Jeffrey Ovall

A function u = u(t,x) describing the behavior of acoustic or
electromagnetic waves in time and space can often be decomposed as an
infinite sum where each term in the sum is a product of a function cn varying only in time and a function  n varying only in space. The spatial functions  n are sometimes referred to as standing waves, and are eigenvectors (eigenfunctions) of a spatial differential operator associated with the medium through
which the waves are propagating. Properties of the medium can cause some
eigenvectors to be strongly spatially localized, and a practical consequence of
this is that waves at certain frequencies can be “trapped” at some location or
“channelled” along some favorable path. Such features are of interest in the
design of structures having desired acoustic or electromagnetic proper- ties:
sound-mitigating outdoor barriers and next generation organic LEDs and
solar cells are examples of this design principle in action. There remain many
open problems related to understanding and exploiting this kind of
localization, and we will discuss a computational approach that we hope will
provide insight. More specifically, we focus on the issue of eigenvector localization,
outlining our computational approach and why it ought to work,
and illustrating this phenomenon and our approach through several examples
(with many pictures). We will also advertise a new NSF-funded Research
Training Group at Portland State University, “Computation- and
Data-Enabled Science”, that will enable a more targeted training of our
students in mathematics, statistics and computational tools needed in these
areas. We are looking for PhD students to join this program in Fall 2023

Refreshments will precede the talk at 3:30pm in Bond Hall 300.