An effectivization of the global reconstruction in the Gel'fand-Calderon inverse problem in three dimensions

On stable determination of potential by boundary measurements

We give new stability estimates for the Gel'fand-Calderon inverse boundary value problem

New global stability estimates for the Calderón problem in two dimensions

We prove a new global stability estimate for the Gel'fand-Calderón inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-Δψ+ v\, ψ= 0$ on $D$, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The principal feature of this estimate is that it shows that the more a potential is smooth, the more its reconstruction is stable, and the stability varies exponentially with respect to the smoothness (in a sense to be made precise). As a corollary we obtain a similar estimate f…

On the Gel'fand-Calderón inverse problem in two dimensions

We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schrödinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive fractional derivatives are in a specific Lorentz space. These spaces are a natural generalization to the usual fractional Sobolev spaces. The thesis consists of two parts. In the first part, we define the generalized fractional Sobolev spaces and prove some of their properties including embeddings and interpolation identities. In particular we sharpen the usual Sobol…

Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data

We prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^\infty$. A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based …

The Calderon problem in transversally anisotropic geometries

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Four…

闭黎曼流形上的分数阶 Calderón 问题

Fractional Calderón problem on a closed Riemannian manifold

本文证明闭黎曼流形上分数阶 Laplace 方程的源到解映射可在任意小观测区恢复流形等距类，并借助 Carlson 定理变体将其归约为 Gel'fand 反谱问题。

Unique determination of a magnetic Schrödinger operator with unbounded magnetic potential from boundary data

We consider the Gel'fand-Calderón problem for a Schrödinger operator of the form $-(\nabla + iA)^2 + q$, defined on a ball $B$ in $\mathbb R^3$. We assume that the magnetic potential $A$ is small in $W^{s,3}$ for some $s>0$, and that the electric potential $q$ is in $W^{-1,3}$. We show that, under these assumptions, the magnetic field $\operatorname{curl} A$ and the potential $q$ are both determined by the Dirichlet-Neumann relation at the boundary $\partial B$. The assumption on $q$ is critical with respect to homogeneity, and the assumption on $A$ is nearly critical. Previous uniqueness theo…

Inverse problems on low-dimensional manifolds

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite…