LOCA, the Library of Continuation Algorithms, is a software library for performing stability analysis of large-scale applications. LOC A enables the tracking of solution branches as a function of a system parameter, the direct tracking of bifurcation points, and, when linked with the ARPACK library, a linear stability analysis capability. It is designed to be easy to implement around codes that already use Newton's method to converge to steady-state solutions. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. This manual presents LOCA's continuation and bifurcation analysis algorithms, and instructions on how to implement LOCA with an application code. The LOCA code is being made publicly available at www . cs . sandia. gov/loca. 3

We present the set of bifurcation tracking algorithms which have been developed in the LOCA software library to work with large scale application codes that use fully coupled Newton's method with iterative linear solvers. Turning point (fold), pitchfork, and Hopf bifurcation tracking algorithms based on Newton's method have been implemented, with particular attention to the scalability to large problem sizes on parallel computers and to the ease of implementation with new application codes. The ease of implementation is accomplished by using block elimination algorithms to solve the Newton iterations of the augmented bifurcation tracking systems. The applicability of such algorithms for large applications is in doubt since the main computational kernel of these routines is the iterative linear solve of the same matrix that is being driven singular by the algorithm. To test the robustness and scalability of these algorithms, the LOCA library has been interfaced with the MPSalsa massively parallel finite element reacting flows code. A bifurcation analysis of an 1.6 Million unknown model of 3D Rayleigh-Bénard convection in a 5 × 5 × 1 box is successfully undertaken, showing that the algorithms can indeed scale to problems of this size while producing solutions of reasonable accuracy.

Mathematics classroom coaching is used across the United States as a means for improving instruction, with the ultimate goal of improving student learning. The job assignments of coaches can vary widely across schools and districts. Regardless of the various forms that coaching epaa aape Education Policy Analysis Archives Vol. 22 No. 53 2 can take, there is the consistent expectation that a coach's day-to-day work will positively influence classroom instruction. The study reported here attempts to gain a more complete picture of the job of an elementary mathematics coach based on the observation of seven coaches in five different districts for a day. We report the variety of ways we observed elementary mathematics coaches interact with teachers, and what roles and responsibilities they take on. Our analysis of these data led us to create a template for conducting observations of mathematics classroom coaching, which could be used by researchers seeking to conduct studies about coaching or by administrators seeking to document the day-to-day work of coaches. Keywords: mathematics coaching; qualitative research Una mirada al interior del entrenamiento en matemáticas: roles, contenidos y dinámicas Resumen: entrenamientos en el área de matemáticas se utilizan extensivamente Estados Unidos con el objetivo final de mejorar el aprendizaje de los estudiantes. Las asignaciones de trabajo de los entrenadores varían ampliamente entre escuelas y distritos. Independientemente de las diversas formas que puede tomar los programas de entrenamiento, existe la expectativa que el trabajo de los entrenadores va a influir positivamente la enseñanza en los salones de clase. Este estudio intenta obtener una imagen más completa del trabajo de los entrenadores de matemáticas utilizando observaciones de siete entrenadores en cinco distritos diferentes. Presentamos la variedad de formas que observamos entrenadores de matemáticas elementales interactuando con docentes, los roles y responsabilidades asumen. Nuestro análisis de estos datos nos llevó a crear una plantilla para la realización de observaciones de entrenamientos de matemáticas, que podría ser utilizada por investigadores o por administradores tratando de documentar el trabajo diario de los entrenadores. Palabras clave: entrenamientos de matemáticas; investigación cualitativa Um olhar dentro da formação em Matemática: funções, conteúdos e dinâmicas Resumo: treinamentos na área da matemática são usados extensivamente nos Estados Unidos com o objetivo final de melhorar a aprendizagem dos alunos. Atribuições de trabalho treinadores variam muito entre as escolas e distritos. Independentemente das várias maneiras que os programas de treinamento podem ter, há uma expectativa de que o trabalho dos treinadores vai afetar positivamente o ensino em sala de aula. Este estudo tenta obter uma imagem mais completa do trabalho dos treinadores de matemática por meio de observações de sete treinadores em cinco distritos diferentes. Apresentamos a variedade de formas que os treinadore...

Locates the onset of oscillatory instability in the fluid flow inside a differentially heated cavity with aspect ratio 2 by computing a steady-state and analyzing the stability of the system via eigenvalue approximation. Discusses the choice of parameters for the Cayley transformation so that the calculation of selected eigenvalues of the transformed system will reliably answer the question of stability. Also presents an argument that due to the symmetry of the problem, the first two unstable modes will have eigenvalues that are nearly identical, and the numerical experiments confirm this. Finally, locates a co-dimension 2 bifurcation signifying where there is a switch in the mode of initial instability. The results were obtained using a parallel finite element CFD code (MPSalsa) along with an Arnoldi-based eigensolver (ARPACK), a preconditioned Krylov method code for the necessary linear solves (Aztec), and a stability analysis library (LOCA).

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We present results for large scale linear stability analysis of buoyancy driven fluid flows using a parallel finite element CFD code (MPSalsa) along with a general purpose eigensolver (ARPACK). The goal of this paper is to examine both the capabilities and limitations of such an approach, with particular focus on solving large problems on massively parallel computers using iterative methods. We accomplish our goal by solving a large variety of two and three dimensional problems of varying difficulty, comparing our results (whenever possible) to semi-analytical results. We also carefully explain how we successfully combined Cayley transformations with an Arnoldi based eigensolver and preconditioned Krylov methods for the necessary linear solves. For problems where the advective terms are not significant, we achieve excellent convergence of the computed eigenvalues as we refine the finite element mesh. We also successfully solve advectively dominated problems, but the convergence is slower. We believe that the main difficulties arise not from problems with the eigensolver, but from the accuracy of the finite element discretization. Therefore, we believe that our results are as reliable as using transient integration but are more efficiently computed. The largest eigenvalue problem we solve has over 16 million unknowns on 2048 processors.

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