Mathematical Sciences: Non-Normal Matrices and Operators: Analysis, Computations, Applications

特雷费滕 研究数学科学中涉及高度非正规矩阵和算子的问题。 近似地说，这意味着矩阵和算子的特征向量或特征函数远离正交。 在应用中出现的大多数矩阵和算子都是正规的或接近正规的，但高度非正规的情况构成了一个重要的少数，传统的基于特征值的方法处理得很差。 这个项目的重点是避免使用特征值的方法。 其中一些利用，而不是某些集在复杂的平面称为伪谱，定义在扰动的频谱或等价的区域所界定的水平曲线的规范的预解式。 调查的应用落在三个领域：非自伴偏微分方程的数值解;大规模矩阵问题的迭代求解;和流体动力学，特别是流体动力学稳定性和过渡到湍流。 这项工作与科学和工程领域正在进行的一些重大发展有关。 一个是流体流动如何变得不稳定并最终成为湍流的流行观点的基本变化，其潜在应用遍及机械，航空和土木工程以及气象学，物理学和其他科学领域的许多分支。 第二个，公众不太了解，但仍然非常重要，是计算科学正在进行的巨大转变，从“直接”到更快的“迭代”方法来解决大规模线性代数问题。 调查小组在上述两个领域的工作都有很大的知名度。 通过上面提到的“伪光谱”的想法，它也有越来越大的教育影响，这是一种在计算机上可视化这些数学问题的方法，许多学生（和教授！）在视觉上和概念上都很吸引人

Trefethen The investigator studies problems in the mathematical sciences involving highly non-normal matrices and operators. Approximately speaking, this means matrices and operators whose eigenvectors or eigenfunctions are far from orthogonal. Most matrices and operators that arise in applications are normal or close to normal, but the highly non-normal cases constitute an important minority, one that has been poorly handled by traditional methods based on eigenvalues. The emphasis of this project is on methods that avoid the use of eigenvalues. Some of these make use instead of certain sets in the complex plane known as pseudospectra, defined in terms of perturbations of the spectrum or equivalently as regions bounded by level curves of the norm of the resolvent. The applications investigated fall in three areas: numerical solution of non-self-adjoint partial differential equations; iterative solution of large-scale matrix problems; and fluid dynamics, specifically hydrodynamic stability and transition to turbulence. This work is connected with some major ongoing developments in science and engineering. One is a basic change in the prevailing view of how fluid flows become unstable and eventually turbulent, with potential applications across many branches of mechanical, aeronautical, and civil engineering as well as in meteorology, geophysics, and other scientific fields. A second, less accessible to the public but still of great importance, is the great shift underway in computational science from "direct" to faster "iterative" methods for solving large-scale linear algebra problems. The work of the investigator's group in both of the areas just mentioned has had considerable visibility. It is also having a growing educational impact through the idea of "pseudospectra" mentioned above, a way of visualizing these mathematical problems on a computer that many students (and professors!) find visually and conceptually appealing.