Applied Mathematics and Machine Learning
Department of Mathematics and Statistics
Texas Tech University
Humans have been discussing the benefits and drawbacks of democratic vs. authoritarian governance for millennia, with perhaps the earliest and most famous discussion favoring enlightened authoritarian rule contained in Plato's 'Republic'. The commonly cited benefits of an enlightened dictatorial regime are the efficiency of governance and long-term horizon in planning due to the independence of frequent election cycles. To analyze the claims of potential superiority of an authoritarian rule, we develop a simple mathematical theory of a dictatorship in the ideal case of a dictator wanting the best outcome for the country, with the additional external noise describing external and internal challenges in the country's path.
We assume the linear proportional feedback control based on the information provided by the output from the advisors. The resulting stochastic differential equations (SDEs) describe the evolution of both the trajectory of a country's well-being and the accuracy of advisors' information. We show the system's inherent instability due to the corruption of the advisor's information provided to the dictator. While the system without noise does possess a large amount of phase space with stable solutions, the noise pushes all solutions to the unstable regime. We show that there is a typical unstable time scale, and describe the long-term evolution of the system using asymptotic solutions, some results from the theory of SDEs and phase space analysis. We also discuss the application of the theory to historical data on grain harvest in the Soviet Union. No previous knowledge of Ito's calculus or theory of SDEs is assumed; I will provide all necessary background from that theory during the lecture.
When: 4:00 pm (Lubbock's local time is GMT -6)
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Surface quasi-geostrophic equations is a special example of an active scalar (other examples include Burgers' and porous media equations) with many applications in geophysics and fluid mechanics. It has caught much attention from mathematicians due to its similarity to the Navier-Stokes/Euler equations in terms of the structures of the equations, as well as the behavior of its solutions according to numerical simulations. Despite the difficulty created by its non-linear term (the Fourier symbol of the velocity is odd in frequency due to Riesz transform), convex integration technique has been successfully applied to the surface quasi-geostrophic equations in the past several years, and we now know that there exist infinitely many weak solutions (e.g., one can construct solutions with prescribed energy). We review these results and remaining open problems in both deterministic and stochastic cases..
When: 4:00 pm (Lubbock's local time is GMT -6)
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The Navier-Stokes equations at high Reynolds are well known to have very chaotic dynamics. One way to measure this chaos is via the top Lyapunov exponent, a number which measures the infintesimal exponential rate of separation of nearby trajectories. A positive Lyapunov exponent is seen as a hallmark of chaos. Despite this conceptually simple characterization, proving positivity of the top Lyapunov exponent is notoriously challenging, especially in infinite or high dimensional dynamical systems. In this talk, I will present some recent progress in this direction for arbitrary Galerkin truncations of the stochastic 2d Navier-Stokes equations on the Torus. Using tools from the theory of random dynamical systems and Hormander's hypoelliptic theory, I will present a new estimate that gives a quantitative lower bound on the top Lyapunov exponent in terms of fractional regularity of a certain invariant probability measure that tracks the statistics of unstable tangent directions. Using computer assisted techniques from algebraic geometry to veryify a certain Lie algebra condition, I will explain how this estimate gives rise to the first rigorous proof of a positive Lyapunov exponent for the Galerkin-Navier-Stokes system for arbitrary large frequency cutoff. The above technique is quite general and can be applied to many other high-dimensional dynamical systems with stochastic forcing. This work is joint with Jacob Bedrossian and Alex Blumenthal.
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Pulsed power is the science and engineering of accumulating large amounts of energy over a long duration and releasing that energy over a very short duration, typically on a timescale ranging from picoseconds to microseconds. Releasing large amounts of electrical energy on such short timescales is associated with tremendous voltages and currents, often kilovolts to megavolts and kilo-amps to mega-amps, with instantaneous power ranging from gigawatts to nearly a petawatt. A biproduct of this extreme environment are very high electric and magnetic fields, the establishment of various classes of plasmas, and ions and electrons accelerated to highly relativistic energies.
Naturally, the pulsed power community faces a number of mathematical and computational challenges. These range from molecular dynamics models of gas, electron, and ion emission from solids and dielectrics, various plasma models (fluid, particle, magnetohydrodynamic, and more), electromagnetic models, and often multi-physics combinations of these models and more. More recently, these models have been combined with modern optimization and machine learning techniques to develop next generation pulsed power systems with state-of-the-art capabilities. This talk will discuss a few of the mathematical and computational challenges in the pulsed power community.
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The construction of robust and accurate numerical methods is essential for simulating complex fluid dynamics. Most partial differential equations that arise in such areas often exhibit certain physical and thermodynamic properties that should be preserved at the discrete level. We call such numerical schemes structure-preserving. Structure-preserving approximation techniques provide theoretical guarantees of reliability for situations where ad-hoc stabilization techniques can fail. In this talk, we give an introduction of relevant PDEs in the field of computational fluid dynamics such as the Shallow Water Equations. We then describe a dispersive extension of the SWEs known as the Serre-Green-Naghdi Equations which are applicable in coastal hydrodynamics. We discuss the respective structure-preserving approximation technique for the two models. We then conclude with a short overview on the verification and validation process in scientific computing.
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Abstract. Although turbulence affects natural and engineered systems from sub-meter to planetary scales, fundamental understanding and predictive modeling of turbulence continue to defy and bedevil scientists and engineers. This talk summarizes our work during the past decade in leveraging disparate data (ranging from sparse observations to full-field data) to enhance Reynolds-averaged Navier-Stokes (RANS) turbulence models in a physics-informed framework. Specifically, I will present (1) using sparse data to infer Reynolds stress fields based on ensemble data assimilation for reducing RANS model uncertainties, and (2) our ongoing work in unifying data assimilation and neural networks for parallel learning of turbulence models with quantified uncertainties.
About the speaker. Dr. Heng Xiao is Professor at the University of Stuttgart, Germany. He holds a bachelor’s degree from Zhejiang University, China, a master’s degree from the Royal Institute of Technology (KTH), Sweden, and a Ph.D. degree from Princeton University, USA. From 2009 to 2012, he worked as a postdoctoral researcher at ETH Zurich, Switzerland. Between 2013 and 2022, he was a faculty member in the Department of Aerospace and Ocean Engineering at Virginia Tech. In 2013 he moved to Stuttgart to take the chaired professorship in Data-Driven Fluid Dynamics. His current research focus on turbulence modeling with data-driven methods, including data assimilation, machine learning, and uncertainty quantification..
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About the Applied Mathematics Seminar: to be notified about TTU's Applied Mathematics Seminar on a weekly basis please send e-mail to igtomas@ttu.edu
Abstract. In this talk, I shall first discuss a newly developed theory of weak fractional (differential) calculus and fractional Sobolev spaces. The focus is the introduction of a weak fractional derivative concept which is a natural generalization of integer order weak derivatives and helps to unify multiple existing fractional derivative concepts. Based on the weak fractional derivative concept, new fractional-order Sobolev spaces can be naturally defined and many essential properties of those Sobolev spaces can also be established. I shall then introduce a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations, the new framework and theory are based on the aforementioned theory of weak fractional derivatives and their associated fractional order Sobolev spaces. It leads to new fractional differential equations, including one-side fractional Laplace operators and future value problems. Finally, if time permits, I shall also briefly introduce some new finite element (and DG) methods for approximating the weak fractional derivatives and the solutions of fractional calculus of variations problems and their associated fractional differential equations.
About the speaker. Dr. Xiaobing Feng is a professor and the department head of the Department of Mathematics at the University of Tennessee. He obtained his Ph.D. degree from Purdue University in Computational and Applied Mathematics in 1992 under the direction of the late Professor Jim Douglas, Jr. His primary research interest is numerical solutions of deterministic and stochastic nonlinear PDEs which arise from various applications including fluid and solid mechanics, subsurface flow and poroelasticity, phase transition, forward and inverse scattering, image processing, optimal control, systems biology, and data assimilation.
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The numerical solution of the radiation transport equation (RTE) is challenging due to the high computational costs and the large memory requirements caused by the high-dimensional phase space. Here we detail an attempt to reduce the memory required, and computational cost of solving RTE by applying the dynamical low-rank (DLR) method, where a memory savings of about an order of magnitude without sacrificing accuracy is observed. The DLR approximation is an efficient technique to approximate the solution to time-dependent matrix differential equations. The desired approximation has three components similar to factors in singular value decomposition (SVD), and each of them is solved by integrating the matrix differential equation projected onto the tangent space of the low-rank manifold. This talk presents our recent work that builds on the established DLR method and aims to enable low-rank schemes for practical radiation transport applications. We propose a high-order/low-order (HOLO) algorithm to overcome the conservation issues in the low-rank scheme by solving a low-order equation with closure terms computed via a high-order solution calculated using DLR. With the properly chosen rank, the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. This improvement goes a long way to making the method robust enough for a variety of physics applications. We also introduce a low-rank scheme with discrete ordinates discretization in angle (SN method). This low-rank-SN system allows for an efficient algorithm called ``transport sweep," which is highly desirable in computation. The derived low-rank SN equations can be cast into a triangular form in the same way as standard iteration techniques.
When: 4:00 pm (Lubbock's local time is GMT -5)
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About the speaker. Dr. Ryan McClarren, Associate Professor of Aerospace and Mechanical Engineering at the University of Notre Dame, has applied simulation to understand, analyze, and optimize engineering systems throughout his academic career. He has authored numerous publications in refereed journals on machine learning, uncertainty quantification, and numerical methods, as well as three scientific texts: Machine Learning for Engineers, Uncertainty Quantification and Predictive Computational Science: A Foundation for Physical Scientists, and Engineers and Computational Nuclear Engineering and Radiological Science Using Python. He was recently named Editor-in-Chief of the Journal of Computational & Theoretical Transport. A well-known member of the computational nuclear engineering community, Dr. McClarren has won research awards from NSF, DOE, and three national labs. Prior to joining Notre Dame in 2017, he was Assistant Professor of Nuclear Engineering at Texas A&M University, and previously a research scientist at Los Alamos National Laboratory in the Computational Physics and Methods group.
Abstract. Our goal is to develop and analyze numerical methods for incompressible flows with variable density and viscosity that can be applied to a large range of problems in engineering and geophysics. We will introduce novel numerical methods that are suitable for high order finite element and spectral methods. Moreover, for computational efficiency, the stiffness matrices of the methods considered are made time independent. First, we present a semi-implicit scheme based on projection methods and the use of the momentum, equal to the density times the velocity, as primary unknown. We analyze the stability and convergence properties of the method and establish a priori error estimates. A fully explicit version of the scheme is then proposed. Its robustness and convergence properties are studied with a pseudo spectral code over various setups involving large ratio of density, gravity and surface tension effects. Then we present a novel method based on artificial compressibility technique and we compare its robustness with the above projection-based method. Applications to magnetohydrodynamics instabilities in industrial setups such as liquid metal batteries and aluminum production cell will be also presented shortly.
About the speaker. In 2015, Loic Cappanera completed his PhD in Fluid Mechanics at Paris-Saclay University. Between 2016 to 2019, he worked as a postdoc at TAMU and Rice University. In 2019, he started to work in the Mathematics Department at the University of Houston as an Assistant Professor.
When: 4:00 pm (Lubbock's local time is GMT -6)
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Abstract. Schrödinger operators with random potentials are very important models in quantum mechanics, in the study of transport properties of electrons in solids. In this talk, we study the approximation of eigenvalues via the landscape theory for some random Schrödinger operators as well as some related models. The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the landscape function u solving Hu=1 for an operator H. The landscape theory has remarkable power in studying the eigenvalue problems for a large class of operators and has led to numerous “landscape baked” results in mathematics, as well as in theoretical and experimental physics. We first give a brief review of the localization landscape theory. Then we focus on some recent progress of the landscape-eigenvalue approximation for operators on general graphs. We show that the maximum of the landscape function is comparable to the reciprocal of the ground state eigenvalue, for Anderson or random hopping models on certain graphs with growth and heat kernel conditions, as well as on some fractal-like graphs such as the Sierpinski gasket. There will be precise asymptotic behavior of the ground state energy for some 1D chain models, as well as numerical stimulations for excited states energies.
About the speaker. Dr. Shiwen Zhang joined the Department of Mathematics & Statistics at University of Massachusetts Lowell as an Assistant Professor in August 2022. Shiwen obtained his PhD from University of California, Irvine in 2016, under the supervision of Svetlana Jitomirskaya. From 2016 to 2022, he worked as a postdoc at Michigan State University, and then at University of Minnesota. Shiwen’s research field is analysis, including Mathematical Physics, dynamical systems, PDEs, and spectral theory of Schrödinger operators. In particular, Shiwen is interested in quantum localization in a disordered medium, dynamical and fractal properties of quasiperiodic operators. More recently, Shiwen works in collaboration with experts in numerical analysis and physicists, on problems of scientific computing of eigenvalues and eigenfunctions, arising from models in physics and semi-conductor devices.
When: 4:00 pm (Lubbock's local time is GMT -5)
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Abstract. This presentation explores an alternate quantum framework in which the wavefunction Ψ(t, x) plays no role. Instead, quantum states are represented as ensembles of real-valued probabilistic trajectories, x(t, C), where C is a trajectory label. Quantum effects arise from the mutual interaction of different trajectories or “worlds,” manifesting as partial derivatives with respect to C. The quantum trajectory ensemble x(t, C) satisfies an action principle, leading to a dynamical partial differential equation (via the Euler-Lagrange procedure), as well as to trajectory-based symmetry and conservation laws (via Noether’s theorem). Several of these correspond to standard laws, e.g. conservation of energy. However, one such trajectory-based law (pertaining to curl-free velocity fields) appears to have no standard analog.
A full understanding of the new trajectory-based conservation law may require relativistic considerations. Whereas an earlier, non-relativistic version of the trajectory-based theory turns out to be mathematically equivalent to the time-dependent Schrödinger equation, the relativistic generalization (for single, spin-zero, massive particles) is not equivalent to the Klein-Gordon (KG) equation—and in fact, avoids certain well-known problems of the latter, such as negative (indefinite) probability density. It is precisely the new trajectory-based conservation law that makes this possible. The new relativistic quantum trajectory equations could in principle be used in quantum chemistry calculations, and otherwise could lead to new physical predictions that could be validated or refuted by experiment.
About the speaker. Dr. Bill Poirier is Chancellor’s Council Distinguished Research Professor and also Barnie E. Rushing Jr. Distinguished Faculty Member at Texas Tech University, in the Department of Chemistry and Biochemistry and also the Department of Physics. He received his Ph.D. in theoretical physics from the University of California, Berkeley, followed by a chemistry research associateship at the University of Chicago. His research interest is in understanding and solving the Schrödinger equation, from both foundational and practical perspectives. He has given well over 100 oral presentations on quantum mechanics, to both scientific and general audiences, and published over 100 papers in this field. He is also the creator of the continuous "many interacting worlds" interpretation of quantum mechanics.
When: 4:00 pm (Lubbock's local time is GMT -5)
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Abstract. The talk focuses on a discontinuous Galerkin method for the time harmonic Maxwell system. This method is based on the use of a finite element grid, but uses plane wave solutions of Maxwell's equations on each element to approximate the global field. Because each basis function satisfies Maxwell's equations, the problem can be reduced to a coupled linear system on the faces of the grid. Arbitrarily high order of convergence can be achieved by taking more planes waves in suitable directions element by element, although ill-conditioning must be carefully controlled. Unfortunately this method has severe deficiencies when applied to some problems involving screens and transmission lines. To remedy this, we have coupled polynomial finite element methods with the plane wave scheme. I shall report on the current state of this effort.
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Abstract. Continuum plasma physics models are used to study important phenomena in astrophysics and in technology applications such as magnetic confinement (e.g. tokamak), and pulsed inertial confinement (e.g. NIF, Z-pinch) fusion devices. The computational simulation of these systems requires solution of the governing PDEs for conservation of mass, momentum, and energy, along with various approximations to Maxwell's equations. The resulting systems are characterized by strong nonlinear coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that these interactions produce. For effective long-time-scale integration of these systems some aspect of implicit time integration is required. These characteristics make scalable and efficient parallel iterative solution, of the resulting poorly conditioned discrete systems, extremely difficult.
This talk will discuss the structure of the implicit continuum fluid models that we are employing for MFC type applications. These include resistive magnetohydrodynamics (MHD), and a partially ionized multifluid electromagnetic (EM) plasma formulation. After discretization by stabilized finite element type methods the strongly coupled highly nonlinear algebraic system is achieved with a fully-coupled Newton nonlinear iterative method. The resulting large-scale sparse linear systems are iteratively solved by a GMRES Krylov method, preconditioned by approximate block factorization (ABF) and physics-based preconditioning approaches. These methods reduce the coupled multiphysics system to a set of simplified sub-systems to which scalable algebraic multilevel methods (AMG) can be applied. A critical aspect of these preconditioners is the development of approximate Schur complement operators that encode the critical cross-coupling physics of the system. To demonstrate the flexibility and performance of these methods we consider application of these techniques to various challenging prototype plasma problems. These include computational results relevant to aspects of magnetic confinement fusion applications. Results are presented on robustness, efficiency, and the parallel and algorithmic scaling of the solution methods. This work is collaborative with Jesus Bonilla, Edward Phillips, Peter Ohm, Michael Crockatt, Ignacio Tomas, Roger P. Pawlowski, R. Tuminaro, Jonathan Hu, Xinazhu-Tang, and Luis Chacon.
About the speaker. John Shadid is a Distinguished Member of the Technical Staff in the Computational Mathematics Department at Sandia National Laboratories, and has an appointment as a National Lab Professor in the Mathematics Department at the University of New Mexico. Dr. Shadid received his Ph.D. in Mechanical Engineering from the University of Minnesota in 1989. John has contributed to the areas of applied math, numerical methods, computational algorithms, and software development, for solution methods of highly-nonlinear coupled multiple-time-scale PDE systems. John was co-PI for the Aztec software library that received a 1997 R&D100 award and was one of the very first scalable parallel iterative solver libraries. He has also been lead-PI for several projects developing robust, scalable, implicit finite element reacting flow, MHD, and multifluid electromagnetic plasma system simulation capabilities in support of DOE Office of Science oriented scientific applications.
In 2018, he was elected as a SIAM Fellow, and in 2019 he received the United States Association of Computational Mechanics (USACM) Thomas J.R. Hughes Medal for Computational Fluid Mechanics.
When: 4:00 pm (Lubbock's local time is GMT -5)
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