Fluid Mechanics and Waves Seminar - Fall 2024

For questions about the seminar schedule, please contact Thi Phong Nguyen.

September 9

Shixu Meng, Virginia Tech

Location: Zoom

Exploring Low Rank Structures for Inverse Scattering Problems

Inverse problems play important roles in various applications, including target identification, non-destructive testing, and parameter estimation. Particularly challenging is the inverse scattering problem in inhomogeneous media, which aims to estimate unknowns based on available measurement data. Given its inherently ill-posed nature, our aim is to address this challenge by exploring the underlying low rank structure that is capable of handling potentially noisy and large-scale measurement data. The unknown is solved in a low-dimensional space comprising disk prolate spheroidal wave functions, which are computed efficiently via a Sturm-Liouville problem. The low rank structure leads to reliable numerical algorithms that demonstrate increasing stability and dimensionality reduction, in the presence of noisy and large-scale measurement data. A stability estimate is proved by leveraging the interplay between a Fourier integral operator and a Sturm-Liouville differential operator. I will also discuss current and future work to explore low rank structures in some inverse problems and PDEs.

Homepage: https://math.vt.edu/people/faculty/meng-shixu.html

September 23

Thuy Le, NC State University

Location: Zoom 

An Inverse Scattering Problem with Experimental Data

We develop the convexification method to solve the 3D inverse scattering problem for the Helmholtz equation. The name convexification is suggested by the fact that we involve a suitable Carleman weight function in the nonconvex least square functional to obtain a new functional, which is strictly convex. This convexifying phenomenon is rigorously proved by employing Carleman estimates. Next, we prove that the minimizer of this convex Carleman weighted functional can be reached by the well-known gradient descent method. More importantly, we prove that this minimizer converges to the true solution of the system of PDEs as the noise contained in the data tends to be 0. Results of numerical studies of both computationally simulated and experimentally collected data are presented. 

Homepage: https://sites.google.com/view/thuy-le/

*Alternate time for this seminar: 1:30PM - 2:30PM

October 7

Christiana Mavroyiakoumou, NYU

Location: CULM 611

Modeling flying formations as flow-mediated matter 

Collective locomotion of flying animals is fascinating in terms of individual-level fluid mechanics and group-level structure and dynamics. In this talk, I will introduce a model of formation flight that views the collective as a material whose properties arise from the flow-mediated interactions among its members. It builds on an aerodynamic model that describes how flapping flyers produce vortex wakes and how they are influenced by others' wakes. Long in-line arrays show that the group behaves as a soft, excitable "crystal" with regularly ordered member "atoms" whose positioning is susceptible to deformations and dynamical instabilities. Perturbing a member produces longitudinal waves that pass down the group while growing in amplitude; with these amplifications even causing collisions. The model explains the aerodynamic origin of the spacing between the flyers, the springiness of the interactions, and the tendency for disturbances to resonantly amplify. Our findings suggest analogies with material systems that could be generally useful in the analysis of animal groups.

Homepage: https://cims.nyu.edu/~cm4291/

October 21

Te-Sheng Lin, National Yang Ming Chiao Tung University, Taiwan

Location: CULM 611

A Feature-Capturing PINN for Interface Problems

In this talk, we introduce feature-capturing physics-informed neural networks designed to solve fluid-structure interaction problems. We first reformulate the governing equations in each fluid domain separately and replace the singular force effect by the traction balance equation between solutions in two sides of the interface. Since the pressure is discontinuous and the velocity has discontinuous derivatives across the interface, we hereby design neural network functions to capture the pressure and velocity behaviors across the interface sharply. Through a series of numerical experiments, the results indicate that the models employed in the current network can achieve high prediction accuracy, and the accuracy is comparable with traditional grid-based methods.

Homepage: https://teshenglin.github.io/ 

November 4

Han Zhou, University of Pennsylvania

Location: CULM 611

A Correction Function-Based Kernel-Free Boundary Integral Method for Solving Interface Problems

Interface problems are ubiquitous in fields such as fluid mechanics, materials science, and biophysics. Traditional finite element and finite volume methods often rely on an interface-fitted mesh, which can be both difficult and expensive to generate with high quality. In this presentation, I will introduce a kernel-free boundary integral method, which serves as a finite difference analogue to the boundary integral approach, for accurately and efficiently solving partial differential equation (PDE) interface problems. This method reformulates boundary and volume integrals as simpler interface problems, which are then solved using a corrected finite difference scheme on an unfitted Cartesian grid. The use of a correction function near the interface significantly simplifies the computation of correction terms, enabling the development of high-order schemes with minimal additional effort. Numerical examples will be presented to demonstrate the accuracy and efficiency of the method.

Homepage: https://www.math.upenn.edu/people/han-zhou 

November 18

Yacine Mokhtari, NJIT

Location: CULM 611

Quadratically-Regularized Distributed Optimal Transport on Graphs

Optimal transport has recently become a powerful tool for addressing complex challenges across diverse fields, including resource allocation, data movement, and network optimization. Its framework for quantifying the "cost" associated with transforming one distribution into another makes it well-suited for structured, interconnected systems.

In this talk, we’ll examine distributed optimization techniques, specifically the Alternating Direction Method of Multipliers (ADMM), to tackle the challenges of distributed optimal transport on graphs . Emphasis will be placed on quadratic regularization to ensure solution uniqueness and enhance stability. We’ll conclude with numerical simulations that highlight the performance of the proposed distributed algorithm, demonstrating its practical effectiveness in real-world applications.

Homepage: https://sites.google.com/view/mokhtariyacine/accueil

Last updated: 11/14/2024