Colloquia
Department of Mathematics and Statistics
Texas Tech University
Stochastic optimization has become a cornerstone of modern data analysis, powering large-scale statistical estimation and machine learning. While computational efficiency has been extensively studied, statistical inference, such as quantifying uncertainty in estimates obtained from stochastic algorithms, remains less explored. In this talk, I will present two related projects that integrate statistical inference into batch-based stochastic optimization algorithms. The first project considers the classical setting where data are independent and identically distributed (i.i.d.). We establish a Central Limit Theorem for batched stochastic gradient descent (SGD) algorithms, which enables the construction of valid confidence intervals. The second project extends these ideas to dependent data, such as time series and reinforcement learning trajectories. A key contribution is to show that in the presence of dependence, batch SGD is essential for valid statistical inference, in contrast to vanilla SGD. Together, these projects establish a unified framework for scalable stochastic optimization with statistical guarantees and advance both the theoretical foundations and practical applications of big data analytics.
Dr. Liu's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Mathematical models provide valuable insight into how and why things change. If we build models in terms of observed behaviors, we can analyze how these behaviors impact overall outcomes. If we incorporate factors that we expect to change or vary, we can explore theoretical scenarios under novel conditions. In this presentation, I will focus on how we apply such modelling principles in mathematical biology, with an emphasis on ecological interactions and insect populations. Insects are highly prevalent and directly impact people in a variety of ways; they can be beneficial (providing services like pollination or biological control) or detrimental (spreading diseases or acting as pests). In either case, mathematical models provide valuable information about what drives those impacts and how we might expect them to change with management choices or weather patterns. I will also briefly discuss my teaching and service experience in the department.
Dr. Laubmeier's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Since the emergence of the Calculus of Variations, understanding the shape of equilibria for certain functionals has played a central role in Differential Geometry and Geometric Analysis.
Geometric variational problems are characterized by energies whose Lagrangians depend on geometric invariants. However, there exist as well other functional arising from physical contexts which are closely related to these geometric variational problems.
In this talk, we will examine the history behind several of these pioneering variational problems and their development up to the present, paying special attention to the speaker's own results in the last three years.
Dr. Pampano's Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Coupled hyperbolic problems are PDE systems that have a hybrid nature. Part of these systems may have a Hilbert space structure; therefore, they are tractable using traditional variational techniques. However, the other portion of these systems usually corresponds to nonlinear wave behavior and cannot be treated with the usual Galerkin methods. In this talk, we provide some background and examples of these systems, discuss what makes them interesting and what makes them relevant. Moving to the central portion of the talk, we discuss a new numerical method in order to solve the compressible magneto-hydrodynamics (MHD) system. In essence, the MHD system is decomposed into two components: a vanishing-viscosity PDE and a purely Hamiltonian PDE. This is a significant point of departure from the dominant body of literature advocating for divergence formulations. Our approach involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom in the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. Similarly, if the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. To the best of our knowledge, it is the first scheme in the literature capable of preserving invariant-domain properties, total energy, and involution constraints exactly.
Dr. Tomas' Tenure and Promotion Colloquium has been archived in the Mediasite catalog.
Abstract pdf available
Everyone is encouraged to attend in person, but this Departmental Colloquium, sponsored by the Differential Geometry, PDE and Mathematical Physics seminar group, may be virtually attended Friday the 3rd at 3:00 PM CDT (UT-5) via this Zoom link.
Meeting ID: 920 3037 7029
Passcode: Suceava