Numerical Linear Algebra for Signal Processing and Integral Equations

信号处理中的许多问题可以公式化为： 正定Toeplitz和酉（或 正交）Hessenberg矩阵。 例如，线性系统 滤波，递归估计，时间序列分析，以及 在图像处理中。 此外，Pisarenko频率估计， 在信号处理中也很重要，可以使用 Toeplitz和酉Hessenberg矩阵如下：a Toeplitz 特征值问题的解决; Schur参数相关联 其中正交多项式由 Toeplitz矩阵，然后用来构造一个特定的酉 Hessenberg矩阵;该矩阵的特征值为 频率估计。 利用多项式理论与 单位圆上的正交和这些问题的线性 代数，研究人员已经开发出竞争力， 特征值和特征向量的数值计算方法 求逆特征值的酉Hessenberg矩阵 问题的酉矩阵，并解决系统的线性 具有正定对称Toeplitz矩阵的方程。 这项工作的目的是进一步加快 算法的上述问题，并调查他们如何可以 在具有向量和并行的计算机上有效地实现 建筑调查人员还希望获得一个更好的 理解数字属性，如稳定性， 的一些算法，以便快速可靠 可以开发计算机程序。 这项工作应导致 已发表的快速稳定的算法对上述问题。 快速Toeplitz解算器的另一个应用是 积分方程 Toeplitz矩阵的迭代格式 作为预处理器也将被研究。 鼓励 平面势积分方程的初步结果 目前存在关于分段平滑曲线的理论。 这是一个积分方程的模型问题， 具有位移核的光滑曲线或曲面。 等 积分方程在边界的应用中是常见的 单元法 该项目将调查当Toeplitz 矩阵是很好的预条件子。 更一般地说，它将 继续研究并开发快速解决方法 利用系统的结构， 离散化后得到的方程。 直接和 将考虑迭代方案。

Many questions in signal processing can be formulated as problems for positive definite Toeplitz and unitary (or orthogonal) Hessenberg matrices. For example, systems of linear filtering, recursive estimation, time series analysis, as well as in image processing. Furthmore, Pisarenko frequency estimates, also of interest in signal processing, can be computed using Toeplitz and unitary Hessenberg matrices as follows: a Toeplitz eigenvalue problem is solved; the Schur parameters associated with the orthogonal polynomials defined implicitly by the Toeplitz matrix are then used to construct a certain unitary Hessenberg matrix; the eigenvalues of this matrix are the frequency estimates sought. Using the close connection between the theory of polynomials orthogonal on the unit circle and these problems of linear algebra, the investigators have already developed competitive numerical methods for computing eignevalues and eigenvectors of unitary Hessenberg matrices for solving the inverse eigenvalue problem for unitary matrices, and for solving systems of linear equations with positive definite symmetric Toeplitz matrices. The purpose of this work is to further speed up the algorithms for the above problems and to investigate how they can be implemented efficiently on computers with vector and parallel architectures. The investigators also wish to gain a better understanding about the numerical properties, such as stability, of some of the algorithms mentioned so that fast reliable computer programs can be developed. The work should result in published fast stable algorithms for the above problems. Another application of fast Toeplitz solvers is the solution of integral equations. Iterative schemes with Toeplitz matrices as preconditioners will also be investigated. Encouraging preliminary results for an integral equation of plane potential theory on a piecewise smooth curve currently exist. This is a model problem for integral equations on a piecewise smooth curve or surface with a displacement kernel. Such integral equations are common in applications of the boundary element method. The project will investigate when Toeplitz matrices make good preconditioners. More generally, it will continue the investigation in and develop fast solution methods for integral equations by exploiting the structure of the system of equations obtained after discretization. Both direct and iterative schemes will be considered.