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

本研究的两个主要目标是：(1)分析和实现由首席研究员（PI）在2005年夏天设计的各种版本的伸缩算法，用于分解可能是不定的对称带矩阵；(2)研究求解逆问题的方法，以确定具有特定光学特性的卷绕光纤的化学成分。该算法保留了矩阵的对称性和带结构，在最坏情况下只需要高斯消去法2/3的空间，理论上只需要1/2的运算次数。该算法使用1 × 1和2 × 2的轴向序列将矩阵变换为块对角线形式，并且变换后的矩阵的元素是有界的。本研究的目的之一是将该算法推广到其他结构化系统。对称带状分解可以用在移位和反转的Lanczos算法中来确定特征值，实际上光纤设计问题涉及到Sturm-Liouville问题的几个特征值，并且是寻找不定带对称分解算法的动力。在光纤设计中，人们希望解决麦克斯韦方程中确定参数的反问题，即偏微分方程-本征值问题，以使本征系统的函数满足一定的准则。需要计算的一个标准是色散，色散是正本征值关于频率的二阶导数的函数，它的梯度与设计参数有关，设计参数决定了光纤各层的折射率分布。这个项目涉及到为一个扩展模型确定计算这些数量的最佳方法，该模型考虑了缠绕在线轴上的纤维。光纤的性能是由组成光纤的各层的化学成分决定的。一种纤维并不适合所有的情况。例如，人们不会在水下传输和局域网中使用相同的光纤。直到2000年，数学模型只被用来确定一个拟议设计的光学特性。2000年，首席研究员是贝尔实验室团队的一员，该团队决定颠倒这一过程，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，就像PI最近设计的算法一样，减少了每个单独系统的计算需求，并提供了可以减少需要解决的系统数量的信息（惯性）。在对线性对撞机的腔体进行建模或对建筑物、石油平台和桥梁进行建模时，解决结构化对称线性系统也是必要的，以帮助防止严重的施工后事件，例如塔科马海峡大桥的倒塌。这个项目的一个重要元素将包括面向本科生的应用子项目，为他们提供他们通常在课堂上无法获得的真实世界的设计和建模经验，然后他们可以在担任中学教师（数学专业）或当地工业（计算机专业）时使用这些经验。

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).