Numerical Analysis Seminar

Date Time 
Location  Speaker 
Title – click for abstract 

09/18 3:00pm 
BLOC 302 
Jeffrey Ovall Portland State University 
Computational tools for approximating eigenvectors having specified properties of interest
Standard eigensolvers for differential operators (or large
matrices) are designed to return eigenpairs for which the
eigenvalues are constrained in some way: the smallest six
eigenvalues (in magnitude), the six eigenvalues nearest a given
target, or all of the eigenvalues lying inside a given complex
contour, for example. In some cases, however, one would rather select
eigenpairs based desired features of the eigenvectors.
For example, one may wish to identify eigenpairs for which the
eigenvectors are strongly spatially localized. Anderson
localization, first posited in the late 1950s, is an important
example of this, and has generated an enormous amount of research in
the intervening years, with significant contributions from the Simons
Collaboration on Localization of Waves in the past decade.
In the first part of this talk, we describe an algorithm for computing
eigenpairs for which the eigenvectors are ``sufficiently close'' to a
userspecified subspace. We provide heuristic, theoretical and
empirical support for this approach. We focus on the magnetic
Schrödinger operator, though many of the results carry
over naturally to other selfadjoint operators. The second part of the
talk concerns a relatively simple modification of the operatora
(canonical) gauge transformthat enables more efficient
approximation of eigenpairs.


10/02 3:00pm 
BLOC 302 
Nicolas Favrie Aix Marseille Université 
Resolution of dispersive equations by the extended Lagrangian method: application to waves and gradient damage
Dispersive systems are encountered in many applications: surface waves
(Serre Green Naghdi equation), ultracold gases, inhomogeneous materials
homogenized to high order, Rayleigh equation, gradient media, etc. When
modeling this type of system, energy no longer depends solely on
internal variables, but also on their temporal or spatial
derivatives. These derivatives change the structure of PDE
systems from hyperbolic to dispersive.
In this talk, I'll introduce the socalled extended Lagrangian
technique, which transforms dispersive equations into hyperbolic
equations with source terms. After a simple (and useless) example on
the massspring system. I'll present this approach for the simulation
of surface waves (tsunami) using the Serre Green Naghdi equations
(time derivative of water height in energy). Then I will present the
extension of this approach to gradient media (defocusing Schrödinger
equations ). I will then discuss the extension of this approach to
bistable ribbon but also gradient damage problems. 

10/16 3:00pm 
BLOC 302 
Ricardo H. Nochetto University of Maryland 
Quasilinear fractional operators in Lipschitz domains: regularity and approximation
Fractional diffusion on bounded domains is notorious for the lack of boundary regularity of solutions regardless of the smoothness of domain boundary. We explore this matter for the homogeneous Dirichlet problem for fractionalorder quasilinear operators with variable coefficients in Lipschitz domains and any dimensions; this includes fractional pLaplacians and operators with finite horizon. We prove lift theorems in Besov norms which are consistent with the boundary behavior of solutions in smooth domains. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional pLaplacians and present several simulations that reveal the boundary behavior of solutions. 

10/23 3:00pm 
BLOC 302 
Guosheng Fu University of Notre Dame 
Finite Element Methods for Optimal Transport and MeanField Control
We first give a brief introduction of the MongeKantorovich optimal transport problem and the closely related Schrodinger bridge problem. The problems can be casted into a (fluid) dynamic formulation through Benamou and Brenier's celebrated work in 2000. It is a linear (PDE) constrained (convex) optimization problem. We show how finite element methods can be naturally used to discretize this optimization problem, which leads to a discrete saddle point optimization problem. Then a firstorder optimization solver, namely, preconditioned PrimalDual Hybrid Gradient, is used to solve this optimization problem. We further extend our discretization approach to more general meanfield control problems for reactiondiffusion systems including control of droplet dynamics in thinfilm lubrication theory. 
The organizer for this seminar is
Bojan Popov.