Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Abstract. We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so–called two–scale methods. We show that this method is convergent, and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms are also considered. Joint work with Wenbo Li (LSEC, Chinese Academy of Sciences).
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Finite element methods for H(curl) interface equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to H(curl) interface problems. In this talk, we will present a novel immersed virtual element method (IVEM) for solving a 3D H(curl) interface problems. The motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence for a class of 3D H(curl) interface problem. To develop a systematic framework, a de Rham complex for H1, H(curl), and H(div) interface problems will be established based on which the Hiptmair-Xu (HX) preconditioner can be adapted to develop a fast solver for the H(curl) interface problem. An efficient polyhedral mesh generator is also provided to generate a polyhedral mesh with an interface fitted boundary triangulation.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Lorentz electrodynamics with point charges is notoriously ill-defined. So is quantum electrodynamics (QED). Already in the 1930s Max Born suggested that the problems could be overcome by changing the ``law of the pure ether'' from Maxwell's and Lorentz' linear identification of E with D and of B with H, into some nonlinear algebraic relation. Subsequently he was joined by L. Infeld who contributed an insight that led to the well-known Born-Infeld (BI) law. In 1940 Fritz Bopp picked up on Born's ideas and suggested a linear higher-derivative law as an alternative. This was independently suggested by Lande and Thomas, and picked up by Podolsky (BLTP), and implemented into QED by Feynman. A few years ago, the joint initial value problem for BLTP electrodynamics was shown to be locally (in time) well-posed. The analogous result for BI electrodynamics is expected but no proof is in sight. The talk will survey the conceptual problems and outline the gist of the proof of the local well-posedness of the BLTP initial value problem.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Mathematical models in the applied sciences often involve an array of application-related parameters whose values are uncertain due to factors such as incomplete data, imprecise measurements, etc. In these cases, it is important to be able to determine how the stochasticity in these values will percolate into a computational approximation of the solution to the model. This is often quantified by sampling different values of the parameters and using this information to estimate statistical moments of the solution. When the model involves the solution of a partial differential equation the sampling process can become computationally expensive. In this talk, motivated by an application in plasma physics, we will explore the use of surrogates and multi-level computations to mitigate the computational cost of sampling. The work was done in collaboration with Howard Elman (University of Maryland) and Jiaxing Liang (Rice University)
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Guidance, navigation, and control of satellites--especially in the study of rendezvous and formation flight--relies heavily on methods that leverage local linear approximations of dynamical systems and measurement functions. This talk focuses mainly on cases in which linear approximations are insufficient to solve dynamics, control, or estimation problems. In such cases, we employ tools from numerical multilinear algebra on tensors arising from higher-order Taylor series. The inherently quadratic nature of some quantities, the linear unobservability of some estimation problems, and the need to quantify the performance of linear methods make these higher-order techniques useful in the setting of guidance, navigation, and control. In particular, tensor eigenvalues will be employed to compute operator norms of tensors associated with the error of linearized dynamics propagation, linearized boundary value problem solutions, and extended Kalman filter measurement updates.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. In today's highly interconnected global economy, financial markets are exposed to a variety of risks, ranging from geopolitical tensions to unexpected market shocks. This talk explores advanced methodologies for managing risk, modeling volatility, and forecasting potential market crashes. Drawing on recent historical events, such as the 2008 financial crisis and the COVID-19 crash, we will examine key risk factors that may precipitate future market downturns. Special focus will be placed on fat-tailed distributions and the importance of considering asymmetric dependencies in financial models. Using tools such as tempered stable models, Expected Tail Loss (ETL), and other advanced risk measures, we will illustrate how these frameworks can provide a more accurate understanding of market behavior, especially in periods of high volatility. The implications for risk management, option pricing, and portfolio optimization will be discussed, offering insights into how investors and financial institutions can better navigate periods of market turbulence
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Vector autoregressive (VAR) models are popular in econometrics due to their flexibility in capturing dynamic relationships between variables. The fact that VARs are also
tailor-made for the determination of Granger-causal relationships, has recently sparked great interest in neuroscience, which seeks to understand how regional brain activation signals connect with complex functional circuitry. In such applications, high-dimensional models are the norm; however these tend to exhibit an inherent sparsity structure. In this study, we propose two different
types of algorithm for sparse VAR model identification, which are able to handle situations where the number of parameters ($m$) is comparable to the sample size ($n$). Both methods rely on individual coefficient p-values as the basis for sparsification. The thresholding method (TLSE) simply declares as non-active those coefficients whose p-value exceeds a cuttoff. The information criterion based method (BLSE), uses the initial p-value ranking as the basis for fitting increasingly larger models in a stepwise manner, identifying as optimal the model with smallest BIC value. We show both methods enjoy an asymptotic oracle property whereby active coefficients are correctly identified in the limit as $n\rightarrow\infty$. Simulations comparing these to existing methods (mostly lasso variants), suggest that the new methods are better at sparsity pattern recovery. The methodology is illustrated on high-dimensional econometric and neuroscience real datasets with the aim of unveiling the strength of Granger-causal flow between nodes.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Pulsed power drivers like the Z-machine at Sandia are effective tools for probing wide ranging material states. Effective modeling tools are required to design experiments and to back out and improve property descriptions. A stripline target may be modeled by a 2D resistive magnetohydrodynamic model but in this geometry the far field must be included. We describe an approach to derive and implement local DtN far field boundary conditions applicable for non-circular boundaries for the scalar vector potential. We also discuss our experience with using a modern software engineering approach for managing computational complexity and reproducibility.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Instantaneous time mirrors (ITMs) were recently introduced by M. Fink and collaborators as a new avenue for time reversal. The latter allows for the focusing of waves, whether acoustic, electromagnetic or elastic, and has found many important applications in medical imaging, non-destructive testing, and telecommunications for instance. The main practical difficulty of standard time reversal is the recording/reversal process which necessitates a quite complex apparatus. ITMs offer on the contrary a simplified experimental alternative that does not require any measurements, provided there is some control over the medium of propagation. We will review in this talk the basics of the time reversal of waves introduced in the nineties, and discuss the recent ITMs and some of their mathematical properties.
When: 3:30 pm (Lubbock's local time is CDT, GMT -5)
Where: room MATH 011 (basement)
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Moving boundary (or often called “free boundary”) problems are ubiquitous in nature and technology. A computational perspective of moving boundary problems can provide insight into the “invisible” properties of complex dynamics systems, advance the design of novel technologies, and improve the understanding of biological and chemical phenomena. However, challenges lie in the numerical study of moving boundary problems. Examples include difficulties in solving PDEs in irregular domains, handling moving boundaries efficiently and accurately, as well as computing efficiency difficulties. In this talk, I will discuss three specific topics of moving boundary problems, with applications to ecology (population dynamics), plasma physics (ITER tokamak machine design), and cell biology (cell movement). In addition, some techniques of scientific computing will be discussed.
Short Bio: Dr. Shuang Liu currently is an assistant professor in the Department of Mathematics at the University of North Texas. During January 2021 and June 2023, she was a SEW Assistant Professor in the Department of Mathematics at University of California, San Diego. Before that, she was a Postdoc Research associate in applied mathematics and plasma physics group at Los Alamos National Laboratory. Dr. Liu received her PhD in 2019 from the Department of Mathematics from the University of South Carolina.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. In multiscale and multiphysics simulation, a predominant challenge is the accurate coupling of physics of different scales, stiffnesses, and dimensionalities. The underlying problems are usually time dependent, making the time integration scheme a fundamental component of the accuracy. Remarkably, most large-scale multiscale or multiphysics codes use a first-order operator split or (semi-)implicit integration scheme. Such approaches often yield poor accuracy, and can also have poor computational efficiency. There are technical reasons that more advanced and higher order time integration schemes have not been adopted however. One major challenge in realistic multiphysics is the nonlinear coupling of different scales or stiffnesses. Here I present a new class of nonlinearly partitioned Runge-Kutta (NPRK) methods that facilitate high-order integration of arbitrary nonlinear partitions of ODEs. Order conditions for an arbitrary number of partitions are derived via a novel edge-colored rooted-tree analysis. I then demonstrate NPRK methods on novel nonlinearly partitioned formulations of thermal radiative transfer and radiation hydrodynamics, demonstrating orders of magnitude improvement in wallclock time and accuracy compared with current standard (semi-)implicit and operator split approaches, respectively.
This talk may be seen by TTU eraider account holders in the Texas Tech Mediasite Catalog.
Abstract. Mathematical models of complex physical and biological systems play a crucial role in understanding real world phenomena and making predictions. Examples include models of weather systems, ocean circulation, ice-sheet dynamics, porous media flow, or spread of infectious diseases. Models governing complex systems typically include parameters that are uncertain and need to be estimated using indirect measurements. This is done by solving an inverse problem that uses the model and measurement data to estimate the unknown parameters. Optimal experimental design (OED) comprises a critical component of parameter estimation: it provides a rigorous framework to guide acquisition of data, using limited resources, to construct model parameters with minimized uncertainty. In this talk, we consider OED for inverse problems governed by partial differential equations (PDEs) with infinite-dimensional inversion parameters. Our focus will be the problem of finding an optimal placement of sensors where data are collected. I will discuss some mathematical and computational aspects of such optimal sensor placement problems, some of the recent advances in the field, as well as some interesting research questions.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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