New Applications and Algorithms that Involve the Kronecker Product

涉及克罗内克积的新应用程序和算法

• 批准号: 9901988
• 负责人: Charles Van Loan
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-09-01 至 2003-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9901988&HistoricalAwards=false
• 关键词: New; Applications; Algorithms; Involve; Kronecker

The Kronecker product has a rich and very pleasing algebra that supports a wide range of fast, elegant, and practical algorithms. Three trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future. First, the application areas where Kronecker products abound are all thriving. These include signal processing, image processing, semidefinite programming, and quantum computing. Second, sparse factorizations and Kronecker products are proving to be the preferred way to look at fast linear transforms. Researchers have taken the Kronecker methodology as developed for the fast Fourier transform and used it to build exciting alternatives. Third, as computers get more powerful, researchers are more willing to entertain problems of high dimension and this leads to Kronecker products whenever low-dimension techniques are "tensored" together. Moreover, Kronecker products are a useful way to keep track of data motion in memory hierarchies that are the hallmark of advanced machines. In light of these developments, the goal of this project is to widen the use of the Kronecker product in numerical linear algebra. The first step is to study the nearest Kronecker product problem because progress in this area will lead to new and interesting ideas for preconditioners. Other applications that require Kronecker product approximations have been identified and many new methods are called for. All of the algorithmic work will culminate in production of Matlab and C++ libraries. The research will heighten the profile of the Kronecker product throughout the field of matrix computations and thereby make it easier for researchers to spot Kronecker "opportunities" in their work. By building an "infrastructure" of effective Kronecker-product algorithms, applications that feature the operation will proliferate at an even greater rate.

Kronecker积有一个丰富而令人愉悦的代数，支持广泛的快速，优雅和实用的算法。科学计算的三个趋势表明，这种重要的矩阵运算在未来将发挥越来越大的作用。首先，克罗内克产品遍布的应用领域都在蓬勃发展。这些包括信号处理，图像处理，半定编程和量子计算。其次，稀疏分解和克罗内克积被证明是查看快速线性变换的首选方法。研究人员已经采用了为快速傅立叶变换开发的克罗内克方法，并使用它来构建令人兴奋的替代方案。第三，随着计算机变得越来越强大，研究人员更愿意考虑高维问题，这导致克罗内克产品时，低维技术“张量”在一起。此外，Kronecker产品是一种有用的方法来跟踪内存层次结构中的数据运动，这是高级机器的标志。鉴于这些发展，本项目的目标是扩大克罗内克积在数值线性代数中的应用。第一步是研究最近的克罗内克积问题，因为这一领域的进展将导致新的和有趣的想法预条件。其他应用程序，需要克罗内克产品近似已被确定和许多新的方法被称为。所有的算法工作将最终产生Matlab和C++库。 这项研究将提高克罗内克积在整个矩阵计算领域的知名度，从而使研究人员更容易在他们的工作中发现克罗内克的“机会”。通过构建有效的克罗内克积算法的“基础设施”，以该操作为特征的应用程序将以更快的速度激增。