Numerical Linear Algebra for Signal Processing and Integral Equations

信号处理中的许多问题都可以归结为正定Toeplitz矩阵和酉阵(或正交阵)的问题。例如，线性滤波、递归估计、时间序列分析以及图像处理中的系统。此外，也是信号处理中感兴趣的Pisarenko频率估计，可以使用Toeplitz和酉Hessenberg矩阵计算如下：求解Toeplitz特征值问题，然后使用与Toeplitz矩阵隐式定义的正交多项式相关联的Schur参数来构造某个酉Hessenberg矩阵；该矩阵的特征值是所寻求的频率估计。利用单位圆上的多项式正交理论与这些线性代数问题之间的密切联系，研究者们已经发展了计算酉Hessenberg矩阵的特征值和特征向量的竞争性数值方法，用于求解酉阵的逆特征值问题，以及求解具有正定对称Toeplitz矩阵的线性方程组。这项工作的目的是进一步加速上述问题的算法，并研究如何在具有向量和并行结构的计算机上高效地实现这些算法。研究人员还希望更好地了解所提到的一些算法的数值性质，如稳定性，以便开发快速可靠的计算机程序。这项工作将为上述问题提供快速稳定的算法。快速Toeplitz求解器的另一个应用是求解积分方程组。还将研究以Toeplitz矩阵为预条件的迭代方案。关于分段光滑曲线上的平面势理论积分方程，目前已有了令人鼓舞的初步结果。这是具有位移核的分段光滑曲线或曲面上积分方程组的模型问题。这种积分方程组在边界元方法的应用中是很常见的。该项目将调查Toeplitz矩阵何时成为良好的预条件。更广泛地说，它将继续研究和发展积分方程组的快速求解方法，利用离散化后得到的方程组的结构。直接方案和迭代方案都将被考虑。

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.