Exponential instability in an inverse problem for the Schrodinger equation

We consider the problem of the determination of the potential from the Dirichlet to Neumann map of the Schrodinger operator.We show that this problem is severely ill posed. The results extend to the electrical impedance tomography.They show that the logarithmic stability results of Alessandrini are optimal

Hausdorff moments in an inverse problem for the heat equation: numerical experiment

In this paper we consider the inverse boundary problem for the heat equation Deltau(x, t) = rho(x)partial derivative(t)u(x, t) in a bounded domain Omega subset of R-2. We develop and test numerically an algorithm of an approximate reconstruction of the unknown p(x). This algorithm is based on the moments' method for the heat equation developed by Kawashita, Kurylev and Soga

Exponential instability in the fractional Calder\'on problem

In this note we prove the exponential instability of the fractional Calder\'on problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calder\'on problem. Here we exploit a close relation between the fractional Calder\'on problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS17}. Finally, in one dimension, we show a close relation between the fractional Calder\'on problem and the truncated Hilbert transform.Comment: 17 page

Stable Determination of the Electromagnetic Coefficients by Boundary Measurements

The goal of this paper is to prove a stable determination of the coefficients for the time-harmonic Maxwell equations, in a Lipschitz domain, by boundary measurements

On the Size Complexity of Non-Returning Context-Free PC Grammar Systems

Improving the previously known best bound, we show that any recursively enumerable language can be generated with a non-returning parallel communicating (PC) grammar system having six context-free components. We also present a non-returning universal PC grammar system generating unary languages, that is, a system where not only the number of components, but also the number of productions and the number of nonterminals are limited by certain constants, and these size parameters do not depend on the generated language

Solution of a class of one-dimensional reaction-diffusion models in disordered media

We study a one-dimensional class of reaction-diffusion models on a $10-$parameters manifold. The equations of motion of the correlation functions close on this manifold. We compute exactly the long-time behaviour of the density and correlation functions for {\it quenched} disordered systems. The {\it quenched} disorder consists of disconnected domains of reaction. We first consider the case where the disorder comprizes a superposition, with different probabilistic weights, of finite segments, with {\it periodic boundary conditions}. We then pass to the case of finite segments with {\it open boundary conditions}: we solve the ordered dynamics on a open lattice with help of the Dynamical Matrix Ansatz (DMA) and investigate further its disordered version.Comment: 11 pages, no figures. To appear in Phys.Rev.

New global stability estimates for the Gel'fand-Calderon inverse problem

We prove new global stability estimates for the Gel'fand-Calderon inverse problem in 3D. For sufficiently regular potentials this result of the present work is a principal improvement of the result of [G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153-172]

Computing Volume Bounds of Inclusions by EIT Measurements

The size estimates approach for Electrical Impedance Tomography (EIT) allows for estimating the size (area or volume) of an unknown inclusion in an electrical conductor by means of one pair of boundary measurements of voltage and current. In this paper we show by numerical simulations how to obtain such bounds for practical application of the method. The computations are carried out both in a 2D and a 3D setting.Comment: 20 pages with figure