Date/Time: Thursday 4/10, 2:30-3:30pm

Location: FAB 1146

Speaker: Jeffrey S. Ovall (Portland State University)

Title: Computational and theoretical tools for exploring eigenvector localization phenomena

Abstract: In complex media, waves (acoustic, electromagnetic, quantum mechanical) can exhibit unusual localization behavior at certain frequencies/energies, wherein the wave magnitude is largely concentrated in a relatively small portion of the physical domain, and this localization behavior persists in time!  Such behavior may be desired or undesired in a particular application.  Regardless, a better understanding of localization, through improved theory and numerical simulation capabilities, can prove useful in guiding design decisions.  As is often the case with space-time partial differential equations, significant insight can be gleaned by considering the associated time-independent eigenvalue problem, and this will be the focus of our discussion.  

After providing some historical context and several examples of localization, we will outline some of the known theoretical results, including our recent work on localization for the magnetic Schr"odinger equation, which will serve as our model operator for the discussion.  We will then shift to computational tools for efficiently exploring localization phenomena.  We will first present an algorithm template for approximating only the eigenvectors of a selfadjoint operator that have certain desired properties, such as being localized in a specific part of the domain and having frequency/energy within some specific band.  We will then present a practical similarity transform for the magnetic Schr"odinger operator that can greatly reduce the computational cost associated with approximating its eigenpairs.