In 1958, motivated by experimental results at Bell Laboratories that could not be explained under the current models of electron transport, Philip Anderson introduced a lattice model of electron transport through an impure/disordered system that showed how electrons might be “trapped” at a lattice location if there was sufficient disorder in the system---work for which he shared the 1977 Nobel prize in physics. In the intervening years, what became known as Anderson localization has evolved to encompass related phenomena for a variety of wave types in complex materials, and has generated an enormous amount of literature. Mathematically, the problem is viewed as one of localization (concentration of mass) of some eigenvectors of a differential operator in relatively small regions. Mathematical insights into the underlying mechanisms driving localization, including domain geometry and "disorder" in the coefficients of the differential operator, have been provided over the last decade, though the picture is far from complete, and there is clear room for development on the algorithmic front. We will briefly describe the two (very recent) computational approaches of which we are aware, before discussing our own contributions. More specifically, we will present an algorithm template for exploring eigenvector localization, and a partial realization of it, that is supported both theoretically and empirically.
BIO: Jeffrey Ovall is a Maseeh Professor of Mathematics at Portland State University, working on numerical analysis and scientific computing for partial differential equations and integral equations. Specific research topics of interest include operator eigenvalue problems, "exotic" discretization schemes, estimation of discretization error, and effective treatment of singular solutions. Jeff received his PhD in mathematics at the University of California, San Diego, in 2004. After postdoctoral positions at the Max Planck Institute in Leipzig and the California Institute of Technology, and a faculty position at the University of Kentucky, Jeff joined faculty at Portland State in 2013.
His favorite linear algebra result is the spectral theorem for normal matrices, and his favorite ODE result is that any solution of a first-order (scalar) autonomous equation must be monotone. He has an extensive collection of Spider-Man comic books.