非线性导电材料层析成像中的单调性原理
Monotonicity Principle in Tomography of Nonlinear Conducting Materials
AI总结：
该研究针对非线性导电材料的逆电导率重构问题，证明了弱假设下狄利克雷能量的单调性原理成立，指出该原理多数情况下无法推广至DtN算子，并提出平均DtN算子以解决此问题，为非迭代实时成像算法提供理论支撑。
摘要：
我们研究了一个逆电导率问题，该问题旨在从稳态电流运行中的边界测量结果出发，重构非线性电导率。在此框架下，单调性原理（Monotonicity Principle）发挥着关键作用，它建立了未知材料属性与（可测量的）Dirichlet-to-Neumann算子（DtN，狄利克雷-诺伊曼算子）之间的单调关系。单调性原理是一类非迭代实时成像方法与算法的基础。本文证明了在较弱假设下，非线性问题中狄利克雷能量（Dirichlet Energy）的单调性原理成立。随后我们表明，除线性情形和$p$-Laplacian（p-拉普拉斯）情形外，无法将该单调性结果从狄利克雷能量推广到DtN算子。为解决这一问题，我们引入了一种新的边界算子，即平均DtN算子（Average DtN operator）。
英文摘要：
We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and $p$-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator.
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