VT Mathematics Colloquium

Abstract: My research program focuses on the learning and teaching of undergraduate mathematics, particularly linear algebra. The two main threads of my research are characterizing various ways that students understand linear algebra concepts, both in mathematics and in quantum mechanics, and developing student-centered curricular materials for introductory linear algebra. In this talk, I will summarize some key research findings from those two threads, mostly focusing on the theme of eigentheory.

Neuroinflammation immediately follows the onset of ischemic stroke. During this process, microglial cells are activated in and recruited to the tissue surrounding the irreversibly injured infarct core, referred to as the penumbra. Microglial cells are activated into two distinct phenotypes; however, the dynamics between the detrimental M1 phenotype and beneficial M2 phenotype are not fully understood. This talk will address recent data-driven modeling approaches to better understand and predict microglial cell dynamics during middle cerebral artery (MCAO)-induced ischemic stroke using phenotype-specific cell count data obtained from experimental studies. Each modeling approach incorporates aspects of inverse problems and uncertainty quantification in making forecast predictions.

Sequential time-stepping using only spatial parallelism is becoming a computational bottleneck because the world's largest parallel computers now have millions of parallel processor cores due to stagnating processor speeds. In this context, parallelization in time can provide additional concurrency leading to further speedups. Parallel-in-time methods have been demonstrated to work well for parabolic PDEs, but remain a challenge for hyperbolic PDEs. In this talk, we present new developments for the multigrid reduction-in-time (MGRIT) parallel-in-time method that solve nonlinear hyperbolic PDEs and hyperbolic systems in a small number of iterations with convergence speed independent of mesh resolution. Crucial ingredients include a careful choice of coarse-grid operators, linearization strategies, and preconditioners for the large algebraic equation systems to be solved. Results are presented for linear advection and linear acoustics, and for nonlinear problems from compressible fluid dynamics.

We will discuss some recent progress on mathematical understanding of score-based generative models, a highly successful paradigm for learning a probability distribution from data and generating further samples, with state-of-the-art performance. We will talk about recent results in convergence analysis of diffusion models and related flow-based methods. In particular, we established convergence of score-based diffusion models applying to any distribution with bounded 2nd moment, relying only on a $L^2$-accurate score estimates, with polynomial dependence on all parameters and no reliance on smoothness or functional inequalities. Time permits, we will also discuss applications and challenges of score-based generative models in scientific computing, in particular conditional generation via score-based priors.

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