PDE Seminar
University of Houston
For further information or to suggest a speaker for the PDE Seminear contact:
Misha Perepelitsa (maperepelitsa AT uh DOT edu)
Gabriela Jaramillo (gtjaramillo AT uh DOT edu)
William Fitzgibbon (fitz AT uh DOT edu)
Speaker: Manuela Girotti (Emory University) Online via Zoom
Title: TBD
Abstract: TBD
Speaker: Andrei Tarfulea (Louisiana State University) Online via Zoom
Title: TBD
Abstract: TBD
Speaker: Christian Glusa (Sandia National Laboratories)
Title: Scalable methods for nonlocal models
Abstract: The naive discretization of nonlocal operators leads to matrices with significant density, as compared to classical PDEs. This makes the efficient solution of nonlocal models a challenging task. In this presentation, we will discuss ongoing research into efficient hierarchical matrix assembly and geometric and algebraic multigrid preconditioners that are suitable for nonlocal models.
Speaker: Tadele Mengesha (The University of Tennessee, Knoxville)
Title: Variational Analysis of a Parametrized Family of Transmission Problems Coupling Nonlocal and Fractional Models
Abstract: I will present a work that examines the coupling of a model based on the regional fractional Laplacian and a nonlocal model employing a position-dependent interaction kernel. Both operators are inherently nonlocal and act on functions defined within their respective domains. The coupling occurs via a transmission condition across a hypersurface interface. The heterogeneous interaction kernel of the nonlocal operator leads to an energy space endowed with a well-defined trace operator. This, combined with well-established trace results of fractional Sobolev spaces, facilitates the imposition of a transmission condition across an interface. The family of problems will be parametrized by two key parameters that measure nonlocality and differentiability. For each pair of parameters, we demonstrate existence of a solution to the resulting variational problems. Furthermore, we investigate the limiting behavior of these solutions as the parameters approach their extreme.
Speaker: Joshua Siktar (TAMU)
Title: Existence of Solutions for Fractional Optimal Control Problems with Minimax Constraint
Abstract:
In this talk we prove the existence of solutions to an optimal control problem where the constraint is an ill-posed, nonlinear equation containing a Fractional Laplacian. For any fixed control data, the constraint equation is known to have multiple solutions by a previous application of the Mountain Pass Theorem. Due to the pointwise nature of the conditions on controls needed to invoke this theorem, we must make substantial adaptations to the usual direct method of calculus of variations in order to prove our main existence result. The main theoretical tools are a thoughtful construction of an admissible set of controls, and a technical lemma that ensures that a minimizing sequence of pairs of controls and corresponding states exhibits convergence to another control-state pair that satisfies the constraint equation.
Speaker: Jimmie Adriazola (SMU)
Title: Data Driven Integrability and Wave Camouflaging
Abstract: This talk will consist of two separate parts both involving some of my recent numerical optimization projects. The first involves the detection of integrability in Hamiltonian dynamical systems. We formulate the discovery of integrability as a numerical optimization problem. That is, we numerically solve the problem of maximizing the compatibility between a pair of Lax operators with the known Hamiltonian of the dynamical system. Our approach is tested on finite degrees of freedom before moving on to the more challenging PDE setting. We find that our approach is robust against non-integrable perturbations of the Hamiltonian and find that, in all examples, our approach reliably confirms or denies the integrability of the dynamics. This is joint work with Panos Kevrekidis (UMass Amherst), Alejandro Aceves (SMU), and Wei Zhu (GT).
I will discuss the problem of camouflaging electromagnetic waves in the second part of the talk. We formulate the problem in terms of optimal control theory as we seek to find a current density that destructively interferes with an outgoing signal subject to Maxwell's equations. I will discuss some of the numerical and functional analytic challenges that can appear in this problem and show how to resolve them to achieve optimal results. This is joint work with Denis Ridzal (Sandia) and Harbir Antil (GMU).