Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Abstract: A distinguishing feature of vehicular traffic flow is that it may exhibit significant wave patterns. We first demonstrate that those frustrating (when stuck in traffic) traffic features possess an intriguing structural beauty (when seen from the outside), rendering phantom traffic jam to be mathematical analogs of detonation waves. We then show how a few well-controlled automated vehicles can mitigate traffic instabilities and waves, first in theory and simulation, then in real-world traffic experiments. These culminate in the CIRCLES (Congestion Impacts Reduction via CAV-in-the-loop Lagrangian Energy Smoothing) project: the largest field test of deployed control vehicles on a fully instrumented highway, carried out by a consortium of mathematicians, engineers, industry partners, and government agencies.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
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* Meeting ID: 949 9288 2213
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Abstract: This talk explores advanced numerical techniques, multigrid methods and acceleration strategies, for solving partial differential equations (PDEs) and nonsmooth optimization problems.
Part I: Advanced Multigrid Methods We begin by introducing multigrid methods alongside local Fourier analysis, a mathematical tool for predicting multigrid performance. Recent interest in poroelasticity has highlighted the challenges of finite-element discretizations and preconditioners, stemming from strong coupling, saddle-point structure, and wide parameter ranges. We present a solver-friendly discretization of the poroelastic Biot model using reduced quadrature, enabling efficient monolithic multigrid methods. Local Fourier analysis guides parameter tuning, and numerical results confirm the robustness and efficiency of the approach.
Part II: Acceleration Techniques for Fixed-Point Iteration The second part focuses on acceleration methods, particularly Anderson acceleration (AA) and nonlinear GMRES (NGMRES), which enhance the convergence of fixed-point iterations. We apply AA to accelerate the alternating least squares method for canonical tensor decomposition and analyze its convergence properties. We propose a generalized alternating Anderson acceleration scheme--a periodic hybrid of fixed-point and AA steps with tunable window sizes--offering flexibility for both linear and nonlinear problems. To demonstrate the applicability of our novel approach, we apply it to accelerate the alternating direction method of multipliers (ADMM) in solving nonlinear, nonsmooth optimization problems. We also investigate NGMRES($m$) applied to Richardson iteration for solving linear systems, establishing theoretical connections with classical Krylov subspace GMRES. We propose an alternating NGMRES, and provide convergence analysis of NGMRES applied to nonlinear systems.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
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* Meeting ID: 949 9288 2213
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Abstract: Operator learning has undergone rapid development in recent years and gradually emerged as a transformative machine-learning paradigm. In this talk, I will provide an overview of the operator-learning framework and explore its applications to computational problems, with a particular focus on quantum many-body dynamics. We will discuss how traditional machine learning models, such as recurrent neural networks (RNNs), can be adapted to learn operators and predict the nonequilibrium dynamics of quantum many-body systems. Additionally, I will highlight how transformer-based neural operators can be employed to model the self-energy of strongly correlated systems. Through these examples, we aim to showcase the potential of operator learning to tackle key challenges in quantum many-body systems, especially when the dynamics are high-dimensional but not necessarily chaotic.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
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* Meeting ID: 949 9288 2213
* Passcode: Applied
Abstract: This talk explores the aerodynamics of complete aircraft geometries at the initial conceptual design stage and the importance of understanding the propulsion-related aerodynamics and performance of ducted fans and open rotors for their integration. A ducted rotor system was used to produce turbulent jets with a Reynolds number up to 5.97x105 and Mach number of 0.222 based on mean streamwise velocity. Three rotors with a diameter of 11.8 cm were manufactured and tested inside a duct with a 1 mm tip clearance at a speed up to 30, 000 revolutions per minute (rpm). three different blade planform shapes were used including a rectangular shape with constant chord, trapezoidal shape with a taper ratio of 0.5, and elliptical shape where the trailing edge of the blade is expressed with an elliptical function. The rotor thrust and electric power were measured, and the thrust coefficient and figure of merit was computed. The flow-field produced by the ducted rotors was measured in the near-field using laser Doppler velocimetry techniques. Time-averaged contours of cross-stream vorticity reveal intense hub and blade tip vortex structures, which are impacted by the shape of the blade, particularly in the blade tip region. Tip vorticity as well as streamwise turbulence intensity and turbulent kinetic energy in this region were mitigated for the rotors with trapezoidal and elliptical blades. In a related study, the blades of a 12 cm diameter ducted rotor system were coated with a sharkskin- inspired surface with diverging tip micropillars. Surfaces containing 40 μm and 70 μm tall micropillars were applied on the rotor blades in order to study their role on fan aerodynamics and downstream jet flow. The effect of the micropillar coatings on the rotor blades marginally increases the mean streamwise velocity and rotor figure of merit due to mitigating boundary layer separation at higher rotor speeds. Moreover, this occurs due to the micropillar’s ability to increase wall-normal turbulence intensity in the boundary layer when the pillar height is scaled appropriately to the boundary layer thickness. The rotor hub and blade tip vortex structures become diffused and undergo breakup into smaller structures accompanied with an acceleration in the decay of absolute mean cross-stream vorticity.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
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Abstract: In this work, we propose a new class of time integration methods, referred to as two-derivative exponential Runge--Kutta (TDexpRK) methods for stiff semilinear parabolic PDEs. Specifically, we construct TDexpRK integrators that inherit the favorable properties of both two-derivative Runge--Kutta (TDRK) methods and explicit exponential Runge--Kutta (ExpRK) methods. In particular, TDexpRK methods treat the stiff linear part exactly via exponential operator, while handling the nonlinear term with a two-derivative correction weighted by exponential $\varphi$-functions of the linear operator. The structure of our TDexpRK schemes enables a local error expansion involving only four stiff order conditions for methods up to fifth order, which is significantly fewer than the sixteen conditions required for ExpRK integrators. Based on this analysis, we rigorously prove convergence up to fifth-order accuracy, with an error bound that remains uniform with respect to the stiffness of the linear operator. As a result, we obtain high-order, explicit, stiffly accurate TDexpRK schemes that exhibit unconditional linear stability and require only a few stages per step (e.g., 2-stage 4th-order scheme). Numerical experiments on PDEs in one and two spatial dimensions confirm the superior accuracy and efficiency of the proposed methods compared with existing exponential Runge--Kutta/Rosenbrock schemes from the literature.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
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* Meeting ID: 949 9288 2213
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Abstract. Electrical Impedance Tomography (EIT) is a PDE-based inverse problem that aims to reconstruct electrical conductivity from boundary measurements of electrical currents and voltages. Mathematically, EIT is modeled by an elliptic equation with a variable conductivity function, and the goal is to infer this function from the Dirichlet-to-Neumann (DtN) map. One fundamental barrier in solving EIT is data scarcity, and in this talk we address this bottleneck by completing full data from partially observed DtN measurements.
Specifically, we train a conditional diffusion model to learn the distribution of DtN data and to infer full measurement vectors given partial observations. Our approach supports flexible source–receiver configurations and can be used as a plug-in preprocessing step with off-the-shelf EIT solvers. Under mild assumptions on the polygonal conductivity class, we derive nonasymptotic end-to-end bounds on the distributional discrepancy between the completed and ground-truth DtN measurements. In numerical experiments, we couple the proposed diffusion-based completion procedure with a deep learning–based inverse solver and compare its performance against the same solver using full measurement data. The results show that diffusion completion enables reconstructions that closely match the full-data baseline while using only 1% of the measurements. In contrast, standard baselines such as matrix completion require 30% of the measurements to achieve similar reconstruction quality.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
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* Meeting ID: 949 9288 2213
* Passcode: Applied
 | Wednesday
Apr. 08
4 PM
Math 011
|
| TBA
Chris Eldred
Sandia National Laboratories, Center for Computing Research
|
Abstract. TBA.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
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* Meeting ID: 949 9288 2213
* Passcode: Applied
 | Wednesday
Apr. 15
4 PM
Math 011
|
| TBA
Yulong Ying
The Ohio State University, Department of Mathematics
|
Abstract: TBA
This talk is co-sponsored with the Analysis seminar group.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room Math 011 (Math Basement)
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* Meeting ID: 949 9288 2213
* Passcode: Applied
 | Wednesday
Apr. 29
4 PM
Math 011
|
| TBA
Abner Salgado
University of Tennessee, Knoxville, Department of Mathematics
|
Abstract. TBA.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this link
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Join Meeting, then you will have to input the ID and Passcode by hand:
* Meeting ID: 949 9288 2213
* Passcode: Applied