Local uniqueness for the inverse boundary problem for the two-dimensional diffusion equation
Local uniqueness for the inverse boundary problem for the two-dimensional diffusion equation
复制标题
二维扩散方程逆边界问题的局部唯一性
DOI：
--
复制
发表时间：
2000
影响因子：
1.9
通讯作者：
N. Grinberg
中科院分区：
文献类型：
--
作者：
N. Grinberg
本文研究了一类扩散方程的边值反问题.我们的动机是，这个方程是一个近似的线性传输方程，并描述了光在高散射介质中的传播。频域中的扩散方程为非自伴椭圆方程div（格拉德u）-（cμa + iω0）u = 0; ω0 <$0，其中D和μa为扩散系数和吸收系数。逆问题是在有界区域内仅使用边界处的测量来重建D和μa。在二维情形下，我们证明了对应于任意一个正频率ω0的Dirichlet-to-Neumann映射唯一地决定了扩散系数和吸收系数，只要它们是充分慢变的。在零背景的情况下，我们估计分析这些系数可以多大，以保证重建的唯一性。
We study an inverse boundary problem for the diffusion equation in ℝ2. Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div(D grad u) - (cμa + iω0) u = 0; ω0 ≠ 0, where D and μa are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and μa inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency ω0, determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.