PRECONDITIONING METHODS FOR THIN SCATTERING STRUCTURES BASED ON ASYMPTOTIC RESULTS

We present a method to precondition the discretized Lippmann–Schwinger integral equations to model scattering of time-harmonic acoustic waves through a thin inhomogeneous scattering medium. The preconditioner is based on asymptotic results as the thickness of the third component direction goes to zero and requires solving a two dimensional formulation of the problem at the preconditioning step

Quantization for Uniform Distributions on Hexagonal, Semicircular, and Elliptical Curves

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon, and then investigated the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form n = 6k for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc, and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n

Social distancing and testing as optimal strategies against the spread of COVID-19 in the Rio Grande Valley of Texas

At the beginning of August 2020, the Rio Grande Valley (RGV) of Texas experienced a rapid increase of coronavirus disease 2019 (abbreviated as COVID-19) cases and deaths. This study aims to determine the optimal levels of effective social distancing and testing to slow the virus spread at the outset of the pandemic. We use an age-stratified eight compartment epidemiological model to depict COVID-19 transmission in the community and within households. With a simulated 120-day outbreak period data we obtain a post 180-days period optimal control strategy solution. Our results show that easing social distancing between adults by the end of the 180-day period requires very strict testing a month later and then daily testing rates of 5% followed by isolation of positive cases. Relaxing social distancing rates in adults from 50% to 25% requires both children and seniors to maintain social distancing rates of 50% for nearly the entire period while maintaining maximum testing rates of children and seniors for 150 of the 180 days considered in this model. Children have higher contact rates which leads to transmission based on our model, emphasizing the need for caution when considering school reopenings

Nivelando el Campo Educativo [Leveling the Educational Field]

The University of Texas Rio Grande Valley is located in south Texas on the border with Mexico. UTRGV’s goals include broadening student success, building students’ self-confidence, and cultivating a sense of pride in the linguistic and cultural heritage of the Rio Grande Valley. This paper describes several programs that seek to level the educational field by building inclusive and supportive academic environments. The narrative provides a template for institutions developing similar inclusive academic initiatives. We describe the role of Spanish and English in the Valley and offer an overview of introductory dual language courses in the department and their impact on students. These courses are part of a Dual Language Certificate currently under development. The peer groups formed in these courses are powerful. They flip the script of English language dominance in the classroom to an environment where both languages are valued. The focus on cultivating equity and inclusivity through multiple modes of communication and interaction continues in peer groups in the Calculus sequence. These groups are part of an effort to build an academic community to support learning and encourage collaborative problem Case Study | 363 solving. Promoting equity through multi-section course coordination is also discussed. Finally, broader components “para nivelar el campo educativo” continue beyond core courses into students’ professional development through the Center of Excellence in STEM Education. Its promotion of pathways through the university and to broader opportunities is designed to increase the number of Latino students attaining STEM degrees and leadership positions across the Nation

The Stability of GMRES Convergence, with Application to Approximate Deflation Preconditioning

How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this approach, we combine our analysis with Stewart's invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors

SHORT-TERM RECURRENCE KRYLOV SUBSPACE METHODS FOR NEARLY HERMITIAN MATRICES ∗

Abstract. The progressive GMRES algorithm, introduced by Beckermann and Reichel in 2008, is a residual-minimizing short-recurrence Krylov subspace method for solving a linear system in which the coefficient matrix has a low-rank skew-Hermitian part. We analyze this algorithm, observing a critical instability that makes the method unsuitable for some problems. To work around this issue we introduce a different short-term recurrence method based on Krylov subspaces for such matrices, which can be used as either a solver or a preconditioner. Numerical experiments compare this method to alternative algorithms. Key words. GMRES, MINRES, nearly Hermitian matrices, low-rank modification