A multilevel sampling method for detecting sources in a stratified ocean waveguide

Abstract: In the reconstruction process of sound waves in a 3D stratified waveguide, a key technique is to effectively reduce the huge computational demand. In this work, we propose an efficient and simple multilevel reconstruction method to locate the accurate position of a point source in a stratified ocean. The proposed method can be viewed as a direct sampling method since no solutions of optimizations or linear systems are involved. The novel method exhibits several strengths: fast convergence, robustness against noise, advantages in computational complexity and applicability for a very small number of receivers.

September 12: Gung-Min Gie, University of Louisville

Finite Volume approximations for some elliptic equations

Abstract:
We construct the Finite Volume (FV) approximations of a class of elliptic equations and perform numerical computations where a 2D domain is discretized by convex quadrilateral meshes. The FV method with Taylor Series Expansion Scheme (TSES), which is properly adjusted from a version widely used in Engineering, is tested in a box, annulus, and a domain including a topography at the bottom boundary. It is verified numerically by simulations that the FV method is a convergent 1st order scheme that manages well the complex geometry. This is a recent joint work with Jung and Nguyen at UNIST, Korea and Temam at Indiana University.

Spring 2016

March 28: Heng Li, University of Louisville

Ritz-Galerkin method for solving a parabolic equation with non-local and time-dependent boundary conditions

Abstract:
The talk is devoted to the investigation of a parabolic partial differential equation with non-local and time-dependent
boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is
implemented by the Ritz-Galerkin method, which is a first attempt at tackling parabolic equation with such non-classical
boundary conditions. In the process of dealing with the difficulty caused by integral term in non-local boundary condition,
we use a trick of introducing the transition function to convert non-local boundary to another non-classical
boundary, which can be handled with the Ritz-Galerkin method. Illustrative examples are included to demonstrate the
validity and applicability of the technique.

March 21: Surina Borjigin, University of Louisville

Variational Mumford-Shah model

Abstract:
From the point of view of approximation theory, the image segmentation problem can be restated as seeking ways to define and compute optimal approximations of a general function f(x,y) by piece-wise smooth functions u(x,y).
I will show how piecewise smooth Mumford-Shah functional approximates the given one-dimensional signal and two dimensional image. I will also show how to derive the Euler-Lagrange equations of M-S energy functional. The 1-D and 2-D level set formulations for M-S minimization problem will be also given. Some numerical experiments will be give to show the approximation results for both 1-D and 2-D case.

February 29: Carlos Paniagua Mejia, University of Louisville

PDEs and the problem of Image Segmentation

Abstract:
We present some Variational-PDE based models for image segmentation,
from their formulation as parametric models to reformulations as geometric ones via the level set method.
Some research trends will be presented--time permitting.

February 22: Yongzhi Steve Xu, University of Louisville

Integral operator method vs Galerkin method

Abstract:
In this talk I continue to discuss Helmholtz equation.
Two methods are considered: integral operator method and Galleria method.
We briefly compare some advantages and disadvantages of the two methods.

February 15: Yongzhi Steve Xu, University of Louisville

Classical solution vs weak solution, integral operator method vs Galerkin method

Abstract:
In this talk I will introduce Helmholtz equation from two aspects, classical solution and weak solution.
Two methods are considered: integral operator method and Galerkin method. We briefly compare
some advantages and disvantages of the two methods.
Numerical integral method and boundary element methods may also be described.

February 1: Changbing Hu, University of Louisville

Shallow water equation as an example for PDE theory

Abstract:
In this talk we will focus on some fundamental notations in the theory of PDE, like the local (global) weak (strong) solutions for PDEs, also how to prove the existence of those solutions by using a priori estimates.
We will use the shallow water equation as the example, roughly following the article by Temam and Rakotoson published in IUMJ.
Some questions will be raised.

January 25: Gung-Min Gie, University of Louisville

Boudary layers of fluid equations I

Abstract:
As a continuation of previous talks,
we discuss the boundary layers of some fluid equations.

Fall 2015

September 21: Heng Li, University of Louisville

Direct and inverse problem for the parabolic equation with initial value and moving boundaries

Abstract: In this paper, we deal with dual problem of a class of non-classical parabolic equations in which the boundaries are moving instead of fixed values, which arise from Ductal carcinoma in situ (DCIS). In the direct problem part, on using the several transformation and heat potential theory, we established the integral form of solution and proved the existence and uniqueness of solution. Then we consider the inverse problem of finding the control parameter of known moving boundaries, which means determining the potential function of model from incisional biopsy information in the view of DCIS. Algorithm and numerical simulation for both problems are included to demonstrate the validity and applicability of solutions.

October 12: Gung-Min Gie, University of Louisville

Recent progresses in boundary layer analysis

Abstract:
In this talk, we review some recent progresses in boundary layer analysis of singular perturbation problems related to the fluids equations.

October 19: Sujeewa Hapuarachchi, University of Louisville

Backward heat equation with time dependent varible coefficient

Abstract:
Backward heat equation with time dependent variable coefficient is severely ill-posed in the sense of Hadamard, so we need regularization. In
this paper we consider Backward heat equation with time dependent variable coefficient and by small perturbing we obtain approximation problem. We
show this approximation problem is well-posed with small parameter. Also we show this approximation system converges to original problem when parameter
goes to zero. Here we use modified-quasi boundary value method to regularize this problem.

October 26: Carlos Paniagua Mejia, University of Louisville

Some mathematical results/questions of a PDE arising from an image segmentation problem

Abstract:
Over the last 25 five years many variational and PDE formulations have been proposed to solve image segmentation problems. These PDEs are often nonlinear and involve special functions that make their formal analysis formidable and much of the emphasis is put on the practical results that the equations provide. In this talk we consider a time dependent PDE that provides satisfactory segmentation results for a wide class of images. After considering its structure and properties, we propose formal questions regarding the existence of solutions, and consider stability and convergence of finite difference approximations.

November 9: Changbing Hu, University of Louisville

An integral equation method in mathematical biology and its application to PDEs

Abstract:
In this talk we first review some classical results in the papers by Thieme (1978, 1979), concerning an integral equations from mathematical biology, which also had been extended to deal with PDE models. We will focus on its existence and some asymptotic behaviors of the solutions. Finally we will introduce some progress of a joint project with Dr. Bingtuan Li.

November 16: Yongzhi Steve Xu, University of Louisville

Variational formulation for a fractional heat transfer model in fire fighter protective clothing

Abstract:
We propose a fractional heat transfer model to simulate the situation with high heat and humidity. The variational formulation for the fractional model is first proposed, which eliminates the singularity of homogeneous problem by multiplying a simple function factor. To prove the existence of weak solution of the variational problem, Galerkin approximation is utilized. The corresponding stability and uniqueness are also derived. To verify the reliability of the proposed model, numerical experiment is carried out, which indicates that our fractional model is appropriate for the heat transfer in firefighter protective clothing.
This is a joint work with Professor Dinghua Xu and his students Yue Yu and Qifeng Zhang, while visiting Zhejiang Sci-Tech
University, Hangzhou, China.