Polynomial eigenproblems are playing an increasingly important role in contemporary engineering design. Indeed, the computation of resonant frequencies arising from extreme designs presents a real numerical challenge, as these designs can lead to very large eigenproblems with poor conditioning. On the other hand the underlying physics of such problems often leads to algebraically structured polynomial eigenproblems, with concomitant symmetries in the spectrum and special properties of the corresponding subspaces. Existing algorithms, unfortunately, often ignore these structural properties. The types of structured polynomial addressed in this project arise in a variety of applications: T-palindromic (analysis of rail noise from high-speed trains, SAW filters), *-palindromic (discrete-time optimal control), K-palindromic (differential delay equations), alternating (corner singularities, gyroscopic systems, continuous-time optimal control), and hyperbolic (overdamped mechanical systems). Polynomial eigenproblems are usually solved by embedding the system into a larger linear system called a ``linearization''. Until recently, the palette of easily available linearizations has been very limited. Recent work, though, has shown how to systematically construct a continuum of linearizations, from which structure-preserving linearizations and linearizations with nearly optimal conditioning can be chosen. This project aims to further this growing body of work by developing algorithms that exploit these new theories, and in turn, develop new theory and insight for the next generation of algorithms in this critical area of scientific computation.
The problems studied in this proposal are ubiquitous in a wide range of important problems in engineering and applied sciences. Numerical methods for their solution are critical in structural mechanics, molecular dynamics, vibrational analysis, the simulation of electrical circuits, elastic deformation of anisotropic materials, and optical waveguide design, to give a few examples. The trend towards extreme designs, such as high speed trains, optoelectronic devices, micro-electromechanical systems, and ``superjumbo'' jets such as the Airbus 380, presents a challenge for the computation of the resonant frequencies of these structures. These extreme designs often lead to computationally sensitive problems, while the physics of the underlying problem leads to algebraic structure that numerical methods should preserve in order to obtain physically meaningful results. The aim of this project is to increase our theoretical understanding of mathematical transformations that preserve these structures and thereby advance the development of computationally effective algorithms. Consequently, this work will have direct benefit to scientists and engineers across a wide range of disciplines.
