On the Approximation of Linear Operators with Applications

DMS-9972591项目摘要：莱斯大学A.C. Antoulas关于线性算子的逼近及其应用-----------------------------------------------------------------------本项目的目的是研究线性算子的逼近问题。所考虑的近似方法是基于所讨论的算子的奇异值分解。更准确地说，我们建议研究(i)有限维非结构化算子和（ii）无限维结构化（Hankel）算子的最优和次优逼近的统一框架的存在性。期望本研究能对一类基于奇异值分解的近似问题有所启发。为了预测动力系统（如CD播放机、化学反应或多层建筑）的行为，并根据给定的规范适当地修改（控制）这些行为，需要一个数学模型。通常，这些动力系统的（详细）模型可以用所谓的有限元方法得到。这类模型的最终复杂性非常高。然而，对于仿真和控制目的，都需要低复杂度模型，一个重要的要求是这些简化模型保留原始高复杂度系统的关键特征。这个项目的目的是研究近似方法，产生低复杂性的模型（理论上）有保证的性质。这是在第二阶段解决这种动态系统的控制问题的必要步骤。预计该项目除了在数学上的意义外，还将对几个工程学科的应用问题产生影响，并对高性能计算的理论基础产生影响。互联网教育子项目---------------------------------------互联网的出现从根本上改变了大学的教学方式。在这方面，一个重要的新概念是“异步学习”，学生可以随时随地使用互联网，按照自己的节奏学习。这种学习模式的一个关键因素是可以访问在线练习，可以为学生和教师提供即时评分和反馈。在传统的课程中，当教学转移到一个新的主题时，教师会在稍后的时间里得到学生对某个概念理解程度的反馈，而重新阐述变得很麻烦。这个问题可以通过提供即时评分的在线练习和测试来解决。然而，需要仔细选择练习，以帮助初学者并挑战高级学习者。我们提议在线性代数和动力系统的广泛领域的课程中研究“问题自动生成和评分”的问题。我们将首先回顾和评估几种现有的基于计算机的作业生成和评分方法，例如：(i)伊利诺伊大学开发的“Mallard”教学环境，（ii）格勒诺布尔大学开发的“Autodidact”教学环境；前者是一个通用平台，原则上可以适应处理许多不同的课程，而后者则面向动态系统和控制课程。随后将决定是否采用这些现有平台之一和/或修改程度以满足所选课程的需要。

Abstract of project DMS-9972591: On the approximation of linear operators with applications by A.C. Antoulas, Rice University-----------------------------------------------------------------------The goal of this project is to investigate problems of approximation of linear operators. The approximation method considered is based on the singular value decomposition of the operator in question. More precisely, we propose to study the existence of a unifying framework for the optimal and sub-optimal approximation of (i) unstructured operators in finite dimensions, and (ii) structured (Hankel) operators in infinite dimensions. It is expected that this study will shed light on a large class of singular-value-decomposition based approximation problems.In order to predict the behavior of dynamical systems such as, a compact disc (CD) player, a chemical reaction, or a multi-story building, and appropriately modify (control) this behavior according to given specifications, a mathematical model is needed. Often, (detailed) models for these dynamical systems can be obtained using so-called finite-elementmethods. The resulting complexity of such models is very high. However, for both simulation and control purposes low-complexity models are needed,an important requirement being that these simplified models retain key features of the original high-complexity systems. The purpose of this project is to study approximation methods which yield low-complexity models with (theoretically) guaranteed properties. This is a necessary step towards addressing, in a second phase, the problem of control of such dynamical systems. It is expected that this project, besides its significance for mathematics, will have an impact on application problems arising in several engineering disciplines, and on the theorerical foundations of high performance computing. Sub-project on internet based education---------------------------------------The advent of the internet is radically changing the way teaching is done at universities. An important new concept in this regard is that of "Asynchronous Learning", where students learn anytime, anywhere, and at their own pace, using the internet. A key ingredient in this mode of learing is access to online exercises, with the capability ofinstantaneous grading and feedback both for the student and for the instructor. In traditional courses, the instructor gets feedback on how well students have understood a certain concept at a later time,when instruction has moved on to a new topic, and re-elaboration becomes cumbersome. This issue can be addressed by making available online excercises and tests which are graded instantaneously. Careful selection of the exercises is required however, so as to help the beginner and challenge the advanced learner. We propose to study the issue of "automatic problem generation and grading" for a course in the broad area of linear algebra and dynamical systems. We will start by reviewing and evaluating several existing methods for computer-based generation and grading of assignemnts, for instance: (i) the "Mallard" instructional environment developed at the University of Illinois, (ii) the "Autodidact" instructional environment developed at the University of Grenoble; the former is a general platform which can in principle be adapted to handle many different courses, while the latter is geared towards courses in dynamical systems and control. It will subsequently be decided whether to adopt one of these existing platforms and/or the degree of modification required to address the needs of the chosen course.