We consider the stability issue for a broad class of inverse problems described by second-order elliptic equations with anisotropic and scalar coefficients that are finite-dimensional. This class of problems encompasses the well-studied conductivity equation, the Helmholtz equation and the Schrödinger equation. The applications of this study range from medicine, for example EIT, where the coefficient to be reconstructed is the conductivity, to the reconstruction of the wave-speed in a medium. It is well known that these inverse problems are ill-posed.

In this talk we prove a logarithmic stability estimate for the inverse problem that regards the determination of an inclusion in terms of local Cauchy data, since the Dirichlet to Neumann map that can encode the data at the boundary is not always available. This talk is based on a joint work with Eva Sincich.

Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache. Inspired by Mandache's idea, in this talk, I would like to refinements of the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. The aim is to derive explicitly the dependence of the instability estimates on key parameters.

The first topic of this talk is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. The second topic is to justify the optimality of increasing stability in determining the near-field of a radiating solution of the Helmholtz equation from the far-field pattern.

We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined conductivities. The argument has three main ingredients: 1. the logarithmic stability of the related inverse problem for the fractional Schrödinger equation by Rüland and Salo; 2. the Lipschitz stability of the exterior determination problem; 3. utilizing and identifying nonlocal analogies of Alessandrini's work on the stability of the classical Calderón problem. The main contribution of the article is the resolution of the technical difficulties related to the last mentioned step. Furthermore, we show the optimality of the logarithmic stability estimates, following the earlier works by Mandache on the instability of the inverse conductivity problem, and by Rüland and Salo on the analogous problem for the fractional Schrödinger equation.

In this talk we investigate the issue of uniqueness and stability for certain inverse problems which forward problem is modelled by a second order elliptic partial differential equation. As is well known, there is a fundamental obstruction to uniquely determine physical properties of anisotropic materials from boundary maps/measurements. In fact, any diffeomorphism of the domain under investigation, that keeps the domain's boundary fixed, changes its material's properties without changing its boundary measurements. In this talk we will provide some positive answers to the issues of uniqueness and stability of certain type of anisotropy in terms to the correspondent boundary measurements.