**Location**: Pickard Hall, 4th Floor, Room 478

411 S. Nedderman Dr., Arlington, TX 76019

**Phone**: 817-272-3261

**Fax**: 817-272-5802

**Email**: math@uta.edu

# Colloquia & Seminars

### Fall 2024

**Title:** **"Boundary Problems In Rough Domains With Data in Weighted Morrey Spaces"**

**Dr. Marcus Laurel**

University of Texas at Arlington

**When:** Friday, October 11, 2024 from 2-3 pm

**Where:** Pickard Hall, Room 305

**Abstract:** The goal of this talk is to present a brief introduction to the method of layer potentials for solving boundary value problems on rough domains. Specifically, we work with the class of weakly elliptic, second-order, homogenous, constant (complex) coefficient systems in Euclidean space. We use singular integrals of layer potential type, which themselves can be defined on the class of uniformly rectifiable sets, the geometric measure theoretic sharp analogue of Lipschitz images. This requires a Calderón-Zygmund theory that works in such rough geometries as well as on the function spaces we have in mind. Specifically, we consider boundary problems where the boundary datum is arbitrarily chosen from a Muckenhoupt-weighted Morrey space (an offshoot of the scale of Muckenhoupt-weighted Lebesgue spaces), in which integrals over balls are bounded by a uniform constant multiplied by a specific power of the radii of the balls. We will see the delicate interplay between harmonic analysis, functional analysis, and geometry that leads to a well-posedness result for the Dirichlet Problem. This is joint work with Professor Marius Mitrea (Baylor University).

**Short Bio:** Dr. Marcus Laurel received his Ph.D. in mathematics in 2024 from Baylor University under the guidance of Professor Marius Mitrea. He works on the confluence of geometry, harmonic analysis, and PDE. His interests lie in layer potential methods to solve boundary value problems for elliptic systems, as well as function space theory in rough geometric settings. He, with Prof. Mitrea, recently published a book titled Weigthed Morrey Spaces: Calderón-Zygmund Theory and Boundary Problem. Currently, Dr. Laurel is an assistant professor of instruction at UT Arlington.

**Title: "Solving linear fractional differential equations with random non-homogeneous parts"**

**Dr. Laura Villafuerte
**The University of Texas at Austin

**When: **Friday, October 4, 2024 from 2-4 p.m.

**Where: **Pickard Hall, Room 311

**Abstract:** Experimental data and algorithms for certain real-world phenomena have shown that fractional order derivatives provide more efficient modeling than integer order derivatives. In these scenarios, using fractional differential equations rather than integer-order differential equations to describe these phenomena seems more appropriate. In addition, to consider the uncertainty arising from measurement errors and the complexity of the phenomena analyzed, randomness is included in the differential equations through their coefficients, initial conditions, and non-homogeneous parts. In this work, we investigate mean square solutions for some families of fractional linear differential equations with random non-homogeneous parts. This approach is based on the mean square Caputo derivative. For the sake of generality, we assume that the initial conditions and coefficients of the equations are random variables satisfying certain mild conditions. For this class of equations, we construct a generalized power series solution by using the mean square Laplace transform. Then, assuming an exponential growth condition on the force term, we show its mean square convergence. As a consequence of the mean square convergence, the convergence of the two first statistical moments, mean and variance, is guaranteed. Several examples are discussed to compare the fractional and integer order random differential equations utilizing its first two moments.

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**

**Title:** **"Bayesian Inversion Using Level Sets in Diffuse Optical Tomography"**

**Dr. Taufiquar Khan**

University of North Carolina at Charlotte

**When:** Friday, September 27, 2024 from 2-4 pm

**Where:** Pickard Hall, Room 311

**Abstract:** In this talk, we will provide an overview of the ill-posed inverse problem in Diffuse Optical Tomography (DOT) at an introductory level. We will discuss several regularization approaches to solve the ill-posed inverse problem in DOT including deterministic, statistical, and machine learning. Then we will present our most recent work using Bayesian Inversion Using level sets for image reconstruction in DOT. The results of image reconstruction will be demonstrated using synthetic data for the recently proposed algorithm. This is joint work with Anuj Abhishek (Case Western Reserve University) and Thilo Strauss (Xi’an Jiaotong-Liverpool University).

**Short Bio:** Taufiquar Khan is currently a Professor and the Chair of the Department of Mathematics and Statistics, University of North Carolina at Charlotte (UNC Charlotte). He was a Professor and an Associate Director of Graduate Studies of the School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA, before joining UNC Charlotte. He is a recipient of the Humboldt Fellowship from Germany. His research interests include machine learning, inverse problems involving ordinary and partial differential equations. His present and past research have been supported through the NSF, DOD, Humboldt Foundation, and the industry.

**Title:** **"A Meta-analysis based Hierarchical Variance Model for Powering One and Two-sample t-tests**

**Jackson Barth, PhD**

Assistant Professor, Department of Statistical Science, Baylor University

**When:** Friday, September 20, 2024 from 3:30-4:20 pm

**Where:** Pickard Hall, Room 110

**Abstract:** Sample size determination (SSD) is essential in statistical inferenceand hypothesis testing, as it directly affects the accuracy and power of the analysis.We propose a SSD methodology for one and two-sample t-tests that ensuresclinical relevance using a pre-determined unstandardized effect size. Our novelapproach leverages Bayesian meta-analysis to account for the uncertainty surroundingthe variance, a common issue in SSD. By incorporating prior knowledge fromrelated studies via a Bayesian gamma-inverse gamma model, we obtain an informativeposterior predictive distribution for the variance that leads to better decisionsabout sample size. For efficient posterior sampling, we propose an empirical Bayesapproach, which is further combined with a quantile simulation approach tofacilitate computation. Simulations and empirical studies demonstrate that ourmethodology outperforms other aggregate approaches (simple average, weightedaverage, median) in variance estimation for SSD, especially in meta-analyses withlarge disparity in sample size and moderate variance. Thus, it offers a robust andpractical solution for sample size determination in t-tests.

### Spring 2024

**Title: **"**Interplay of Linear Algebra, Machine Learning, and High Performance Computing**"

**Dr. Xiaoye Sherry Li**

Lawrence Berkeley National Laboratory

**When: **Friday, April 5, 2024 from 3-4 pm

**Where:** Pickard Hall, Room 110

**Abstract: **In recent years, we have seen a large body of research using hierarchical matrix algebra to construct low complexity linear solvers and preconditioners. Not only can these fast solvers significantly accelerate the speed of large scale PDE based simulations, but also they can speed up many AI and machine learning algorithms which are often matrix-computation-bound. On the other hand, statistical and machine learning methods can be used to help select best solvers or solvers' configurations for specific problems and computer platforms. In both of these fields, high performance computing becomes an indispensable cross-cutting tool for achieving real-time solutions for big data problems. In this talk, we will show our recent developments in the intersection of these areas.

**Short Bio:** Dr. Xiaoye S. Li is a Senior Scientist in the Computational Research Division, Lawrence Berkeley National Laboratory. Dr. Li earned her Ph.D. in Computer Science from UC Berkeley in 1996, MS in Math & Computer Science from Penn State Univ. and B.S. in Computer Science from Tsinghua Univ. She has worked on diverse problems in high performance scientific computations, including parallel computing and sparse matrix computations. She has authored over 130 publications, and is the lead developer of SuperLU, a widely-used sparse direct solver, and has contributed to the development of several other mathematical libraries, including LAPACK and XBLAS. She has served on the editorial boards of the ACM Trans. Math. Software, IJHPCA, and SIAM J. Scientific Comput., as well as many program committees of the scientific conferences. She is a Fellow of SIAM and a Senior Member of ACM.

** **

**Title: **"**How do Immune Cells Kill Tumor Cells?”**"

**Ami E. Radunskaya, PhD**

Lingurn H. Burkhead Professor of Mathematics at Pomona College, CA

**When: **Friday, February16, 2024 from 10-11 am

**Where:** Pickard Hall, Room 311

**Abstract: **The immune system is able to fight cancer by mustering and training an army of effector “killer” cells. Mathematical models of tumor-immune interactions must describe the proliferation, recruiting and killing rates of immune cells. Earlier work surprisingly showed that the functions describing the kill rates distinguish between two types of immune cells. The mechanisms behind these differences have been a mystery, however. In an attempt to unravel this mystery, we have created a cell-based fixed-lattice model that simulates immune cell and tumor cell interaction involving MHC recognition, and two killing mechanisms. These mechanisms play a big role in the effectiveness of many cancer immunotherapies. Results from model simulations, along with theories developed by ecologists, can help to illuminate which mechanisms are at work in different conditions.

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**Title: **"**What is Liutex: Examples of Hurricane and Tornado Vortex Visualization using Liutex**"

**Oscar Alvarez**

University of Texas at Arlington

**When: **Friday, February 2, 2024, from 2-3 pm

**Where:** Pickard Hall, Room 311

**Abstract: **A vortex is a common phenomenon that occurs in fluid flow, especially when studying turbulence. It is to say, understanding vortices is essential knowledge for scientists, researchers, and engineers doing work in the field of fluid mechanics. In the past, scientists such as Helmholtz (1858) had a desire to understand the physical nature of vortices and popularized the idea of using vorticity to define a vortex in a fluid field. Vorticity became the default for studying vortices and vortex identification methods were created that were built on the idea of vorticity. This went on for a long time until only recently, Dr. Chaoqun Liu discovered the Liutex method in 2018. The Liutex method started a new generation of vortex identification methods. Liutex is based directly on the rotation of a fluid as opposed to vorticity which also contains shear. It has been shown that vorticity cannot distinguish between shear and rotation. An immediate counterexample for the invalidity of vorticity to define a vortex can be found near the boundary wall of a boundary layer where vorticity is large but there is no rotation or no vortex. Liutex has now become a well-known vortex identification method. Researchers around the world agree that Liutex mathematically defines what a vortex is. In this presentation I hope to explain what Liutex is. I will also show some examples of Liutex being applied using real/experimental hurricane data provided by the National Oceanic and Atmospheric Administration (NOAA) and simulated high-resolution tornado data with 250 billion grid points provided by a senior research scientist in University of Wisconsin.

**Short Bio:** Oscar Alvarez is a Research Scientist 1 at the University of Texas at Arlington Research Institute (UTARI) in Fort Worth, Texas. He also works under the supervision of Dr. Chaoqun Liu as a mathematics PhD student studying Liutex.