论文标题
非线性导电材料断层扫描中的单调性原理
Monotonicity Principle in Tomography of Nonlinear Conducting Materials
论文作者
论文摘要
我们处理一个反电导率问题，该问题涉及从稳定电流操作中边界测量开始的非线性电导率的重建。在此框架中，单调性原理起着关键作用，该原理建立了一个单调关系，该关系将未知的材料属性连接到（测量的）dirichlet to-neumann运算符（DTN）。单调性原理是一类非著作和实时成像方法和算法的基础。在本文中，我们证明了非线性问题中的dirichlet能量的单调性原理在轻度假设下。然后，我们表明，除了线性和$ p $ -laplacian的案例外，不可能将这种单调性从Dirichlet Energy转移到DTN操作员。为了克服这个问题，我们介绍了一个新的边界运算符，该运算符被确定为普通DTN操作员。
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.
