CAM Seminar

Computational and Applied Mathematics Seminar

Spring 2024

Mondays at 2:15-3:05p.m  (ZOOM/WebEx or in-person (Room Carver 401) talks)

The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the area of computational and applied mathematics.


  • April 15

       Title:  On the Study of Noise-Induced Transitions over Periodic Boundaries in Non-Gradient Systems

        Emmanuel Fleurantin, George Mason University

Abstract:   Noise-induced tipping (N-tipping) emerges when random fluctuations prompt transitions from one (meta)stable state to another, potentially as a rare event. In this talk, we delineate new techniques for determining Most Probable Escapes Paths (MPEPs) in stochastic differential equations over periodic boundaries. We utilize a dynamical system approach to unravel MPEPs for the intermediate noise regime. We discuss the framework for computing the MPEPs by first looking at intersections of stable and unstable manifolds of invariant sets of a Hamiltonian system derived from the Euler-Lagrange equations of the Freidlin-Wentzell (FW) functional. The Maslov index helps identify which critical points of the FW functional are local minimizers and assists in explaining the effects of the interaction of noise and the deterministic flow. The Onsager-Machlup functional, which is treated as a perturbation of the FW functional, will provide a selection mechanism to pick out the MPEP. We will illustrate our approach and compare our theoretical prediction with Monte Carlo simulations in the Inverted Van der Pol system and a carbon cycle model. 

  • April 22



  • April 29





  • January 22



  • January 29

       Title:  Training data studies for the cell-average-based neural network method for linear hyperbolic and parabolic equations

       Tyler Kroells, Iowa State University

Abstract:  In this talk, we present the training data studies for the cell-average-based neural network (CANN) method for linear hyperbolic and parabolic equations.  We enlarge the training data set to include representative solution trajectories such as Fourier basis functions and determine if or not the method will be improved. A perturbed stochastic gradient descent method is proposed and is verified to improve the method's stability after long-time simulations. Robustness of the method to contaminated training data is investigated. For the linear parabolic equation, the CANN method can be generalized to solve a large group of initial value problems, especially those involving high-frequency modes, while being an explicit method with a large time step size applied to evolve the solution forward in time.

This is a collaboration work with Ethan Schmidt, Changxin Qiu and Jue Yan

  • February 05





  • February 12

       Title:  High order methods for wave propagation problems

       Ian Morgan, Iowa State University

Abstract:   When dealing with the wave equation in high spatial frequencies, accurate solutions can be difficult to achieve. Asymptotic approximations may only offer localized solutions, while finite element or finite difference schemes can be susceptible to pollution errors. We are currently working on developing Fourier pseudospectral methods that are efficient, high-order, and free from pollution. These methods involve time-stepping schemes that utilize a modified Helmholtz equation, which can be solved using either a rapidly converging Krylov subspace scheme or by structuring the inverse operator as a functional that can be evaluated numerically.


  • February 19

       Title:  Local and Global Convergence of General Burer-Monteiro Tensor Optimizations

       Shuang Li, Iowa State University

Abstract:   Tensors, a multi-dimensional generalization of vectors and matrices, provide natural representations for multi-way datasets and find numerous applications in signal processing and machine learning. In this work, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a non-convex optimization problem using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of a best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.


  • February 26

       Title:  Adaptive Preconditioned Gradient Descent with Energy

       Xuping Tian, Iowa State University

Abstract:   We propose an adaptive time step with energy for a large class of preconditioned gradient descent methods, mainly applied to constrained optimization problems. Our strategy relies on representing the usual descent direction by the product of an energy variable and a transformed gradient, with a preconditioning matrix, for example, to reflect the natural gradient induced by the underlying metric in parameter space or to endow a projection operator when linear equality constraints are present. We present theoretical results on both unconditional stability and convergence rates for three respective classes of objective functions. In addition, our numerical results shed light on the excellent performance of the proposed method on several benchmark optimization problems. ( )


  • March 04

       Title:  On the geometry of diffusion

       Thialita M. Nascimento,  Iowa State University

Abstract:   In this talk we explore a new geometric approach in the analysis of diffusive partial differential equations (PDEs). This new approach has its roots in the theory of minimal surfaces and, more generally, free boundary problems. Thus, the geometric insights behind the idea of diffusion gives a powerful toolbox to analyze diffusive elliptic equations, in which degenerate points of ellipticity are seen as ”non- physical” free boundary points.

  • March 11 (Spring Break)



  • March 18 (ZOOM only)

       Title:  Diffusion Models: Theory and Applications (in PDEs)

       Yulong Lu, University of Minnesota

Abstract:   Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis establishes a groundbreaking result: SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by low-complexity structures. This theoretical underpinning sheds light on the robust capabilities of SGMs in learning and sampling complex distributions.


In the second part of the talk, we shift our focus to the practical realm, demonstrating the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present the development of a physics-guided diffusion model designed for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation. 


  • March 25 (In-Person Carver 401, also with a ZOOM link)

       Title:  First principles investigation of magnetic interaction and excitation in quantum materials

       Zhenhua Ning, Ames Lab

Abstract:   Permanent magnets, Fe-based superconductors, magnetic 2D van der Waals materials, and topological magnetic materials play important roles in various applications. Understanding and controlling magnetic interactions and excitations in these materials are crucial for realizing desired properties. This presentation delves into the calculation and manipulation of magnetic interactions and excitations using first principles tools, illustrated through two examples. Firstly, the magnetic interactions in the antiferromagnetic (AFM) Dirac semimetal candidate SrMnSb$_2$ are investigated using \textit{ab initio} linear response theory and inelastic neutron scattering (INS). A rigid-band model indicates that electron doping fills the minority spin channel and results in a decrease in the AFM coupling strength for both nearest exchange $J_1$ and next-nearest exchange $J_2$. Secondly, we introduce a many-body Hamiltonian to effectively describe strong correlation effects in $4f$ electrons of rare-earth (RE) elements. Subsequently, we discuss magnetocrystalline anisotropy in RE-transition metals, uncovering the significant role of crystal field parameters.


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  • April 01



  • April 08