Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem
Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem
复制标题
逆椭圆边值问题的因式分解方法及混合型包含
DOI：
10.3934/ipi.2008.2.355
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发表时间：
2008
影响因子：
1.3
通讯作者：
Nuutti Hyvönen
中科院分区：
文献类型：
--
作者：
B. Gebauer;Nuutti Hyvönen
在各种成像问题中，任务是利用椭圆边值问题解的柯西数据来重建相应的偏微分方程的系数。通常被测试的对象具有已知的背景性质，但受到引起系数函数扰动的非同质性的污染。Kirsch分解法为定位这类内含物提供了一种工具。本文在散度型强制椭圆型偏微分方程的框架下研究了因子分解技术，先前已经证明了因子分解算法可以重建对前阶系数的严格正（负）定摄动的支持，或者如果前阶系数保持无摄动，可以重建对零阶系数的严格正（负）定摄动的支持。在这项工作中，我们表明，这两种类型的非均质性，事实上，可以同时定位。与先前关于分解方法的文章不同，我们的包含可能具有不相关的补体，并且我们还削弱了该方法的一些其他先验假设。我们的理论发现是补充二维数值实验，是在光学层析成像的扩散近似的框架提出的。
In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the co- efficients of the corresponding partial differential equation. Often the exam- ined object has known background properties but is contaminated by inhomo- geneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly pos- itive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) per- turbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have discon- nected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.