Computational Tools for Exploring Eigenvector Localization

Portland State University, Portland OR

Investigators

Abstract

Vibrations in complex systems are best understood in terms of superpositions of stationary waves whose amplitudes vary in time. These stationary waves are eigenvectors (modes of vibration) of an associated time-independent differential operator, and their corresponding eigenvalues represent energies associated with these modes of vibration. A better understanding of where such eigenvectors are likely to localize, and for which eigenvalues this localization occurs, is of practical interest in the design of structures having desired acoustic or electro-magnetic properties: sound-mitigating outdoor barriers and next generation organic LEDs and solar cells are examples of this design principle in action. Computational tools for exploring eigenvector localization phenomena are an essential component of simulations guiding design decisions, but such tools are very new and very few. The work supported by this grant will provide new techniques for a much broader exploration of localization phenomena than is currently feasible, allowing for exploration much deeper into the spectrum for a broader class of operators/models, with built-in mechanisms for accepting/rejecting localization claims based on measurable quantities. This project focuses on the following fundamental task: Given a subdomain, a small tolerance and a finite interval, find all eigenvalue/eigenvector pairs (eigenpairs) of the operator whose eigenvalue is in the interval and whose eigenvector is localized in the subdomain to within the given tolerance. A complex and compact perturbation of the original operator is constructed such that any eigenpair of the original operator satisfying the conditions above will have an "echo" for the new operator, i.e. there will be an eigenpair of the new operator whose eigenvalue is in a well-defined "target region" of the complex plane determined by the localization conditions, and most(!) eigenpairs of the original operator not satisfying the localization condition will not have such an echo. A contour-integral based eigenvalue solver is then used to efficiently identify eigenpairs of the new operator whose eigenvalues are within the target region. This constitutes the first phase of the method, and empirical evidence already strongly indicates that most unlikely candidates are filtered out in this phase, and all likely candidates are maintained. The remaining candidates are then (cheaply) post-processed to determine the eigenpairs of the original operator of which they were echos. A final simple check of these eigenpairs determines which are ultimately accepted. Several aspects of this approach are embarrassingly parallel, and that fact will be exploited to conduct more thorough investigations of localization than have been previously attempted. The project involves the development of software that will be made publicly available. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.