Mathematical Sciences: Preconditioning by Fast Transforms

数学科学：通过快速变换进行预处理

• 批准号: 8703313
• 负责人: Gilbert Strang
• 金额: $ 18万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703313&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Preconditioning; Fast; Transforms

The analytical problem in this proposal is the relation between a given operator and its preconditioner. Given is a differential or integral or convolution operator with general boundary conditions. The preconditioner has particularly convenient domain and boundary conditions. After discretization it can be inverted by the Fast Fourier Transform, or in the case of the Dirichlet problem by a Fast Sine Transform. In the matrix case one applies the preconditioned conjugate gradient method and the distribution of the eigenvalues is crucial. A small number of extreme eigenvalues is acceptable if all other eigenvalues are clustered - that is what appeared in numerical experiments. The conjecture on clustering depends on establishing that the eigenvectors decay exponentially, away from the boundary. The analysis should connect problems in operator theory to problems in numerical analysis, and may give a new direction in the class of equations for which preconditioning is essential. This research falls into the general area of numerical analysis, numerical computing, computational mathematics and software design. The problem of Toeplitz operators is important in signal processing. The problem of numerical computations associated with Toeplitz matrices arises in many problems of communications engineering and its solution is of considerable importance.

该提案中的分析问题是给定运算符及其先进器之间的关系。 给定的是具有一般边界条件的差分或积分或卷积操作员。 预处理具有特别方便的域和边界条件。离散化后，它可以被快速的傅立叶变换或在Dirichlet问题的情况下通过快速正弦变换而倒置。 在矩阵情况下，一个应用于预处理的共轭梯度法，特征值的分布至关重要。 如果所有其他特征值都聚集在一起，则可以接受少数极端特征值 - 这就是数值实验中出现的。 集群上的猜想取决于确定特征向量衰减远离边界的衰减。 该分析应将操作者理论中的问题与数值分析中的问题联系起来，并可能在预处理至关重要的方程式中给出一个新的方向。 这项研究属于数值分析，数值计算，计算数学和软件设计的一般领域。 Toeplitz操作员的问题在信号处理中很重要。 与Toeplitz矩阵相关的数值计算问题是在通信工程的许多问题中出现的，其解决方案非常重要。