On the scattering for the $\bar{\partial}$- equation and reconstruction of convection terms

守恒律和对流扩散方程的三阶半离散中心格式

A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

提出一种三阶半离散中心方法求解守恒律和对流扩散方程，基于CWENO重构，实现高精度、高分辨率和鲁棒性。

Derivation of a Convection Process in a Steady Diffusion-Transfer Problem by Homogenization

We study the homogenization of a steady diffusion equation in a highly heterogeneous medium made of two subregions separated by a periodic barrier through which the flow is proportional to the jump of the temperature by a layer conductance of the same order of magnitude of the materials in consideration. The macroscopic governing equations and the effective conductivity of the homogenized model are obtained by means of the two scale convergence technique. We show that under some hypothesis the homogenized systems contain convective terms of order one.

The Behavior of the Free Boundary for Reaction-Diffusion Equations with Convection in an Exterior Domain with Neumann or Dirichlet Boundary Condition

Let \begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} \end{equation*} be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, \begin{equation*} \begin{aligned} &Lu=Λu^p\ \text{in}\ \mathbb{R}^d-D;\\ &\nabla u\cdot \bar n=-h\ \text{on}\ \partial D;\\ &u\ge0 \ \text{is minimal}, \end{aligned} \end{equation*} where $p\in(0,1)$, $d\ge2$, $h$ and $Λ$ are continuous positive functions, $D\subset R^d$ is bounded, and $\bar n$ is the unit inward normal to the domain…

3D EIT Reconstructions from Electrode Data using Direct Inversion D-bar and Calderon Methods

The first numerical implementation of a D-bar method in 3D using electrode data is presented. Results are compared to Calderón's method as well as more common TV and smoothness regularization-based methods. D-bar methods are based on tailor-made non-linear Fourier transforms involving the measured current and voltage data. Low-pass filtering in the non-linear Fourier domain is used to stabilize the reconstruction process. D-bar methods have shown great promise in 2D for providing robust real-time absolute and time-difference conductivity reconstructions but have yet to be used on practical ele…

二维复电导率重建的直接D-bar重构算法

A direct D-bar reconstruction algorithm for recovering a complex conductivity in 2-D

提出一种基于Francini唯一性证明的D-bar直接重构算法，用于从Dirichlet-to-Neumann数据重建二维有界Lipschitz域中的复电导率，并包含数值模拟的胸部体模重建结果。

An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions

By developing the d-bar approach to global "inverse scattering" at zero energy we give a principal effectivization of the global reconstruction method for the Gel'fand-Calderon inverse boundary value problem in three dimensions. This work goes back to results published by the author in 1987, 1988 and proceeds from recent progress in the d-bar approach to inverse scattering in 3D published by the author in 2005, 2006.