Stability estimates for some anisotropic inverse problems

This doctoral thesis is devoted to the study of the stability of some inverse problems concerning media with anisotropic properties. Specifically, we have studied the conditional stability, which refers to the dependence of the unknown parameters or defects on known measurement data when the unknowns satisfy certain constraints. Inverse problems are known to be ill-posed in the Hadamard sense, making it difficult to obtain reliable numerical reconstructions of the parameters of interest. In this thesis, we discuss the stability issue for two types of inverse problems: the coefficient identification problem and the determination of inclusions. The underlying mathematical model is given by a second-order linear elliptic equation of the form: \begin{equation}\label{eqnAb: elliptic} \text{div}(\sigma\nabla u) + q\:u=0,\qquad \text{in }\Omega, \end{equation} with prescribed Cauchy data $\{u|_{\partial\Omega},\sigma\nabla u\cdot \nu|_{\partial\Omega}\}$ or the Dirichlet-to-Neumann (DtoN) map, when the latter is defined. The leading-order term exhibits an anisotropic behaviour, which is described by a coefficient of the form \begin{equation}\label{eqnAb: conductivity} \sigma(x)=\gamma(x)A(x), \end{equation} where $\gamma$ is a suitably regular scalar function and $A$ is a matrix-valued function. We address the stability issue for the inverse conductivity problem, when $q=0$ in \eqref{eqnAb: elliptic}, of determining the conductivity $\sigma$ of the form \eqref{eqnAb: conductivity}, where $\gamma$ is a piecewise affine function, from the knowledge of the boundary data. We construct an ad-hoc misfit functional to encode the error arising from the comparison with two different known boundary data. We prove a H\"older stability estimate in terms of the misfit functional and then we derive a Lipschitz stability estimate in terms of the local Dirichlet to Neumann map. In the general case, when $q\neq 0$ and no sign or spectrum condition on $q$ is assumed in the generalised Schr\"odinger equation \eqref{eqnAb: elliptic}, we address the inverse problem of determining an anisotropic inclusion in a medium $\Omega$ from the knowledge of the local Cauchy data and the problem of the simultaneous determination of the two coefficients in \eqref{eqnAb: elliptic}. For the first problem, we provide a log-type stability estimate which holds under mild constraints on the unknown inclusion. For the latter, we derive a global Lipschitz stability estimate which holds for the coefficients $\sigma$ and $q$ in \eqref{eqnAb: elliptic} with a piecewise affine scalar part on a known partition of the domain.

This doctoral thesis is devoted to the study of the stability of some inverse problems concerning media with anisotropic properties. Specifically, we have studied the conditional stability, which refers to the dependence of the unknown parameters or defects on known measurement data when the unknowns satisfy certain constraints. Inverse problems are known to be ill-posed in the Hadamard sense, making it difficult to obtain reliable numerical reconstructions of the parameters of interest. In this thesis, we discuss the stability issue for two types of inverse problems: the coefficient identification problem and the determination of inclusions. The underlying mathematical model is given by a second-order linear elliptic equation of the form: \begin{equation}\label{eqnAb: elliptic} \text{div}(\sigma\nabla u) + q\:u=0,\qquad \text{in }\Omega, \end{equation} with prescribed Cauchy data $\{u|_{\partial\Omega},\sigma\nabla u\cdot \nu|_{\partial\Omega}\}$ or the Dirichlet-to-Neumann (DtoN) map, when the latter is defined. The leading-order term exhibits an anisotropic behaviour, which is described by a coefficient of the form \begin{equation}\label{eqnAb: conductivity} \sigma(x)=\gamma(x)A(x), \end{equation} where $\gamma$ is a suitably regular scalar function and $A$ is a matrix-valued function. We address the stability issue for the inverse conductivity problem, when $q=0$ in \eqref{eqnAb: elliptic}, of determining the conductivity $\sigma$ of the form \eqref{eqnAb: conductivity}, where $\gamma$ is a piecewise affine function, from the knowledge of the boundary data. We construct an ad-hoc misfit functional to encode the error arising from the comparison with two different known boundary data. We prove a H\"older stability estimate in terms of the misfit functional and then we derive a Lipschitz stability estimate in terms of the local Dirichlet to Neumann map. In the general case, when $q\neq 0$ and no sign or spectrum condition on $q$ is assumed in the generalised Schr\"odinger equation \eqref{eqnAb: elliptic}, we address the inverse problem of determining an anisotropic inclusion in a medium $\Omega$ from the knowledge of the local Cauchy data and the problem of the simultaneous determination of the two coefficients in \eqref{eqnAb: elliptic}. For the first problem, we provide a log-type stability estimate which holds under mild constraints on the unknown inclusion. For the latter, we derive a global Lipschitz stability estimate which holds for the coefficients $\sigma$ and $q$ in \eqref{eqnAb: elliptic} with a piecewise affine scalar part on a known partition of the domain.