2011 — 2014 |
Szyld, Daniel [⬀] Xue, Fei |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Eigenvalues Problems, Krylov Subspace Methods, and Subspace Recycling
The project concerns the development, analysis, refinement and testing of efficient numerical algorithms for the solution of algebraic eigenvalue problems and of systems of linear equations arising from a variety of applications. The PI's research is concentrated on the following two classes of problems: 1. Interior eigenvalues of generalized non-Hermitian eigenvalue problems. These arise in many scientific and engineering applications, such as stability analysis of steady flows of incompressible fluid and evaluation of passivity in control systems and circuit networks. 2. Sequences of linear systems of equations. These arise, e.g., in iterative methods for nonlinear problems, such as inexact Newton's method for Riccati equations, inexact eigenvalue algorithms, and interior-point methods for convex optimization. A goal of the project is the development of rapidly convergent and robust Krylov subspace methods to efficiently solve both classes of problems. For the first class of problems, this entails the study of convergence properties, subspace expansion and extraction, and preconditioning techniques that take advantage of the structure of the problems. For the second problem class, the aim is to reduce the iteration counts and computational effort needed for the solution of each linear system by using a properly recycled subspace obtained from the iterative solution of a preceding linear system in the sequence. The study of both problem classes also entails extensive computational experimentation on benchmark problems.
The problems to be studied in this project include the efficient computation of a group of eigenvalues and the solution of sequences of linear systems. Eigenvalue calculations include analysis of vibration frequencies in structures including buildings, to make sure, for example, that they are far from the earthquake band. Fast algorithms for generalized eigenvalue problems also contribute to the design and analysis of electronic integrated circuit and micro-electro-mechanical systems (MEMS), and the detection of potential presence of turbulent fluid flows. Efficient solution of a sequence of linear systems facilitates modeling of fatigue and fracture via finite element analysis, and the stability analysis of linear systems through the solution of Riccati equations. The two problems mentioned are fundamental in the field of numerical linear algebra as well as many relevant areas such as fluid and solid mechanics, system and control theory, and numerical optimization. Although numerical algorithms have been developed and studied for some of these problems, efficient solution of large-scale applications remains a major computational challenge. Development and refinement of these computational methods have potential broader impact in engineering and science.
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0.961 |
2014 — 2017 |
Xue, Fei |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Fast Algorithms For Large-Scale Nonlinear Algebraic Eigenproblems
This project concerns development and analysis of new numerical algorithms for large-scale algebraic eigenproblems with nonlinearity in eigenvalues, eigenvectors, and parameters. These eigenproblems arise in electronic structure calculation, design of accelerator cavities, delay differential equations, vibration analysis of complex structures, and many more. Structure-preserving linearization techniques that have been developed recently are competitive for small or medium polynomial and rational eigenproblems, but they entail high computational costs for large-scale simulations due to the significantly enlarged dimension of linearized problems. In addition, linearization introduces considerable complications for the development of preconditioners, and it is not applicable to eigenproblems with full nonlinearity.
The PI shall develop novel iterative projection methods that are accurate, robust and efficient, for the solution of large-scale truly nonlinear eigenproblems. This goal can be achieved in part by exploration of special properties of different types of nonlinear eigenproblems that enable solution strategies similar to those for linear eigenproblems. This investigation is focused on ( 1) new preconditioned eigensolvers, including conjugate-gradient-like and minimal-residual-like methods, for efficient solution of a large number of extreme and interior eigenvalues of problems with nonlinearity in eigenvalues, with and without the variational principle; (2) fast inexact Newton-like methods to solve parameter-dependent degenerate eigenproblem for the study of (in)stabilities of dynamical systems; (3) efficient algorithms for solving eigenproblems with nonlinearity in eigenvectors arising from condensed matter physics and electronic structure calculation. The research will develop a systematic and unified treatment of mathematical theory and development of numerical software.
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0.957 |
2017 — 2020 |
Xue, Fei Miller, David (co-PI) [⬀] Schroeder, Larissa Mcgivney-Burelle, Jean (co-PI) [⬀] Haruta, Mako |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Supporting and Sustaining Scholarly Mathematics Teaching
The Supporting and Sustaining Scholarly Mathematics Teaching (SSSMT) project at the University of Hartford seeks to improve teaching and learning in undergraduate mathematics courses. Recent projections suggest that the United States needs to dramatically increase the annual award rate of STEM degrees by 34% in the next decade to remain competitive internationally. A critical step towards improving the recruitment, retention and success of STEM majors is the adoption of research-based teaching practices in first-year and second-year college mathematics courses. These introductory courses often act as gatekeepers that filter students out of STEM majors. There is mounting evidence that suggests active learning can be much more effective than direct instruction in engaging students. Despite this evidence, there continues to be an overwhelming reliance on lecture as the primary method of delivering content in college mathematics courses. The goals of the Supporting and Sustaining Scholarly Mathematics Teaching (SSSMT) project are: 1) to form a collaborative community of faculty who are committed to implementing active learning and flipping pedagogy in a variety of first-year and second-year mathematics courses with diverse student populations; 2) to studying the effectiveness of this pedagogy; and 3) to sharing the results with the larger mathematics community. Collaboratively and collectively, SSSMT faculty participants will develop a shared understanding of how to best leverage these strategies and processes to support a more widespread adoption of engaged teaching and learning strategies locally, regionally, and nationally.
Drawing on existing research on active learning, flipping pedagogy, and faculty development, the SSSMT project seeks to create a multi-institutional network of faculty to implement its goals in these areas. The network is comprised of college mathematics faculty at different stages of their careers, from a variety of institutions, who are interested in implementing active learning and becoming scholarly teachers who conduct and publish research on their own teaching and their students' learning. In the short term, this project promises to identify and disseminate best practices regarding the use of engaged learning strategies in a variety of mathematics courses populated by diverse student populations. The project will produce comprehensive sets of resources for faculty interested in using active learning and flipping pedagogy in courses ranging from college algebra to introduction to proofs and will develop a more robust understanding of the challenges, opportunities and necessary supports for mathematics faculty who have different levels of experience with the scholarship of teaching and learning. Project outcomes will be investigated using a mixed-methods approach involving qualitative and quantitative data collected from participants through surveys, reflective journals, and teaching practice inventories. The longer-term goal of this project is to expand the cross-institutional network, inviting in and involving the departmental colleagues of project participants, as well as faculty at other institutions, thereby broadening and enriching the virtual community of faculty engaged in this work of enhancing the teaching and learning of undergraduate mathematics.
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0.961 |