Lefschetz Center for Dynamical Systems Seminar
Applied Mathematics Colloquium
Abstract: In this talk, I will describe some recent results on the local and global well-posedness of the full water wave equations and give brief ideas on the proofs.
Scientific Computing Seminar
Abstract: In 1907 Erhard Schmidt published a paper where he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process (CGS). Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P. Gram in 1883. The Schmidt version was the first to become popular and widely used. The two algorithms produce the same results when carried out in exact arithmetic, however, the Gram version, now known as the modified Gram-Schmidt process (MGS), produces superior results when carried out in finite precision arithmetic. In actuality, Gram rediscovered a method that first appears in an 1812 treatise by P. S. Laplace. While the MGS algorithm has been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthonormalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary we present a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Among the topics covered are: the early papers on orthogonality and least squares, loss of orthogonality, reorthogonalization, super orthogonality, iterative refinement, rank revealing factorizations, and applications to Krylov subspace methods. Joint work with Walter Gander, ETH Zurich, Ake Bjorck, Linkoping University, and Julien Langou, University of Colorado at Denver.
PDE Seminar
Abstract: This talk concerns recent results on the time-dependent Ginzburg-Landau equations on a smooth, bounded 2-D domain, subject to both an applied magnetic field and an applied boundary current. The boundary current is modeled with a gauge-invariant inhomogeneous Neumann boundary condition. After a quick discussion of well-posedness, I will present a result that controls the growth of the energy of the solutions. I will then turn to the study of the dynamics of the vortices of the solutions in the limit as the Ginzburg-Landau parameter vanishes. In the original time scale, the vortices do not move, and the solutions undergo a phase relaxation. In an accelerated time scale, the vortices do move, and I derive their dynamical law, identifying a novel Lorentz force term induced by the applied boundary current.
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