Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Abstract. Sparse grids is a common strategy for mitigating the curse of dimensionality for problems with moderate number of dimensions. First developed in the contest of multidimensional quadrature, the approach has been very successful when handling problems of interpolation, regression and finding the solutions to partial differential equations. We will look at sparse grid interpolation and the theoretical results that guarantee convergence for different classes of problems, and we will then consider the challenges in extending the approach to a context of approximating discontinuous functions. Finally, we will conclude with some of our latest results in extending sparse grids to finite element and discontinuous Galerkin settings.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. We present the progress toward understanding the solution structures of transonic problems in multidimensional conservation laws. In applications such as the airflow near the wing tip for a high-speed aircraft or space shuttle, the flow makes an abrupt change in its direction, creating shock waves and turbulences depending on the contact angles and the speed of the aircraft. We have a well-known system, the compressible Euler system, to explain such applications and recent advancements in high-performing computing shed light on the shock configurations to understand such abrupt changes of the airflow near the wing. For some shock configurations, the governing system can be reduced in self-similar coordinates, making the problem more tractable to develop mathematical analysis. On the other hand, the system now changes it type, hyperbolic far from the locus of characteristics and mixed near the locus. I will present the mathematical advancements in understanding transonic problems for reduced systems.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. 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.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. This presentation outlines recent developments in advancing multiphase fluid dynamics simulations using Flash-X, with a focus on notable performance improvements resulting from its integration with AMReX. The talk provides an overview of Flash-X’s composable software architecture designed for modeling diverse simulations, including solid-liquid-gas interactions, phase transitions, and chemical transport, and compares its capabilities with those of existing open-source tools and commercial products. Additionally, this presentation outlines computational workflows centered around Flash-X, emphasizing its integration with scientific machine learning models. It identifies potential research directions that leverage Flash-X to establish a robust infrastructure for composability and performance portability in incompressible multiphase flows. The ultimate goal is to apply these advancements to address real-world engineering problems.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. In drug-resistant epilepsy, neurophysiological abnormalities have been identified in resting-state brain imaging data during seizure intervals (ictal), as well as between seizures (interictal), where the background brain activity is altered by abnormal brain discharges. These abnormalities have been linked to the epileptogenic zone, the brain area that is indispensable for the generation of seizures. During this presentation, we elaborate on the ability of coherent brain patterns, identified from intracranial electroencephalography (iEEG) data through signal processing and machine learning tools, to automatically delineate the epileptogenic zone and predict surgical outcome in children with drug-resistant epilepsy. The talk will provide a background about presurgical evaluation process and describe the proposed mathematical formulation that automates the estimation of the epileptogenic zone. We finally validate the proposed approach by statistically evaluating a cohort of epilepsy patients and discussing the clinical value of our methodology.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction of experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity.
More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties in preserving causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids.
In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues. This talk is intended for a general Math and Physics audience.
When: 4:00 pm (Lubbock's local time is GMT -6)
Where: room MATH 011 (basement)
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Abstract. Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. For the numerical solution of a general linear partial differential equation on a domain, Schwarz iterative methods use a decomposition of the domain into two or more (possibly overlapping) subdomains. In essence one is introducing new artificial boundary conditions on the interfaces between these subdomains. In the classical formulation, these artificial boundary conditions are of Dirichlet type. Given an initial approximation, the method progresses by solving for the PDE restricted to each subdomain using as boundary data on the artificial interfaces the values of the solution on the neighboring subdomain from the previous step. This procedure is inherently parallel, since the (approximate) solutions on each subdomain can be performed by a different processor. In the case of optimized Schwarz, the boundary conditions on the artificial interfaces are of Robin or mixed type. In this way one can optimize the Robin parameter(s) and obtain a very fast method.
Instead of using this method as a preconditioner, we use it as a solver, thus avoiding the pitfall of synchronization required by the inner products. In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. A coarse grid correction is added and one obtains a scalable method. Several theorems show convergence for particular situations. Numerical results are presented on large three-dimensional problems illustrating the efficiency of the proposed
asynchronous parallel implementation of the method. The main application shown is the calculation of the gravitational potential in the area around the Chicxulub crater, in Yucatan, where an asteroid is believed to have landed 66 million years ago contributing to the extinction of the dinosaurs.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. In decision-making applications where multiple forward simulations are needed, such as parameter study, design optimization, optimal control, uncertainty quantification, and inverse problems, we need to repeatedly solve forward problems. However, subject to the model complexity and the fineness of the discretization, the computational cost of forward simulations can be high. It may take a long time to complete a single forward simulation with the available computing resource. In this talk, we will introduce various reduced order modeling techniques, which aim to lower the computational complexity and maintain a good accuracy, including projection-based intrusive nonlinear model reduction and non-intrusive model reduction approaches. We will demonstrate the implementation of these reduced order modeling techniques in libROM (www.librom.net) and its application to numerical solvers for solving various physics problems.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. In recent years, there is an increasing interest in applying deep learning to geophysical/medical data inversion. However, direct application of end-to-end data-driven approaches to inversion have quickly shown limitations in the practical implementation. Indeed, due to the lack of prior knowledge on the objects of interest, the trained deep learning neural networks very often have limited generalization. In this talk, we introduce a new methodology of coupling model-based inverse algorithms with deep learning for two typical types of inversion problems. In the first part, we present an offline-online computational strategy of coupling classical least-squares based computational inversion with modern deep learning based approaches for full waveform inversion to achieve advantages that can not be achieved with only one of the components. In the second part, we present an integrated data-driven and model-based iterative reconstruction framework for joint inversion problems. The proposed method couples the supplementary data with the partial differential equation model to make the data-driven modeling process consistent with the model-based reconstruction procedure. We also characterize the impact of learning uncertainty on the joint inversion results for one typical inverse problem.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
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Abstract. Had Sir Isaac Newton possessed access to an NVIDIA GPU, he would not have wasted his time creating calculus. The realm of continuum mathematics presents formidable challenges to undergraduate students, whereas discrete mathematics offers a notably more intuitive approach that doesn't demand years of advanced study to comprehend. It is imperative that we equip our students with the tools necessary for conducting research at the outset of their academic journeys. Such an approach ignites their enthusiasm to tackle more demanding coursework and to direct their ambitions toward attacking the world's most challenging problems. Delaying their engagement with open problems until their graduate studies is a missed opportunity. Here we show how to integrate n-body techniques with NVIDIA GPUs to confront this very challenge.
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room MATH 011 (basement)
ZOOM details:
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