RUI: Solving Symmetric Banded Linear Systems and Other Problems in Fiber Optic Design

RUI：解决对称带状线性系统和光纤设计中的其他问题

• 批准号: 0611574
• 负责人: Linda Kaufman
• 金额: --
• 依托单位: William Paterson University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-10-01 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0611574&HistoricalAwards=false
• 关键词: RUI; Solving; Symmetric; Banded; Linear

There are two main objectives of this research :(1) analyzing and implementing various versions of the retraction algorithm devised in the Summer of 2005 by the Principal Investigator (PI) for factoring a symmetric band matrix which may be indefinite, and (2) studying methods for solving the inverse problem to determine the chemical composition of a spooled optical fiber with particular optical properties. The retraction algorithm preserves symmetry and the band structure of a matrix and requires in the worst case 2/3 the space of Gaussian elimination for banded unsymmetric matrices and theoretically about 1/2 the operation count. The algorithm uses a sequence of 1 x 1 and 2 x 2 pivots to transform the matrix to block diagonal form, and the elements of the transformed matrix are bounded. One aim of this research is to extend the algorithm to other structured systems. . The symmetric banded factorization may be used within a shift-and- invert Lanczos algorithm for determining eigenvalues, and in fact the fiber optics design problem involved finding several eigenvalues of a Sturm-Liouville problem and was the impetus for searching for an indefinite band symmetric factorization algorithm. In fiber optics design one wishes to solve an inverse problem of determining parameters in Maxwell's equation, a partial differential equation-eigenvalue problem, so that functions of the eigensystem meet certain criteria. One such criterion that needs to be computed is the dispersion, a function of the second derivative of the positive eigenvalues with respect to frequency and its gradient with respect to the design parameters which determine the refractive index profile of the various layers of the fiber. This project involves determining the best method for calculating these quantities for an extended model that takes into consideration fibers that are wrapped around a spool.The properties of an optical fiber are determined by the chemical composition of the layers that compose the fiber. One fiber is not suitable for all situations. For example, one would not use the same fiber for underwater transmission and for a local area network. Until 2000 mathematical models were used only to determine the optical properties of a proposed design. In 2000 the Principal Investigator was part of a team at Bell Labs which decided to invert the process and to predict the chemical composition of the fiber to meet certain optical specifications. The modeling tool required the solution of thousands of systems of linear equations with symmetric and banded matrices. Traditionally one would ignore the symmetry, but taking symmetry into consideration, as in the algorithm recently devised by the PI, decreases the computational requirements for each individual system and provides information (the inertia) that could decrease the number of systems that need to be solved. Solving structured symmetric linear systems is also necessary when modeling the cavity of a linear collider or when modeling buildings, oil platforms and bridges to help prevent serious post-construction events, such as the collapse of the Tacoma Narrows Bridge. An important element of this project will include application-oriented subprojects for undergraduate students to give them the real world design and modeling experiences they would not normally receive in the classroom and which they will then be able to use when they undertake careers as secondary or middle school teachers (math majors) or in local industry (computer majors).

本研究有两个主要目标：（1）分析和实施各种版本的收缩算法设计的2005年夏天的主要研究者（PI）的对称带矩阵，这可能是不确定的，和（2）研究方法，解决反问题，以确定具有特定光学性能的缠绕光纤的化学成分。该算法保留了矩阵的对称性和带状结构，在最坏的情况下，带状非对称矩阵需要2/3的高斯消元空间，理论上需要约1/2的运算次数。该算法利用一系列1 × 1和2 × 2的主元将矩阵变换为块对角形式，变换后的矩阵元素是有界的。本研究的一个目的是将该算法扩展到其他结构化系统。.对称带状因式分解可以在用于确定本征值的移位和反转Lanczos算法内使用，并且实际上光纤设计问题涉及找到Sturm-Liouville问题的几个本征值，并且是寻找不定带状对称因式分解算法的动力。在光纤设计中，人们希望解决确定麦克斯韦方程中的参数的逆问题，即偏微分方程-本征值问题，使得本征系统的函数满足某些准则。需要计算的一个这样的标准是色散，色散是正本征值相对于频率的二阶导数及其相对于确定光纤各层折射率分布的设计参数的梯度的函数。该项目涉及确定计算扩展模型的这些量的最佳方法，该模型考虑了缠绕在线轴上的光纤。光纤的特性由组成光纤的层的化学成分决定。一种光纤并不适用于所有情况。例如，水下传输和局域网不会使用相同的光纤。直到2000年，数学模型仅用于确定拟议设计的光学特性。2000年，贝尔实验室的一个团队决定逆转这一过程，并预测光纤的化学成分，以满足某些光学规格。建模工具需要解决数千个具有对称和带状矩阵的线性方程组。传统上，人们会忽略对称性，但考虑到对称性，如PI最近设计的算法，降低了每个单独系统的计算要求，并提供了可以减少需要解决的系统数量的信息（惯性）。在线性对撞机腔体建模或建筑物、石油平台和桥梁建模时，求解结构对称线性系统也是必要的，以帮助防止严重的施工后事件，如塔科马海峡大桥的倒塌。该项目的一个重要组成部分将包括面向本科生的应用子项目，为他们提供他们通常不会在课堂上获得的真实的世界设计和建模经验，然后他们将能够在担任中学或中学教师（数学专业）或当地工业（计算机专业）时使用。