6,645 research outputs found
Into the Fray: Novice Teachers Tackle Standards-Based Mathematics
This article tracks twenty-one graduates of a refom-based mathematics teacher education program for two years as they begin teaching mathematics in public elementary schools in New York City. Using surveys, classroom observations, and interviews, it examines the extent to which these beginning teachers were able to implement standards-based mathematics instruction in their classes. Results of the study were mixed. The novice teachers generally demonstrated an adequate understanding of the underlying mathematics principles and strong intentions of teaching mathematics for understanding.They were generally able to engage children in learning, and most performed at the “beginning stages of effective instruction” in their first year. However, they still struggled to engage students in higher order thinking and knowledge construction. ln their second year their abilities improved, but they were still hampered by local factors such as insufficient in-service support, the restrictions of high-stakes testing, and the overall school climate
Heavy Meson Physics: What have we learned in Twenty Years?
I give a personal account of the development of the field of heavy quarks.
After reviewing the experimental discovery of charm and bottom quarks, I
describe how the field's focus shifted towards determination of CKM elements
and how this has matured into a precision science.Comment: This talk was presented during the ceremony awarding the Medalla 2003
of the Division of Particles and Fields of The Mexican Phsyical Society, at
the IX Mexican Workshop on Particles and Fields; submitted for proceedings; 9
pages, 9 figures; replacement: fix multiple typo
Multilevel convergence analysis of multigrid-reduction-in-time
This paper presents a multilevel convergence framework for
multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid
estimates. The framework provides a priori upper bounds on the convergence of
MGRIT V- and F-cycles, with different relaxation schemes, by deriving the
respective residual and error propagation operators. The residual and error
operators are functions of the time stepping operator, analyzed directly and
bounded in norm, both numerically and analytically. We present various upper
bounds of different computational cost and varying sharpness. These upper
bounds are complemented by proposing analytic formulae for the approximate
convergence factor of V-cycle algorithms that take the number of fine grid time
points, the temporal coarsening factors, and the eigenvalues of the time
stepping operator as parameters.
The paper concludes with supporting numerical investigations of parabolic
(anisotropic diffusion) and hyperbolic (wave equation) model problems. We
assess the sharpness of the bounds and the quality of the approximate
convergence factors. Observations from these numerical investigations
demonstrate the value of the proposed multilevel convergence framework for
estimating MGRIT convergence a priori and for the design of a convergent
algorithm. We further highlight that observations in the literature are
captured by the theory, including that two-level Parareal and multilevel MGRIT
with F-relaxation do not yield scalable algorithms and the benefit of a
stronger relaxation scheme. An important observation is that with increasing
numbers of levels MGRIT convergence deteriorates for the hyperbolic model
problem, while constant convergence factors can be achieved for the diffusion
equation. The theory also indicates that L-stable Runge-Kutta schemes are more
amendable to multilevel parallel-in-time integration with MGRIT than A-stable
Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material
Density of Surface States in Discrete Models
We consider a simple quantum model with a surface and prove the existence of a surface density of states. We show that the energy spectrum of the model is the union of the support of the bulk densities of states of the media forming the surface and the support of the surface density of states
Space-Time Block Preconditioning for Incompressible Resistive Magnetohydrodynamics
This work develops a novel all-at-once space-time preconditioning approach
for resistive magnetohydrodynamics (MHD), with a focus on model problems
targeting fusion reactor design. We consider parallel-in-time due to the long
time domains required to capture the physics of interest, as well as the
complexity of the underlying system and thereby computational cost of long-time
integration. To ameliorate this cost by using many processors, we thus develop
a novel approach to solving the whole space-time system that is parallelizable
in both space and time. We develop a space-time block preconditioning for
resistive MHD, following the space-time block preconditioning concept first
introduced by Danieli et al. in 2022 for incompressible flow, where an
effective preconditioner for classic sequential time-stepping is extended to
the space-time setting. The starting point for our derivation is the continuous
Schur complement preconditioner by Cyr et al. in 2021, which we proceed to
generalise in order to produce, to our knowledge, the first space-time block
preconditioning approach for the challenging equations governing incompressible
resistive MHD. The numerical results are promising for the model problems of
island coalescence and tearing mode, with the overhead computational cost
associated with space-time preconditioning versus sequential time-stepping
being modest and primarily in the range of 2x-5x, which is low for
parallel-in-time schemes in general. Additionally, the scaling results for
inner (linear) and outer (nonlinear) iterations are flat in the case of fixed
time-step size and only grow very slowly in the case of time-step refinement.Comment: 25 pages, 4 figures, 3 table
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