Iterative Methods for Transmission Eigenvalues
Iterative Methods for Transmission Eigenvalues
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DOI：
10.1137/100785478
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发表时间：
2011-09
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影响因子：
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通讯作者：
Jiguang Sun
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文献类型：
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作者：
Jiguang Sun
传输本征值在逆散射理论中有重要应用。它们可以用来获得有用的信息的物理特性，如折射率，散射目标。尽管相当大的努力致力于传输特征值的存在性和估计，数值处理是有限的。由于问题是非标准的，经典的有限元方法导致非埃尔米特矩阵特征值问题。在本文中，我们着重于计算的几个最低传输本征值，这是实际的重要性。而不是一个非埃尔米特问题，我们的工作在一系列广义埃尔米特问题。首先，我们使用四阶重的传输特征值问题，以构建涉及相关的广义特征值问题的功能。这些函数的根是传输本征值。然后，我们采用迭代方法来计算传输特征值。我们证明了数值格式的收敛性。使用各种数值例子证明了方法的有效性。
Transmission eigenvalues have important applications in inverse scattering theory. They can be used to obtain useful information of the physical properties, such as the index of refraction, of the scattering target. Despite considerable effort devoted to the existence and estimation for the transmission eigenvalues, the numerical treatment is limited. Since the problem is nonstandard, classical finite element methods result in non-Hermitian matrix eigenvalue problems. In this paper, we focus on the computation of a few lowest transmission eigenvalues which are of practical importance. Instead of a non-Hermitian problem, we work on a series of generalized Hermitian problems. We first use a fourth order reformulation of the transmission eigenproblem to construct functions involving an associated generalized eigenvalue problem. The roots of these functions are the transmission eigenvalues. Then we apply iterative methods to compute the transmission eigenvalues. We show the convergence of the numerical schemes. The effectiveness of the methods is demonstrated using various numerical examples.