A&S Graduate Studies
[PAST EVENT] Dissertation Proposal - Heather Maria Switzer, Computer Science
Abstract:
Scientific Computing is a multidisciplinary field that intersects Computer Science, Mathematics, and another discipline to address complex problems utilizing computational systems. Numerical Linear Algebra (NLA), a subfield within Scientific Computing, develops and analyzes numerical algorithms for tasks involving linear operators such as matrices and their transformations. This dissertation proposal focuses on developing kernels for two specific areas of NLA.
Firstly, we explore variance reduction methods for trace approximation. In Lattice Quantum Chromodynamics (LQCD), computing the trace of a matrix inverse is crucial for investigating interactions among quarks and gluons in subatomic space. Directly computing a matrix inverse is computationally expensive, motivating the use of stochastic methods to estimate the trace directly. However, higher accuracy models result in a high variance of the trace estimator. Techniques such as Probing leverage matrix structure in LQCD to reduce this variance. More recently, the development of so-called ``disconnected diagrams'' in LQCD necessitate the sum of certain off-diagonal elements in the matrix inverse, rendering Probing ineffective. We propose an extension to the Probing method to mitigate the variance of the trace estimator in such scenarios.
Subsequently, we explore the viability of a new technique that uses randomized subspace projections in Krylov-based iterative methods to approximate a subset of the eigenpairs of a large matrix. A bottleneck in iterative methods is the need to reorthogonalize the basis to extract accurate approximations. Randomized subspace projections, commonly used to solve least-squares problems, reduce the problem to a more manageable size. These random projection methods, often referred to as "sketching" methods, can be integrated into iterative methods to extract information from a non-orthogonal basis with minimal accuracy lost. We investigate the efficiency of these sketching methods using two iterative methods, Lanczos and Generalized Davidson, with and without restarting, within the high-performance software library PRIMME.
Bio:
Heather Switzer is a Ph.D. Candidate working under the supervision of Prof. Andreas Stathopoulos in the Department of Computer Science at William & Mary. Her research interests lie in the field of Scientific Computing, specifically Numerical Linear Algebra, where she focuses on variance reduction techniques for trace approximation problems in Lattice Quantum Chromodynamics and randomized subspace embeddings for approximating the eigenpairs of sparse, symmetric matrices. Her research has been accepted into SIAM CISC 2021. She is also the recipient of the VSGC Graduate Research Fellowship. Previously, she obtained her Associate of Science in 2015 at Richard Bland College, her Bachelor of Science in 2018 for Mathematics and Computer Science at Longwood University, and her Master of Science in 2020 for Computer Science with a specialization in Computational Science from William & Mary.
Sponsored by: Computer Science