Elimination and Counting via Thomas Decomposition

通过托马斯分解进行消除和计数

• 批准号: 171336561
• 负责人: Professor Dr. Wilhelm Plesken
• 金额: --
• 依托单位: Lehrstuhl für Algebra und Darstellungstheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171336561?language=en
• 关键词: Elimination; Counting; via; Thomas; Decomposition

The Thomas decomposition based on a formal triangulation algorithm by J. M. Thomas in the 1930s, now implemented in its full generality for the first time, decomposes the set of solutions of a system of equations and inequations into disjoint subsets corresponding to so called simple systems. The system might be either polynomial or polynomial differential. In the polynomial case one obtains a counting polynomial for the solutions, which depends slightly on the chosen coordinate system. Here the challenge is to study this polynomial for generic and nongeneric coordinates (where methods from representation theory of algebraic groups are to be used), to extract more refined combinatorial information from the Thomas decomposition, and to improve the implementation by getting structural insight into the decomposition. In the differential case the aim is to define counting polynomials for free Taylor coefficients via a common generalization of the algebraic case and Janet’s approach to linear PDEs. We also expect applications to the algebraic analysis of numerical methods for solving nonlinear PDEs.

Thomas分解基于J.M.Thomas在20世纪30年代提出的正式三角剖分算法，现在首次实现其完全通用，它将一组方程和不等式的解分解成与所谓的简单系统相对应的不相交的子集。系统可以是多项式的，也可以是多项式的。在多项式的情况下，我们得到解的计数多项式，这与所选择的坐标系略有关系。这里的挑战是研究这种通用和非通用坐标的多项式(其中将使用代数群表示理论的方法)，从Thomas分解中提取更精细的组合信息，并通过对分解的结构洞察来改进实现。在微分情形下，其目的是通过代数情形的共同推广和Janet的线性偏微分方程组的方法来定义自由Taylor系数的计数多项式。我们还期待着它在求解非线性偏微分方程组的数值方法的代数分析中的应用。

项目成果

Algorithmic Thomas decomposition of algebraic and differential systems

• DOI: 10.1016/j.jsc.2011.12.043
• 发表时间: 2011-08
• 期刊: J. Symb. Comput.
• 作者: T. Bächler;V. Gerdt;Markus Lange-Hegermann;D. Robertz

Numerical flux functions for Reynolds‐averaged Navier–Stokes and kω turbulence model computations with a line‐preconditioned p‐multigrid discontinuous Galerkin solver

使用线预处理 pâmultigrid 不连续 Galerkin 求解器进行雷诺平均纳维斯托克斯和 kÏ 湍流模型计算的数值通量函数

• DOI: 10.1002/fld.3702
• 发表时间: 2013
• 期刊: International Journal for Numerical Methods in Fluids
• 影响因子: 1.8
• 作者: M. Wallraff;T. Leicht;M. Lange-Hegermann
• 通讯作者: M. Lange-Hegermann