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

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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.