A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions

We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain

Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT

In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions

New global stability estimates for the Calder\'on problem in two dimensions

We prove a new global stability estimate for the Gel'fand-Calder\'on inverse problem on a two-dimensional bounded domain or, more precisely, the inverse boundary value problem for the equation $-\Delta \psi + v\, \psi = 0$ 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 for the Calder\'on problem for the electrical impedance tomography.Comment: 18 page

Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions

17 pages.International audienceWe study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this paper we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy, for low and high energies (in modulus). In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases when increasing the energy (in modulus)

What Future for the Renewable Energy

       In 1973, after the Kippur war, a considerable increase of the petroleum price occurred, because, Arabian countries decided to reduce the extraction and export of this raw material. It was the first time that petroleum price was imposed not by the market but as a consequence of a political unilateral decision. The governments of the occidental countries reacted to this situation by promoting researches on the use of coal as a possible substitute of petroleum. In Italy, for example, a national “Energy†project based on the use of coal had been launched and funded covering a period of about 10 years.  Although, after some years the situation turned to the normality and the price of petroleum was consistently lowered rendering not convenient the use of coal, thanks to the performed studies,  the scientists have learned to obtain from coal all the necessary components to satisfy the energetic needs. Coal was chosen as a possible alternative for the abundance of this raw material and, because, at that time the consume of petroleum was relatively limited and the environmental problems, deriving from a majestic use of petroleum, were not so important as today. In the last century, the petroleum consumption increased exponentially and the growing economy of the countries under development greatly contributed and still contributes to this increment. Gradually, the anxiety for the depletion of petroleum as raw material, which remedy could be clearly the use of coal, was substituted by the anxiety for the sustainability of the consequences of a continuous increase of the petroleum consumption on both the environment and the quality of the life. Under this aspect, coal is worse than petroleum and cannot be considered as a possible substitute.       Sustainability gradually became the cultural driving force for the development of the “Renewable Energy Sources†and new concepts like Bio-Refineries and Renewable Feedstock were diffused in a lot of publications and books and the governments funded many researches on the subject. However, it is opportune to remember that the consume of petroleum in the world is greater than 4 billion tons per year with only 4-5% destined to the chemicals production. The big economic interests, accompanying the commerce of petroleum, hold-down the development of any possible alternative. The recent intensive production of shale oil in United States, for example, has been contrasted by a strong decrease of the petroleum price produced in Arabia. In the meantime, a large availability of petroleum at low cost strongly reduced or blocked the production of bio-based fuels as biodiesel and bio-ethanol. The conclusion is that nowadays, Renewable Energy Sources cannot compete with petroleum for both the volume of production and price of the energy but remain the only feasible alternative if inserted in a revolutionary change of the way of life in which the man look for an equilibrium with the nature without renouncing to the wellness but certainly changing the previous habits. This is, at the same time a cultural, scientific but above all a political challenge to save the planet. Citation: Santacesaria, E. (2015). "What future for the Renewable Energy." Trends in Renewable Energy, 1(2), 57-58. DOI: 10.17737/tre.2015.1.2.001

Positive-energy D-bar method for acoustic tomography: a computational study

A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results

Neural networks for classification of strokes in electrical impedance tomography on a 3D head model

We consider the problem of the detection of brain hemorrhages from three dimensional (3D) electrical impedance tomography (EIT) measurements. This is a condition requiring urgent treatment for which EIT might provide a portable and quick diagnosis. We employ two neural network architectures -- a fully connected and a convolutional one -- for the classification of hemorrhagic and ischemic strokes. The networks are trained on a dataset with $40\,000$ samples of synthetic electrode measurements generated with the complete electrode model on realistic heads with a 3-layer structure. We consider changes in head anatomy and layers, electrode position, measurement noise and conductivity values. We then test the networks on several datasets of unseen EIT data, with more complex stroke modeling (different shapes and volumes), higher levels of noise and different amounts of electrode misplacement. On most test datasets we achieve $\geq 90\%$ average accuracy with fully connected neural networks, while the convolutional ones display an average accuracy $\geq 80\%$. Despite the use of simple neural network architectures, the results obtained are very promising and motivate the applications of EIT-based classification methods on real phantoms and ultimately on human patients.Comment: 17 pages, 11 figure

Revisiting the Role of Mass and Heat Transfer in Gas–Solid Catalytic Reactions

The tremendous progress in the computing power of modern computers has in the last 20 years favored the use of numerical methods for solving complex problems in the field of chemical kinetics and of reactor simulations considering also the effect of mass and heat transfer. Many classical textbooks dealing with the topic have, therefore, become quite obsolete. The present work is a review of the role that heat and mass transfer have in the kinetic studies of gas–solid catalytic reactions. The scope was to collect in a relatively short document the necessary knowledge for a correct simulation of gas–solid catalytic reactors. The first part of the review deals with the most reliable approach to the description of the heat and mass transfer outside and inside a single catalytic particle. Some different examples of calculations allow for an easier understanding of the described methods. The second part of the review is related to the heat and mass transfer in packed bed reactors, considering the macroscopic gradients that derive from the solution of mass and energy balances on the whole reactor. Moreover, in this second part, some examples of calculations, applied to chemical reactions of industrial interest, are reported for a better understanding of the systems studie