Partial Differential Equations and Applications

Link for registration

https://docs.google.com/forms/d/1ra3TSRYtuXxZFPOYhLLyfezzCw6tIWoPHRwnRtjJ-fM

Detailed Program

(Abstracts below)

Day 1

Day 2

Break time or Chat with Speakers

Speakers and Abstracts

The Carleman estimates, the convexification method, and its applications in Hamilton-Jacobi equations and the inverse scattering problem.

Abstract:

The talk consists of three parts. In the first part of the talk, we propose a globally convergent numerical method, called the convexification, to numerically compute solutions to nonlinear partial differential equations (PDEs). By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the PDE under consideration. We discuss the use of Carleman estimates to rigorously prove the strict convexity of this functional. We also prove that the unique minimizer of this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the desired solution to the PDE as the noise contained in the boundary data tends to zero. Then in the second part, we apply the convexification method to numerically solve Hamilton-Jacobi equations. In the third part of the talk, we present numerical solutions to the inverse scattering problem obtained by the convexification method.

Imaging of defects in complex media

Abstract:

Inverse scattering problems arise in many real life applications such as non-destructive evaluation, medical imaging, geophysical exploration, etc. For instance, illuminating a probed domain with some (acoustic, electromagnetic, elastic) waves and measuring the response (scattered waves) at some distance, the inverse problem is to identify the presence of defects, such as cracks, and if possible, finding their location and reconstructing their shape. In this talk, I will discuss the study of the so-called Sampling Methods for solving such inverse problems and their application for two particular projects. The first one is the reconstruction of closed fractures inside a finite solid body. The second one is related to the reconstruction of local defects in an infinite periodic layer without any prior knowledge on the periodic structure.

Full-field photoacoustic tomography

Abstract:

Photoacoustic tomography (PAT) is a hybrid method of biomedical imaging. The object of interest is scanned with laser light. Due to the photo-elastic effect, the object slightly expands. The expansion produces ultrasound signals propagating throughout the space. By measuring the external ultrasound signals, one aims to reconstruct the object's image. In this talk, I will discuss an emerging setup of PAT, called full-field PAT. I will present its mathematical problem and our approach. .

On the existence and instability of solitary water waves with a finite dipole

Abstract:

This paper considers the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is air and that the waves are acted upon by gravity with surface tension effects on the air-water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted delta-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of 0. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis–Shatah–Strauss method recently introduced by Varholm, Wahlén, and Walsh.

Some Recent Results On Wave Turbulence: Derivation, Analysis, Numerics and Physical Application

Abstract:

Wave turbulence describes the dynamics of both classical and non-classical nonlinear waves out of thermal equilibrium. Recent mathematical interests on wave turbulence theory have their roots from the works of Bourgain, Staffilani and Colliander-Keel-Staffilani-Takaoka-Tao. In this talk, I will present some of our recent results on wave turbulence theory. In the first part of the talk, I will discuss our rigorous derivation of wave turbulence equations. The second part of the talk is devoted to the analysis of wave turbulence equations as well as some numerical illustrations. The last part concerns some physical applications of wave turbulence theory. The talk is based on my joint work with Staffilani (MIT), Soffer (Rutgers), Pomeau (ENS Paris), and Walton (PhD student at SMU).

Gradient flows of modified Wasserstein distances and homogeneous porous media fractional equations.

Abstract:

We study families of homogeneous porous media fractional equations. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the $L^p$ norms. This is joint work with Quoc Hung Nguyen.

g-expectations and g-stochastic orderings with applications to option price comparisons

Abstract:

Stochastic ordering under the linear expectation setting has found a wide range of applications in various fields such as reliability theory, economics, actuarial sciences, operation research, risk management, biology, option evaluation, etc. In this work, we introduce a novel g-stochastic ordering under nonlinear g-expectations and derive sufficient conditions for the convex, increasing convex, and monotonic g-stochastic orderings of diffusion processes. Our approach relies on comparison results for forward backward stochastic differential equations (FBSDEs) and on several extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated semi linear parabolic partial differential equations. Applications to option price comparison under different hedging portfolio constraints are also provided.

Created by ICASM team