Abstract

For electrodynamic equations with permittivity specified by a symmetric matrix \(\varepsilon (x) = ({{\varepsilon }_{{ij}}}(x),i,j = 1,2,3)\), the inverse problem of determining this matrix from information on solutions of these equations is considered. It is assumed that the permittivity is a given positive constant \({{\varepsilon }_{0}} > 0\) outside a bounded domain \(\Omega \subset {{\mathbb{R}}^{3}}\), while, inside \(\Omega \), it is an anisotropic quantity such that the differences \({{\varepsilon }_{{ij}}}(x) - {{\varepsilon }_{0}}{{\delta }_{{ij}}} = :{{\tilde {\varepsilon }}_{{ij}}}(x),\)\(i,j = 1,2,3,\) are small. Here, \({{\delta }_{{ij}}}\) is the Kronecker delta. The inverse problem is studied in the linear approximation. The structure of the solution to a linearized direct problem for the electrodynamic equations is investigated, and it is proved that all elements of the matrix \(\tilde {\varepsilon }(x) = {{\tilde {\varepsilon }}_{{ij}}}(x),\;i,j = 1,2,3\), can be uniquely determined by special observation data. Moreover, the problem of recovering the diagonal components \({{\tilde {\varepsilon }}_{{ij}}}(x),\;i = 1,2,3,\) leads to a usual X-ray tomography problem, so these components can be efficiently computed. The recovery of the other components leads to a more complicated algorithmic procedure.

中文翻译：

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各向异性介质电动力学的反问题：线性近似

摘要

对于具有介电常数由一个对称矩阵指定的电动方程\（\ varepsilon（X）=（{{\ varepsilon} _ {{IJ}}}（x）中，I，J = 1,2,3）\） ，逆考虑了从这些方程的解的信息确定该矩阵的问题。假设介电常数是有界域\（\ Omega \ subset {{\ mathbb {R}} ^ {3}}外的给定正常数\（{{\ varepsilon __ {0}}> 0 \）\），而在\（\ Omega \）内，它是一个各向异性的量，因此差异\（{{\ varepsilon} _ {{ij}}}（x）-{{\ varepsilon} _ {0}} {{\ delta} _ {{ij}}} =：{{\波浪线{\ varepsilon}} _ {{ij}}}（x），\）\（i，j = 1,2,3，\）很小。在这里，\（{{\ delta __ {{ij}}} \）是克罗内克三角洲。在线性逼近中研究反问题。研究了电动方程的线性化直接问题解的结构，并证明矩阵\（\ tilde {\ varepsilon}（x）= {{\ tilde {\ varepsilon}} _ { {ij}}}（x），\; i，j = 1,2,3 \）可以由特殊的观测数据唯一地确定。此外，恢复对角分量\（{{\ tilde {\ varepsilon}} _ {{ij}}}（x），\; i = 1,2,3，\）的问题会导致通常的X射线层析成像问题，因此可以有效地计算这些分量。其他组件的恢复导致更复杂的算法过程。