On the Approximation of Linear Operators with Applications

DMS-9972591项目摘要：关于线性算子的近似及其应用 关于AC Antoulas，Rice大学-考虑的近似方法是基于奇异值分解的运营商的问题。更确切地说，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）算子的最佳和次佳近似的统一框架的存在。为了预测动态系统（如CD播放器、化学反应或多层建筑）的行为，并根据给定的规范对这些行为进行适当的修改（控制），需要一个数学模型。通常，这些动力系统的（详细）模型可以使用所谓的有限元方法获得。这种模型的复杂性非常高。然而，为了仿真和控制的目的，低复杂度的模型是必要的，一个重要的要求是，这些简化的模型保留原来的高复杂度系统的关键功能。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）保证的属性。这是一个必要的步骤，解决，在第二阶段，这样的动态系统的控制问题。预计这个项目，除了其意义的数学，将产生影响的应用问题中出现的几个工程学科，高性能计算的理论基础。基于互联网的教育子项目-在这方面，一个重要的新概念是“异步学习”，即学生使用互联网随时随地以自己的速度学习。这种学习模式的一个关键因素是在线练习，同时为学生和教师提供即时评分和反馈。在传统的课程中，教师会在稍后的时间得到学生对某个概念的理解程度的反馈，当教学转移到一个新的主题时，重新阐述变得很麻烦。这个问题可以通过提供在线练习和即时评分的测试来解决。然而，需要仔细选择练习，以帮助初学者并挑战高级学习者。我们建议在线性代数和动力系统的广泛领域中研究“自动问题生成和分级”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）伊利诺斯大学开发的“Mallard”教学环境，（ii）格勒诺布尔大学开发的“Autodidact”教学环境;前者是一个通用的平台，原则上可以适用于处理许多不同的课程，而后者则面向动力系统和控制课程。随后将决定是否采用这些现有平台之一和/或为满足所选课程的需要所需的修改程度。

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.