On the Approximation of Linear Operators with Applications

项目摘要-9972591：关于线性算子的逼近及A.C.Antoulas的应用，University-----------------------------------------------------------------------The这个项目的目标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。更确切地说，我们建议研究(I)有限维非结构算子和(Ii)无限维结构(Hankel)算子的最优和次最优逼近的统一框架的存在性。为了预测CD播放机、化学反应或多层建筑等动力系统的行为，并根据给定的规范对这种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)模型可以用所谓的有限元方法得到。由此产生的这类模型的复杂性非常高。然而，为了仿真和控制的目的，需要低复杂度的模型，一个重要的要求是这些简化的模型保留了原始高复杂性系统的关键特征。这个项目的目的是研究近似方法，以产生具有(理论)保证性质的低复杂性模型。这是在第二阶段解决此类动力系统控制问题的必要步骤。预计这个项目除了对数学具有重要意义外，还将对几个工程学科中出现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。基于互联网的子项目education---------------------------------------The互联网的出现正在从根本上改变大学的教学方式。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访问在线练习，能够对学生和教师进行即时评分和反馈。在传统课程中，当教学转移到一个新的主题时，教师会在稍后的时间获得关于学生理解某个概念的程度的反馈，重新阐述变得很麻烦。这个问题可以通过提供在线练习和测试来解决，这些测试和测试可以即时评分。然而，需要仔细选择练习，以帮助初学者和挑战高级学习者。我们建议对线性代数和动力系统这门广泛领域的课程的“自动问题生成和评分”问题进行研究。我们将首先回顾和评价几种现有的以计算机为基础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“自学”教学环境；前者是一个通用平台，原则上可以适用于许多不同的课程，而后者是针对动力系统和控制课程的。随后将决定是否采用这些现有平台中的一种和/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.