Applied and Computational Mathematics Seminars

COSAM Departments Mathematics & Statistics Research Seminars Applied and Computational Mathematics



Upcoming Applied and Computational Mathematics Seminars

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Past Applied and Computational Mathematics Seminars
DMS Applied and Computational Mathematics Seminar
Feb 21, 2025 02:00 PM
328 Parker Hall


weizhu

Speaker: Wei Zhu (Georgia Tech)  

Title: Symmetry-Preserving Machine Learning: Theory and Applications

 

Abstract: Symmetry underlies many machine learning and scientific computing tasks, from computer vision to physical system modeling. Models designed to respect symmetry often perform better, but several questions remain. How can we measure and maintain approximate symmetry when real-world symmetries are imperfect? How much training data can symmetry-based models save? And in non-convex optimization, do these models truly converge to better solutions? In this talk, I will share my work on these challenges, revealing that the answers are sometimes surprising. The approach draws on applied probability, harmonic analysis, differential geometry, and optimization, but no specialized background is required.

DMS Applied and Computational Mathematics Seminar
Feb 14, 2025 02:00 PM
328 Parker Hall


zhong
 
Speaker: Yimin Zhong (Auburn)
 
Title: Fast solvers for radiative transfer and beyond
 
 
Abstract:  Despite the tremendous developments in recent years, constructing efficient numerical solution methods for the radiative transfer equation (RTE) is still challenging in scientific computing. In this talk, I will present a simple yet fast computational algorithm for solving the RTE in isotropic media in steady-state and time-dependent settings. The algorithm we developed has two steps. In the first step, we solve a volume integral equation for the angularly averaged solution using iterative schemes such as the GMRES method. The computation in this step is accelerated with a variant of the fast multipole method (FMM). In the second step, we solve a scattering-free transport equation to recover the angular dependence of the solution. The algorithm does not require the underlying medium to be homogeneous. We present numerical simulations under various scenarios to demonstrate the performance of the proposed numerical algorithm for both homogeneous and heterogeneous media. Then I will extend the formulation to the time-domain and anisotropic scattering media and analyze the possibility of applying the fast algorithm. 
DMS Applied and Computational Mathematics Seminar
Nov 22, 2024 01:00 PM
328 Parker Hall


liu

Speaker: Yi Liu (Auburn University)

Title: Convergence Analysis of the ADAM Algorithm for Linear Inverse Problems

 

Abstract:  The ADAM algorithm is one of the most popular stochastic optimization methods in machine learning. Its remarkable performance in training models with massive datasets suggests its potential efficiency in solving large-scale inverse problems. In this work, we apply the ADAM algorithm to solve linear inverse problems and establish the sub-exponential convergence rate for the algorithm when the noise is absent. Based on the convergence analysis, we present an a priori stopping criterion for the ADAM iteration when applied to solve inverse problems at the presence of noise. The convergence analysis is achieved via the construction of suitable Lyapunov functions for the algorithm when it is viewed as a dynamical system with respect to the iteration numbers. At each iteration, we establish the error estimates for the iterated solutions by analyzing the constructed Lyapunov functions via stochastic analysis. Various numerical examples are conducted to support the theoretical findings and to compare with the performance of the stochastic gradient descent (SGD) method. 

DMS Applied and Computational Mathematics Seminar
Nov 15, 2024 02:00 PM
ZOOM


 
Speaker: Patrizio Bifulco (FernUniversität in Hagen, Germany)
 
Title: Comparing the spectrum of Schrodinger operators on metric graphs using heat kernels
 
 
Abstract: We study Schrodinger operators on compact finite metric graphs subject to \(\delta\)-coupling and standard boundary conditions often known as Kirchoff-Neumann vertex conditions. We compare the \(n\)-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of the eigenvalue deviations which represents a generalization to a recent result by Rudnick, Wigman and Yesha obtained for domains in \(\mathbb{R}^2\) to the setting of metric graphs. We start this talk by introducing the basic notion of a metric graph and discuss some basic properties of heat kernels on those graphs afterwards. In this way, we are able to discuss a so-called local Weyl law which is relevant for the proof of the asymptotic main result. If time permits, we will also briefly discuss the case of \(\delta'\)-coupling conditions and some possible generalizations on infinite graphs having finite total length.
 
This talk is based on joint works with Joachim Kerner (Hagen) and Delio Mugnolo (Hagen).
DMS Applied and Computational Mathematics Seminar
Nov 08, 2024 02:00 PM
328 Parker Hall


watson

Speaker: Alexander Watson (University of Minnesota Twin Cities)

Title: Multiple-scales perspective on moiré materials 

 

Abstract: In recent years, experiments have shown that twisted bilayer graphene and other so-called "moiré materials" realize a variety of important strongly-correlated electronic phases, such as superconductivity and fractional quantum anomalous Hall states. I will present a rigorous multiple-scales analysis justifying the (single-particle) Bistritzer-MacDonald PDE model, which played a critical role in the prediction of these phases in twisted bilayer graphene. The significance of this model is that it has moiré-periodic coefficients even when the underlying material is aperiodic at the atomic scale. This allows moiré materials to be studied via Floquet-Bloch band theory, a variant of the Fourier transform. I will then discuss generalizations of this model and other mathematical questions related to moiré materials.

DMS Applied and Computational Mathematics Seminar
Oct 29, 2024 02:00 PM
228 Parker Hall


deang

Speaker: Dr. Jennifer Deang (Lockheed Martin; affiliated faculty member of DMS) 

Title: On the Mathematical Perspective of the Missile Defense System

 

Abstract: We first provide an overview of the systems, weapons, and technology needed for detection, tracking, interception, and destruction of attacking missiles. Then we will outline the current research areas sought by the MDA to advance and solve complex technological problems, ultimately contributing to a more robust Missile Defense System (MDS). 

DMS Applied and Computational Mathematics Seminar
Oct 25, 2024 01:00 PM
328 Parker Hall


musslimani

Speaker: Ziad Musslimani (Florida State University)

Title: Space-time nonlocal integrable systems

 

Abstract: In this talk I will review past and recent results pertaining to the emerging topic of integrable space-time nonlocal integrable nonlinear evolution equations. In particular, we will discuss blow-up in finite time for solitons and the physical derivations of many integrable nonlocal systems.

DMS Applied and Computational Mathematics Seminar
Oct 04, 2024 11:00 AM
328 Parker Hall


Please note the special time of the seminar. 

Ju

Speaker: Dr. Lili Ju (University of South Carolina)

Title: Level Set Learning with Pseudo-Reversible Neural Networks for Nonlinear Dimension Reduction in Function Approximation

 

Abstract: Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

DMS Applied and Computational Mathematics Seminar
Sep 27, 2024 01:00 PM
328 Parker Hall


curran
 
Speaker: Dr. Mitch Curran (Auburn). 
 
Title: Hamiltonian spectral theory via the Maslov index
 
 
Abstract:  As Arnol’d pointed out, Sturm’s 19th century theorem regarding the oscillation of solutions to a second-order selfadjoint ODE has a topological nature: it describes the rotation of a straight line in the phase space of the equation. The topological ingredient here is the Maslov index, a homotopy invariant which counts the (signed) intersections of a path of Lagrangian planes with a codimension-one subset of the set of all Lagrangian planes. In this talk, I’ll begin with a discussion of Sturm’s theorem, the main idea being that one can glean spectral information from the geometric structure of an eigenfunction. I'll then show how these ideas translate to study the eigenvalues of a class of Hamiltonian differential operators that I studied in my PhD, which do not enjoy the selfadjointness property of the operators that Sturm studied. In particular, by viewing the problem symplectically, one can use the Maslov index to give a lower bound for the number of positive real eigenvalues in terms of the Morse indices of two related selfadjoint operators, as well as a mysterious correction term. The Hamiltonian operators in focus here arise, for example, when determining the (spectral) stability of standing waves in NLS type equations; if time permits, I’ll go through some applications to such problems on both bounded and unbounded domains.
 
Part of this talk was joint work with Graham Cox, Yuri Latushkin, and Robby Marangell.
DMS Applied and Computational Mathematics Seminar
Sep 20, 2024 01:00 PM
328 Parker Hall


doan

Speaker: Cao-Kha Doan (Auburn) 

Title: Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations.

 

Abstract: In this work, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we reformulate the original equations into an equivalent system that incorporates the energy evolution process. First- and second-order dynamically regularized Lagrange multiplier (DRLM) schemes are derived based on the backward differentiation formulas and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a scalar quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting the accuracy and stability of the numerical solutions. Various numerical experiments including the Taylor-Green vortex problem, lid-driven cavity flow, and Kelvin-Helmholtz instability are carried out to demonstrate the performance of the DRLM schemes. Extension of the DRLM method to the Cahn-Hilliard-Navier-Stokes system will also be discussed.

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