Symbolic linear algebra, symbolic-numeric computation and applications

符号线性代数、符号数值计算及应用

• 批准号: RGPIN-2015-04168
• 负责人: Labahn, George
• 金额: $ 3.13万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612815
• 关键词: Symbolic; linear; algebra; symbolic; numeric

For users symbolic computation is often viewed as having two main components. In the first instance, symbolic computation is a field for mathematical computation where algorithms work over exact (or inexact) arithmetic and where symbols or parameters are first class objects. Second is the equally important ability to transform symbolic expressions. This proposal involves elements of both. Specifically  I propose research in three distinct subareas of symbolic computation:  symbolic linear algebra, sparse polynomial computation in numerical environments and applications of computer algebra. In the area of symbolic linear algebra I propose to study algorithms for exact arithmetic of matrices of differential operators.  Our algorithms will focus on those domains  where growth of coefficient size becomes a fundamental concern. One goal will be to find fast algorithms for computation of matrix normal forms  for matrices of differential operators.  These normal forms are useful in a variety of applications  of mathematical, scientific and  engineering applications.   We are interested in sparse problems in multivariate polynomial interpolation along with the sparse decomposition of a signal built as a linear mixture of complex exponentials. This problem is closely connected to the shape from moments problem in the plane. In these cases efficient algorithms need to depend on the number of terms in the sparse representation rather than the possible number of terms found in a dense representation. Sparse representations are common in applications. Examples of of problems which will be addressed in this proposal include the numeric conditioning of  multi-dimensional sparse polynomial interpolation problems, generalizations to higher dimensional exponential interpolation and representations in terms of alternate bases. Applications include inverse problems such as the higher dimensional shape from moments problem and determination of sparse data in exponential, frequency data.   Our third research area focuses less on arithmetic but more on core  applications of computer algebra. Here we mention two directions for research : algorithms for manipulation and reduction of  higher order linear differential systems and algorithms for symmetry reduction of polynomial and dynamical systems  invariant under finite group actions or which require parameter or variable reduction.  In the first case  such algorithms should provide a new way for dealing with higher order systems of linear differential systems. In the second case the algorithms will impact the ability of solve dynamical systems and multivariate systems of polynomial equations.  Software is part of our approach, as this enables us to verify that our theory agrees with actual practice. The long term goal is to ensure that symbolic computation and symbolic algorithms become more important problem solving tools for scientists and engineers.

对于用户，符号计算通常被视为具有两个主要组件。首先，符号计算是用于数学计算的字段，其中算法可以通过精确（或不精确）算术和符号或参数为第一类对象。第二是转化符号表达式的同样重要的能力。该建议涉及两者的要素。 具体而言，我在三个不同的符号计算下群体中进行了研究：符号线性代数，数值环境中的稀疏多项式计算以及计算机代数的应用。 在符号线性代数I的领域，提出了研究算法的精确算法，以确切的差异算子的材料。我们的算法将集中在核心规模增长成为基本问题的领域上。一个目标是找到用于计算差异操作员材料的基质正常形式的快速算法。这些正常形式在数学，科学和工程应用的各种应用中很有用。 我们对多元多项式插值中的稀疏问题感兴趣，以及作为复杂指数的线性混合而构建的信号的稀疏分解。这个问题与飞机上的矩问题紧密连接到形状。在这些情况下，有效的算法需要取决于稀疏表示中的术语数量，而不是在密集表示中发现的术语数量。稀疏表示在应用中很常见。本提案中将解决的问题的示例包括多维稀疏多项式插值问题的数字条件，对较高维指数插值的概括以及在替代基础方面的表示。应用程序包括逆问题，例如矩问题较高的维度形状，并确定指数，频率数据中的稀疏数据。 我们的第三个研究领域的重点较少，而更多地关注计算机代数的核心应用。在这里，我们提到了两个研究方向：用于操纵和减少高阶线性差异系统和算法的算法，用于对称降低多项式和动态系统在有限的组动作下不变或需要降低参数或可变的降低。在第一种情况下，这种算法应为处理线性差异系统的高阶系统提供一种新的方法。在第二种情况下，算法将影响求解多项式方程的动态系统和多元系统的能力。 软件是我们方法的一部分，因为这使我们能够验证我们的理论是否与实际实践一致。长期目标是确保符号计算和符号算法成为科学家和工程师更重要的解决问题工具。