Mark Embree
Rice University
Network
Latest external collaboration on country level. Dive into details by clicking on the dots.
Publication
Featured researches published by Mark Embree.
Communications in Mathematical Physics | 2008
David Damanik; Mark Embree; Anton Gorodetski; Serguei Tcheremchantsev
We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as
Siam Review | 2005
Christopher A. Beattie; Mark Embree; Danny C. Sorensen
Siam Review | 2003
Mark Embree
\lambda \to \infty, {\rm dim} (\sigma(H_\lambda)) \cdot {\rm log} \lambda
Siam Review | 1999
Mark Embree; Lloyd N. Trefethen
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | 1999
Mark Embree; Lloyd N. Trefethen
converges to an explicit constant,
arXiv: Mathematical Physics | 2015
David Damanik; Mark Embree; Anton Gorodetski
SIAM Journal on Scientific Computing | 2001
Mark Embree; Lloyd N. Trefethen
{\rm log}(1+\sqrt{2})\approx 0.88137
Siam Review | 2012
Steven J. Cox; Mark Embree; Jeffrey Mattson Hokanson
SIAM Journal on Matrix Analysis and Applications | 2009
Mark Embree
. We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.
SIAM Journal on Scientific Computing | 2016
Danny C. Sorensen; Mark Embree
Krylov subspace methods have led to reliable and effective tools for resolving large-scale, non-Hermitian eigenvalue problems. Since practical considerations often limit the dimension of the approximating Krylov subspace, modern algorithms attempt to identify and condense significant components from the current subspace, encode them into a polynomial filter, and then restart the Krylov process with a suitably refined starting vector. In effect, polynomial filters dynamically steer low-dimensional Krylov spaces toward a desired invariant subspace through their action on the starting vector. The spectral complexity of nonnormal matrices makes convergence of these methods difficult to analyze, and these effects are further complicated by the polynomial filter process. The principal object of study in this paper is the angle an approximating Krylov subspace forms with a desired invariant subspace. Convergence analysis is posed in a geometric framework that is robust to eigenvalue ill-conditioning, yet remains relatively uncluttered. The bounds described here suggest that the sensitivity of desired eigenvalues exerts little influence on convergence, provided the associated invariant subspace is well-conditioned; ill-conditioning of unwanted eigenvalues plays an essential role. This framework also gives insight into the design of effective polynomial filters. Numerical examples illustrate the subtleties that arise when restarting non-Hermitian iterations.