Higher-order Runge--Kutta type schemes based on the Method of Characteristics for hyperbolic equations with crossing characteristics

The Method of Characteristics (MoC) is a well-known procedure used to find the numerical solution of systems of hyperbolic partial differential equations (PDEs). The main idea of the MoC is to integrate a system of ordinary differential equations (ODEs) along the characteristic curves admitted by the PDEs. In principle, this can be done by any appropriate numerical method for ODEs. In this thesis, we will examine the MoC applied to systems of hyperbolic PDEs with straight-line and crossing characteristics. So far, only first- and second-order accurate explicit MoC schemes for these types of systems have been reported. As such, the purpose of this thesis is to develop MoC schemes which are of an order greater than two. The order of the global truncation error of an MoC scheme goes hand-in-hand with the order of the ODE solver used. The MoC schemes which have already been developed use the first-order Simple Euler (SE) and second-order Modified Euler (ME) methods as the ODE solvers. The SE and ME methods belong to a larger family of numerical methods for ODEs known as the Runge--Kutta (RK) methods. First, we will attempt to develop third- and fourth-order MoC schemes by using the classical third- and fourth-order RK methods as the ODE solver. We will show that the resulting MoC schemes can be strongly unstable, meaning that the error in the numerical solution becomes unbounded rather quickly. We then turn our attention to the so-called pseudo-RK (pRK) methods for ODEs. The pRK methods are at the intersection of RK and multistep methods, and a variety of third- and fourth-order schemes can be constructed. We show that when certain pRK schemes are used in the MoC, at most a weak instability, or no instability at all, is present, and thus the resulting methods are suitable for long-time computations. Finally, we present some numerical results confirming that the MoC using third- and fourth-order pRK schemes have the desired accuracy

Apodized vortex coronagraph designs for segmented aperture telescopes

Current state-of-the-art high contrast imaging instruments take advantage of a number of elegant coronagraph designs to suppress starlight and image nearby faint objects, such as exoplanets and circumstellar disks. The ideal performance and complexity of the optical systems depends strongly on the shape of the telescope aperture. Unfortunately, large primary mirrors tend to be segmented and have various obstructions, which limit the performance of most conventional coronagraph designs. We present a new family of vortex coronagraphs with numerically-optimized gray-scale apodizers that provide the sensitivity needed to directly image faint exoplanets with large, segmented aperture telescopes, including the Thirty Meter Telescope (TMT) as well as potential next-generation space telescopes.Comment: To appear in SPIE proceedings vol. 991

Segmented coronagraph design and analysis (SCDA): an initial design study of apodized vortex coronagraphs

The segmented coronagraph design and analysis (SCDA) study is a coordinated effort, led by Stuart Shaklan (JPL) and supported by NASA's Exoplanet Exploration Program (ExEP), to provide efficient coronagraph design concepts for exoplanet imaging with future segmented aperture space telescopes. This document serves as an update on the apodized vortex coronagraph designs devised by the Caltech/JPL SCDA team. Apodized vortex coronagraphs come in two flavors, where the apodization is achieved either by use of 1) a gray-scale semi-transparent pupil mask or 2) a pair of deformable mirrors in series. Each approach has attractive benefits. This document presents a comprehensive review of the former type. Future theoretical investigations will further explore the use of deformable mirrors for apodization.Comment: White Paper (2016-2017

La preparación de los maestros en la Universidad de Virginia

The purpose of this article is to describe how teacher education candidates at The University of Virginia’s College at Wise (UVa-Wise) in the United States are prepared and become eligible for professional teaching licenses in grades 6-12 in the Commonwealth of Virginia. The authors first offer a brief overview of the College and the collaborative efforts between the UVa-Wise Department of Education and other academic disciplines. The authors identify specific program endorsement areas and grades levels in which UVa-Wise TEP candidates pursue licensure. Respective curricula in endorsement areas that candidates are required to complete to become eligible for their professional licenses in their respective teaching endorsement areas are summarized. Topics of discussion include: 1.) Requirements that candidates must meet for admission into the UVa-Wise teacher education program; 2.) the professional dispositions expectations for admission into and continuation in the UVa-Wise teacher education program; and 3.) the admission requirements to the student teaching internship, which is required of all program completers. Additionally, the roles of the Virginia Department of Education and the Virginia Board of Education in the teaching endorsement approval process for all teacher education programs in Virginia are discussed. Finally, the process of how teacher education programs can become recognized by a national accrediting organization (more specifically, the Teacher Education Accreditation Council) is briefly examined.El objetivo de este artículo es describir la formación de los estudiantes de Formación del Profesorado en la Universidad de Virginia de Estados Unidos y cómo llegan a ser docentes para la Enseñanza Primaria (Grados 6-12) en el Estado de Virginia. Los autores presentan, en primer lugar, una breve visión general del College y los esfuerzos colaborativos entre el Departamento de Educación de UVA-Wise y otras disciplinas académicas. Los autores identifican las áreas específicas de apoyo a los programas y las etapas y grados de los candidatos a la Formación del Profesorado en UVA-Wise para licenciarse. Asimismo, se resumen los respectivos planes de estudio que deben completar los estudiantes para adquirir la licencia profesional en sus correspondientes áreas. Los temas de la presente exposición incluyen: 1. Requisitos que los candidatos necesitan para ser admitidos en el programa de Formación del Profesorado en UVA-Wise; 2. Las expectativas de los profesionales en cuanto a su admisión y continuación en el mencionado programa; y 3. Los requisitos de admisión al periodo de prácticas, el cual es requerido para completar el programa. Se expone además, el papel que tiene el Departamento de Educación de Virginia y el Consejo de Educación para la aprobación de los programas dirigidos a la Formación de Profesores en Virginia. Finalmente, ha sido brevemente examinado, el reconocimiento otorgado a los programas de formación de profesores por parte de una organización nacional de acreditación (en concreto, el Consejo de Acreditación de Educación de Profesores)

Cosmological Parameters from CMB Maps without Likelihood Approximation

We propose an efficient Bayesian MCMC algorithm for estimating cosmological parameters from CMB data without use of likelihood approximations. It builds on a previously developed Gibbs sampling framework that allows for exploration of the joint CMB sky signal and power spectrum posterior, P(s,Cl|d), and addresses a long-standing problem of efficient parameter estimation simultaneously in high and low signal-to-noise regimes. To achieve this, our new algorithm introduces a joint Markov Chain move in which both the signal map and power spectrum are synchronously modified, by rescaling the map according to the proposed power spectrum before evaluating the Metropolis-Hastings accept probability. Such a move was already introduced by Jewell et al. (2009), who used it to explore low signal-to-noise posteriors. However, they also found that the same algorithm is inefficient in the high signal-to-noise regime, since a brute-force rescaling operation does not account for phase information. This problem is mitigated in the new algorithm by subtracting the Wiener filter mean field from the proposed map prior to rescaling, leaving high signal-to-noise information invariant in the joint step, and effectively only rescaling the low signal-to-noise component. To explore the full posterior, the new joint move is then interleaved with a standard conditional Gibbs sky map move. We apply our new algorithm to simplified simulations for which we can evaluate the exact posterior to study both its accuracy and performance, and find good agreement with the exact posterior; marginal means agree to less than 0.006 sigma, and standard deviations to better than 3%. The Markov Chain correlation length is of the same order of magnitude as those obtained by other standard samplers in the field.Comment: 9 pages, 3 figures, Published in Ap

Performance and sensitivity of vortex coronagraphs on segmented space telescopes

The detection of molecular species in the atmospheres of earth-like exoplanets orbiting nearby stars requires an optical system that suppresses starlight and maximizes the sensitivity to the weak planet signals at small angular separations. Achieving sufficient contrast performance on a segmented aperture space telescope is particularly challenging due to unwanted diffraction within the telescope from amplitude and phase discontinuities in the pupil. Apodized vortex coronagraphs are a promising solution that theoretically meet the performance needs for high contrast imaging with future segmented space telescopes. We investigate the sensitivity of apodized vortex coronagraphs to the expected aberrations, including segment co-phasing errors in piston and tip/tilt as well as other low-order and mid-spatial frequency aberrations. Coronagraph designs and their associated telescope requirements are identified for conceptual HabEx and LUVOIR telescope designs

Optimization of coronagraph design for segmented aperture telescopes

The goal of directly imaging Earth-like planets in the habitable zone of other stars has motivated the design of coronagraphs for use with large segmented aperture space telescopes. In order to achieve an optimal trade-off between planet light throughput and diffracted starlight suppression, we consider coronagraphs comprised of a stage of phase control implemented with deformable mirrors (or other optical elements), pupil plane apodization masks (gray scale or complex valued), and focal plane masks (either amplitude only or complex-valued, including phase only such as the vector vortex coronagraph). The optimization of these optical elements, with the goal of achieving 10 or more orders of magnitude in the suppression of on-axis (starlight) diffracted light, represents a challenging non-convex optimization problem with a nonlinear dependence on control degrees of freedom. We develop a new algorithmic approach to the design optimization problem, which we call the ”Auxiliary Field Optimization” (AFO) algorithm. The central idea of the algorithm is to embed the original optimization problem, for either phase or amplitude (apodization) in various planes of the coronagraph, into a problem containing additional degrees of freedom, specifically fictitious ”auxiliary” electric fields which serve as targets to inform the variation of our phase or amplitude parameters leading to good feasible designs. We present the algorithm, discuss details of its numerical implementation, and prove convergence to local minima of the objective function (here taken to be the intensity of the on-axis source in a ”dark hole” region in the science focal plane). Finally, we present results showing application of the algorithm to both unobscured off-axis and obscured on-axis segmented telescope aperture designs. The application of the AFO algorithm to the coronagraph design problem has produced solutions which are capable of directly imaging planets in the habitable zone, provided end-to-end telescope system stability requirements can be met. Ongoing work includes advances of the AFO algorithm reported here to design in additional robustness to a resolved star, and other phase or amplitude aberrations to be encountered in a real segmented aperture space telescope