Fast Reliable Algorithms for Structured Computations

这个项目涉及的是计算算法的设计，它融合了结构矩阵问题的效率和数值精度的实用优点。速度和精度这两个要求通常被认为是相互竞争的，因此，在许多情况下，设计大规模结构线性矩阵方程的快速和数值可靠的算法一直是一个重要的开放问题。结构化矩阵方程出现在过滤、估计和控制领域的几个应用中。这样，拟议工作的结果将进一步在解决这些领域理论结果与实际结果之间一直存在的一些令人讨厌的差异方面发挥具体作用。这些差异往往是由于有限精度实现的缺陷，特别是在涉及数字和模拟组件互连的系统中，解决或改善其中许多困难的研究早就应该进行了。这项研究将解决将数值精度和结构开发技术结合到一个统一框架中的挑战。该项目将以最近完成的工作为基础，将结果和框架扩展到更广泛类别的结构化矩阵问题，并将探索在过滤、估计和控制方面的应用。

This project deals with the design of computational algorithms that blend together the practical virtues of efficiency and numerical accuracy for matrix problems with structure. The two requirements of speed and accuracy have often been regarded as competitive and, for this reason, the design of fast and numerically reliable algorithms for large-scale structured linear matrix equations has remained a significant open issue in many instances. Structured matrix equations arise in several applications in the disciplines of filtering, estimation, and control. In this way, the results of the proposed work will further play a concrete role in addressing some of the annoying discrepancies that have always existed in these areas between the theoretical findings and the practical results. These discrepancies are often due to the pitfalls of finite-precision implementations, especially in systems that involve interconnections of digital and analog components, and studies to resolve or ameliorate many of these difficulties have long been overdue. This investigation will address the challenge of combining numerical accuracy and structure exploiting techniques into a unifying framework. The project will build on the work recently done, will extend the results and the framework to a wider class of structured matrix problems, and will explore applications in filtering, estimation and control.