Mathematical Sciences: Preconditioning by Fast Transforms

这个建议中的分析问题是给定的算子和它的预条件子之间的关系。给定的是具有一般边界条件的微分、积分或卷积算子。该预条件子具有特别方便的区域和边界条件。离散化后，可以用快速傅里叶变换求逆，对于狄里克莱特问题，可以用快速正弦变换求逆。在矩阵情形下，采用预条件共轭梯度法，特征值的分布是至关重要的。如果所有其他特征值都聚集在一起--这就是数值实验中出现的情况，那么少量的极端特征值是可以接受的。关于聚类的猜想依赖于确定特征向量在远离边界处以指数方式衰减。分析应该将算子论中的问题与数值分析中的问题联系起来，并可能在预处理必不可少的方程类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算子问题是信号处理中的一个重要问题。与Toeplitz矩阵相关的数值计算问题存在于许多通信工程问题中，其解具有相当重要的意义。

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.