Numerical linear algebra: large sparse matrix computations.

数值线性代数：大型稀疏矩阵计算。

• 批准号: 9236-2012
• 负责人: Paige, Christopher
• 金额: $ 1.38万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=544614
• 关键词: Numerical; linear; algebra; large; sparse

In order to solve many scientific, engineering, business and other problems using the computer, we often convert the problems to subproblems involving matrices --- rectangular arrays of numbers. Because of this, matrix computations and its theory (often called numerical linear algebra) lies at the heart of most scientific computing. The key to solving many smaller matrix problems is to find some appropriate factorizations of matrices --- so for example a given matrix can be described as the product of two or more such factors. However many problems are large or complicated, and the resulting matrix problems are extremely large --- matrices having millions of rows and millions of columns are not at all unusual. And such large matrices are nearly always very sparse --- the vast majority of elements being zero. Factorization methods can then become far too costly, and iterative methods may be necessary for solving such problems. This research develops numerically reliable and efficient algorithms for iteratively solving such large sparse matrix problems. It carries out analyses of such algorithms to prove their efficiency and reliability, and sensitivity analyses to show what effects changes in the data (for example caused by finite precision computation, or uncertainty in the data) will have on the final results. These analyses lead to greater understanding of individual problems and their computed answers, and improved general purpose and specific area algorithms.

为了使用计算机解决许多科学、工程、商业和其他问题，我们经常将问题转换为涉及矩阵——数字的矩形数组的子问题。因此，矩阵计算及其理论（通常称为数值线性代数）是大多数科学计算的核心。 解决许多较小矩阵问题的关键是找到一些适当的矩阵因式分解——例如，给定的矩阵可以描述为两个或多个此类因式的乘积。 然而，许多问题都很大或很复杂，并且产生的矩阵问题非常大——具有数百万行和数百万列的矩阵并不罕见。 如此大的矩阵几乎总是非常稀疏——绝大多数元素为零。 因式分解方法可能会变得过于昂贵，并且可能需要迭代方法来解决此类问题。 这项研究开发了数值可靠且高效的算法，用于迭代解决此类大型稀疏矩阵问题。 它对此类算法进行分析，以证明其效率和可靠性，并进行敏感性分析，以显示数据变化（例如由有限精度计算或数据不确定性引起的变化）将对最终结果产生何种影响。 这些分析可以更好地理解各个问题及其计算答案，并改进通用和特定区域算法。