)的推广。在其他项目中，研究人员进一步发展了对称性约化方法，并研究了隐藏对称性，仅为‘几乎’等变问题的最优等变预条件，通过对称性的区域约简，非线性等变方程的数值解和分叉现象。初步的数字代码被进一步发展成用于一系列重要应用的代码集合。几何对称性在理论物理和应用数学的某些分支中一直扮演着重要的角色。然而，利用这些对称性直到最近才在数值分析和计算数学领域得到发展。这个项目的目的是使用几何和代数技术来给出有效的算法来解决具有这种对称性的大型数值问题。非常大的问题(通过对称性考虑)被分解为可以在高性能计算的现代体系结构上独立且更有效地处理的较小问题。在工程中(例如，在弹性问题或流动问题中，或在结构力学中)，只有“近乎”等变的问题(即某些对称结构有缺陷)经常发生。有效地利用这些不完善的对称性将减少解决这类问题的计算工作量。这有两个直接的好处：第一，可以更快地解决中等规模的问题，可能是实时的；第二，通过标准方法无法解决的非常大的问题可以用这种对称约化方法解决。

9403392 Allgower Many problems in science display symmetries with respect to a group of transformations. The equations describing such problems are then said to be `equivariant' with respect to this symmetry group. The focus of this project is to continue to study systematic techniques for exploiting symmetry in the numerical solution of such equivariant equations. This exploitation leads to more stable solution methods and to a greatly increased numerical efficiency. One of the methods studied by the investigators (the symmetry reduction method) can be viewed as a generalization of Fourier Transform methods (which exploit symmetry under a cyclic group). Among other projects, the investigators further develop the symmetry reduction method and study hidden symmetries, optimal equivariant preconditioners for problems that are only `nearly'equivariant, domain reduction via symmetries, the numerical solution of nonlinear equivariant equations and bifurcation phenomena. Preliminary numerical codes are further developed into a collection of codes for a range of important applications. Geometric symmetries have always played an important role in theoretical physics and some branches of applied mathematics. However, exploiting these symmetries has only lately been developed in the areas of numerical analysis and computational mathematics. The aim of this project is to use geometric and algebraic techniques to give efficient algorithms for solving large numerical problems that enjoy such symmetries. Very large problems are broken down (via symmetry considerations) into smaller problems that can be handled independently and more efficiently on modern architectures of high-performance computing. Problems that are only `nearly' equivariant (i.e., where certain symmetry structures have imperfections) occur quite frequently in engineering (e.g., in elasticity or flow problems, or in structural mechanics). An efficient exploitation of these imperfect sym metries would reduce the computational effort for solving such problems. This has two direct benefits: firstly, one can solve moderately sized problems much faster, possibly in real time; secondly, very large problems that are inaccessible via standard methods can be solved with this symmetry reduction method.