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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688716
• 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.*** **

对于用户来说，符号计算通常被视为有两个主要组成部分。首先，符号计算是一个用于数学计算的领域，其中算法在精确（或不精确）算术上工作，其中符号或参数是第一类对象。其次是同样重要的转换符号表达的能力。这个提议包含了这两方面的因素。我特别建议在符号计算的三个不同的子领域进行研究：符号线性代数、数值环境中的稀疏多项式计算和计算机代数的应用。******在符号线性代数领域，我建议研究微分算子矩阵的精确算术算法。我们的算法将集中在系数大小的增长成为一个基本问题的领域。一个目标将是找到快速的算法来计算矩阵的正规形式的矩阵微分算子。这些标准形式在数学、科学和工程应用的各种应用中都很有用。***我们感兴趣的是多元多项式插值中的稀疏问题，以及作为复指数线性混合的信号的稀疏分解。这个问题与平面上的矩形问题密切相关。在这些情况下，有效的算法需要依赖于稀疏表示中的项数，而不是密集表示中可能找到的项数。稀疏表示在应用程序中很常见。将在本提案中解决的问题的示例包括多维稀疏多项式插值问题的数值条件，推广到高维指数插值和替换基的表示。应用包括逆问题，如高维形状矩问题和确定稀疏数据的指数，频率数据。******我们的第三个研究领域不太关注算术，而是更多地关注计算机代数的核心应用。本文提出了两个研究方向：高阶线性微分系统的操作约简算法和有限群作用下多项式和动力系统不变量或需要参数或变量约简的对称约简算法。在第一种情况下，这种算法应该为处理线性微分系统的高阶系统提供一种新的方法。在第二种情况下，算法将影响求解动力系统和多项式方程的多元系统的能力。***软件是我们方法的一部分，因为它使我们能够验证我们的理论与实际实践相一致。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问题解决工具。* * * * *