Special Matrix Problems in Signals, Systems and Control

在过去的几十年里，线性代数在信号、系统和控制领域取得的进展中发挥了重要作用。最深刻的影响是在计算和实现方面，在这些方面，数值线性代数算法强烈影响了解决问题的方式。诸如向量处理器和分布式处理器阵列等特殊计算体系结构的出现也强调了用于这些应用领域的基本线性代数模块的并行和实时处理。这个项目是关于数值线性代数和在信号、系统和控制中的应用，特别强调在并行体系结构上的实现方面。重点是“特殊矩阵”问题，即稀疏、图案化或结构化的矩阵。对于这类矩阵，最近引入了数值稳定性和灵敏度的适当定义。上述应用领域的数值方法应该在有限意义上具有数值稳定性，同时应该利用矩阵的结构来提高计算复杂性和并行化能力。这将被应用于这些领域的挑战性问题，如目标跟踪、鲁棒控制和大型对象的模型降阶。

During the last few decades linear algebra has played a fundamental role in advances being made in the area of signals, systems and control. The most profound impact has been in the computational and implementational aspects, where numerical linear algebraic algorithms have strongly influenced the ways in which problems are being solved. The advent of special computing architectures such as vector processors and distributed processor arrays has also emphasized parallel and real-time processing of basic linear algebra modules for these application areas. This project is about numerical linear algebra and applications in signals, systems and control, with special emphasis on implementation aspects on parallel architectures. The focus is on "special matrix" problems, i.e. matrices which are either sparse, patterned or structured. For such matrices appropriate definitions of numerical stability and sensitivity have recently been introduced. Numerical methods for the above application areas ought to be numerically stable in this restricted sense and at the same time ought to exploit the structure of the matrices to improve computational complexity and parallelizability. This will be applied to challenging problems in these areas such as target tracking, robust control and model reduction of large scale plants.