Stable determination of polygonal inclusions in Calderón's problem by a single partial boundary measurement

Differentiability of the Dirichlet to Neumann map under movements of polygonal inclusions with an application to shape optimization

In this paper we derive rigorously the derivative of the Dirichlet to Neumann map and of the Neumann to Dirichlet map of the conductivity equation with respect to movements of vertices of triangular conductivity inclusions. We apply this result to formulate an optimization problem based on a shape derivative approach.

Reconstruction of singular and degenerate inclusions in Calderón's problem

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In …

Calderón's problem for p-Laplace type equations

We investigate a generalization of Calderón's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in la…

Enclosure method for the p-Laplace equation

We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.

具有两对 Cauchy 数据的反导电率问题的 Lipschitz 稳定性

Lipschitz stability of an inverse conductivity problem with two Cauchy data pairs

本文证明了具有两对 Cauchy 数据的反导电率问题中线性化前向映射的单射性，推导出非线性反问题解的 Lipschitz 稳定性，并指出绝缘包含物仅需单对数据即可。

弹性卡德隆问题 via 共振硬包含物：N-D 映射的线性化与密度重构

Elastic Calderón Problem via Resonant Hard Inclusions: Linearisation of the N-D Map and Density Reconstruction

通过共振硬包含物构造有效介质，实现弹性反问题中密度的线性化重构与全局恢复。

Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

This work presents a new constructive uniqueness proof for Calderón's inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call "piecewise constant layered conductivities" (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem …

非线性包含的逆障碍问题

The inverse obstacle problem for nonlinear inclusions

针对非线性异常体逆障碍问题缺乏定量物理成像方法的现状，本文基于单调性原理开展理论研究，证明了相关成像方法的噪声稳定性、重构收敛性及极限性质，结果适用于多类非线性情况，为该方法的实际应用提供理论支撑。