论文标题
树图的Calderón类型的反问题
A Calderón type inverse problem for tree graphs
论文作者
论文摘要
我们从与Laplacian相关的Dirichlet到Neumann矩阵的知识中研究了恢复树图的逆问题以及其边缘上的重量（等效于公制树）。我们证明了一个明确的公式，该公式将此矩阵与树的叶子的成对加权距离联系起来，因此可以恢复加权树。在PDE分析中，可以将此结果视为Calderón问题的对应物。与公制图的反问题的早期结果相反，我们仅假设对固定能量的dirichlet到neumann矩阵的知识，而不是整个矩阵值值的函数。
We study the inverse problem of recovering a tree graph together with the weights on its edges (equivalently a metric tree) from the knowledge of the Dirichlet-to-Neumann matrix associated with the Laplacian. We prove an explicit formula which relates this matrix to the pairwise weighted distances of the leaves of the tree and, thus, allows to recover the weighted tree. This result can be viewed as a counterpart of the Calderón problem in the analysis of PDEs. In contrast to earlier results on inverse problems for metric graphs, we only assume knowledge of the Dirichlet-to-Neumann matrix for a fixed energy, not of a whole matrix-valued function.
