Vrushali Bokil

Professor, Department of Mathematics

Oregon State University (OSU)

Title: Energy Stable Discretizations for Nonlinear Electromagnetic Models 

Abstract: In this talk, we discuss a variety of nonlinear electromagnetic phenomena with applications to the areas of nonlinear optics and photonics, describing the behavior and interaction of light with nonlinear media. In such media, the polarization density responds non-linearly to the electric field. We present mathematical models and energy stable compatible discretizations for the simulation of the nonlinear optical phenomenon considered. These models incorporate instantaneous nonlinear effects, linear and nonlinear dispersion and multiple scales in space and time. Applications of nonlinear optics and photonics are widespread including in optical computing, optical switches, telecommunication, medicine and the design of new physical and biological materials.

Maryam Fazel

Moorthy Family Professor, Department of Electrical and Computer Engineering

University of Washington (UW)

Title: Global Convergence of Gradient EM for Over-Parameterized Gaussian Mixtures

Abstract: Learning Gaussian Mixture Models (GMMs) is a fundamental problem in machine learning, and the Expectation-Maximization (EM) algorithm and its variant gradient-EM are the most widely used algorithms in practice. When the ground-truth GMM and the learning model have the same number of components, m, a line of prior work has attempted to establish rigorous recovery guarantees; however, this has been shown only for the case of m=2, and EM methods are known to fail to recover the ground truth when m>2. 

This talk considers the "over-parameterized" case, where the learning model uses n>m components to fit an m-component GMM. I will show that gradient-EM converges globally: for a well-separated GMM, I prove that with only mild over-parameterization n = \Omega(m log m), randomly initialized gradient-EM converges to the ground truth at a polynomial rate with polynomial samples. The analysis relies on novel techniques to characterize the geometric landscape of the likelihood loss. This is the first global convergence result for EM methods beyond the special case of m=2.  

David Ketcheson

Professor

Applied Mathematics and Computational Science Computer 

Electrical and Mathematical Science and Engineering Division

King Abdullah University of Science and Technology (KAUST)

Title: Hyperbolic PDEs with spatially periodic coefficients

Abstract: Nonlinear hyperbolic PDEs in one dimension generically develop shock discontinuities in finite time, and their behavior for long times is characterized by entropy decay and irreversibility.  However, as first demonstrated in a landmark 2003 paper by LeVeque & Yong, dramatically different solution behavior is possible in the presence of a spatially-periodic variable medium.  I will review what has been learned since then about such waves.  First, solutions of moderate amplitude tend to behave like solutions to dispersive nonlinear wave equations: there are solitary wave solutions that seem to be globally attracting, while energy seems to be conserved for arbitrarily long times.  Waves of large amplitude can still lead to shock formation, and I will describe a rudimentary theory describing the conditions for shock formation, along with open questions.  More recently, it has been shown that this behavior is typical for a wide variety of hyperbolic systems, including the shallow water equations, isothermal gas equations, and the Euler equations of gas dynamics, and can manifest in different ways in more spatial dimensions.

The homogenization techniques applied originally by LeVeque and Yong can be extended and systematically applied to all of these — and many other — hyperbolic systems, to derive effective equations that are more amenable to both mathematical analysis and rapid numerical solution.

Keh-Ming Shyue

Professor, Department of Mathematics

National Taiwan University (NTU)

Title: Solutions of nonlinear dispersive shallow water equations: analytical and numerical study

Abstract: Our goal in this presentation is to discuss analytical and numerical solutions of nonlinear dispersive shallow water waves. We will use the Benjamin-Bona-Mahony (BBM) equation and the Serre-Green-Naghdi (SGN) equations as our model systems. To solve the model numerically, we employ a natural hyperbolic-elliptic splitting, where the hyperbolic step is solved by a high-order wave propagation algorithm with an adaptive reconstruction. For the SGN model, the elliptic step is related to the inversion of a scalar elliptic operator for the non-hydrostatic pressure [3]. To validate the accuracy of the numerical method, in one dimension, we derive the (singular) solitary limit of the Whitham modulation system for each of the BBM and the SGN equations, and compare the corresponding numerical and analytical solutions for the interaction of a solitary wave and a slowly varying mean background flow [1, 2, 4]. In two dimensions, we compare our SGN results with the hyperbolized SGN model for problems with and without the bottom topography. This is a joint work with T. Congy, G. El, S. Gavrilyuk, M. Hoefer, B. Nkonga, and L. Truskinovsky.

References

• [1]  Gavrilyuk, S., Nkonga, B., Shyue, K.-M., and Truskinovsky, L., Stationary shock-like transition fronts in dispersive systems, Nonlinearity, 33: 5477–5509, 2020.

• [2]  Gavrilyuk, S. and Shyue, K.-M., Singular solutions of the BBM equation: analytical and numerical study Nonlinearity, 35:388–410, 2022.

• [3]  Gavrilyuk, S. and Shyue, K.-M., 2D Serre-Green-Naghdi equations over topography: elliptic operator inversion method, J. Hydraul. Eng., 150: 04023054, 2024.

• [4]  Congy, T., El, G., Gavrilyuk, S., Hoefer, M., and Shyue, K.-M., Solitary wave-mean flow interaction in strongly nonlinear dispersive shallow water waves J. Nonlinear Waves,1:e5,1-31, 2025.