广义阻尼边界条件下Laplace方程反问题的非线性积分方程法

Nonlinear Integral Equation Method for the Laplace Inverse Problem with Generalized Impedance Boundary Condition

研究由Laplace方程边值问题对应的边界上的柯西数据重构内部障碍物的形状的问题,其物理背景是由导电介质对应的边界上的电压和电流信息确定介质内部腔体形状的问题。利用格林公式以及双层位势的边界跳跃关系得到一组非线性边界积分方程,从而将边值问题转化为了求解非线性方程组。通过计算非线性积分方程组关于未知数的Frechet导数构造一种迭代算法重构出内部障碍物的形状。最后给出了数值例子,证明了该迭代方法的有效性。

Consider the reconstruction of the shape of an inclusion within conducting medium from voltage and current measurements on the accessible boundary of the medium. This problem can be mathematically modeled as an inverse boundary value problem for the Laplace equation. More specifically, our goal is to reconstruct the boundary shape from a knowledge of measured Cauchy pairs on an accessible boundary containing the inclusion inside. By Green＇s formula and the jump relation of the double-layer potential we get the nonlinear boundary integral equations. Then the boundary value problem is solved in terms of this system of the nonlinear integral equations. We compute the Frechet derivative of the system of nonlinear integral equations with respect to the unknowns and then develop an iterative algorithm to reconstruct the shape of the interior obstacle. The numerical results are presented to show the validity of the proposed scheme.