# Faculty Research Talks - Spring 2024

Talks will be held at 2:30PM on every other Monday (M) at 2:30PM in CULM 611 unless otherwise noted.

**February 26**

**Location:** CULM 611

**Leapfrogging and Scattering of Point Vortices**

The interaction among vortices is a fundamental process in the motion of fluids. The n-vortex problem, which describes the motion of a finite number of vortices in a two-dimensional inviscid fluid, has been studied since the late 1800s but remains of interest due to its deep connection with the dynamics of quantum fluids. A founding document in this field is the 1877 doctoral dissertation of Walter Gröbli. We consider two problems with roots in this work. First, we consider the linear stability and the nonlinear dynamics of the so-called leapfrogging orbit of four vortices, bringing in modern Hamiltonian reductions and a numerical visualization technique called Lagrangian descriptors. Second, we consider the scattering of vortex dipoles, introducing a new coordinate system based on a non-standard formulation of Hamiltonian mechanics called the Nambu bracket. While some of the ideas are technical, the talk will be centered around a series of interesting and informative images like the one included below:

**March 25**

**Location:** CULM 611

*Inverse Obstacle Scattering with Dissipative and Trapping Media*

I will review some general principles in inverse scattering and present some results from my recent (mis)adventures in numerical methods for these problems.

Inverse obstacle scattering is the recovery of the boundary of a homogeneous object given scattering measurements far from the object. This can be contrasted with inverse medium scattering where some continuously varying (and compact) material parameter is to be recovered. The obstacle recovery problem has the benefit that the governing PDEs are generally homogeneous boundary value problems; standard boundary integral representations reduce the dimension of the PDE discretization by one. However, the obstacle setting introduces some difficulties. The space of solutions is non-convex and the natural regularizations of the problem are non-linear. In this talk, we'll review some numerical methods for the obstacle problem that utilize ideas originally applied to the medium problem by Yu Chen. Then we'll present some numerical methods and results of two recent papers that concern the obstacle scattering problem for trapping and dissipative media, respectively, and discuss some remaining challenges. This work is in collaboration with Carlos Borges, Jeremy Hoskins, and Manas Rachh.

**April 8**

**Professor Zoi-Heleni Michalopoulou**

**Location:** CULM 611

**Inverse Problems in Ocean Acoustics**

In this talk, we will present the nature of inverse problems in ocean acoustic problems. We will focus on localizing a sound transmitting source and om estimating properties related to the propagation medium. New work in our group features denoising acoustic signals for better inversion results and developing Machine Learning algorithms for seabed classification. We will describe open problems in the field and seek solutions

**April 22**

**Location:** CULM 611

**About Non Overlapping Domain Decomposition Methods for the Solution of Wave Equations**

In this talk we describe a procedure based on domain decomposition algorithms in order to derive an efficient iterative solver for the solution of the Helmholtz equation. We will explain several types of transmission conditions as well as the approach used to deal with the cross-points problem. Various numerical tests are presented.