Numerical linear algebra: large sparse matrix computations.

为了使用计算机解决许多科学、工程、商业和其他问题，我们经常将这些问题转化为涉及矩阵的子问题——矩形数字数组。正因为如此，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。解决许多较小的矩阵问题的关键是找到一些适当的矩阵分解——例如，一个给定的矩阵可以被描述为两个或多个这样的因子的乘积。然而，许多问题都很大或很复杂，由此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。而且这样大的矩阵几乎总是非常稀疏——绝大多数元素都是零。因式分解方法可能会变得过于昂贵，而迭代方法可能是解决此类问题所必需的。本研究为迭代求解此类大型稀疏矩阵问题开发了数值可靠且高效的算法。对这些算法进行分析，以证明它们的效率和可靠性，并进行敏感性分析，以显示数据的变化（例如有限精度计算引起的，或数据的不确定性）对最终结果的影响。这些分析可以更好地理解单个问题及其计算答案，并改进通用和特定区域算法。****本研究的结果对科学计算的大多数实践者是有用的。在加拿大使用这些技术的一些特别重要的应用领域是航空航天工业、气候变化和天气预报、发电和配电以及资源发现和开采。****

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.****The results of the research are useful for most practitioners in scientific computation. Some application areas of particular importance in Canada that use these techniques are the aerospace industry, climate change and weather prediction, power generation and distribution, and resource discovery and extraction.****