POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Ready for Transition: Factors that facilitate transfer to undergraduate engineering programs among Black African and American students

This study examines the factors facilitating the transfer admission of students broadly classified as Black from a single community college into a selective engineering college. The work aims to further research on STEM preparation and performance for students of color, as well as scholarship on increasing access to four-year institutions from two-year schools. Factors illuminating Underrepresented Racial and Ethnic Minority (URM) student pathways through Science, Technology, Engineering, and Mathematics (STEM) degree programs have often been examined through large-scale quantitative studies. However, this qualitative study complements quantitative data through demographic questionnaires, as well as semi-structured individual and group. The backgrounds and voices of diverse Black transfer students in four-year engineering degree programs were captured through these methods. Major findings from this research include evidence that community college faculty, peer networks, and family members facilitated transfer. Other results distinguish Black African from Black American transfers; included in these distinctions are depictions of different K-12 schooling experiences and differences in how participants self-identified. The findings that result from this research build upon the few studies that account for expanded dimensions of student diversity within the Black population. Among other demographic data, participants’ countries of birth and years of migration to the U.S. (if applicable) are included. Interviews reveal participants’ perceptions of factors impacting their educational trajectories in STEM and subsequent ability to transfer into a competitive undergraduate engineering program. This study is inclusive of, and reveals an important shifting demographic within the United States of America, Black Africans, who represent one of the fastest-growing segments of the immigrant population.