Resumo em Inglês:

This paper is a reprint of the original work by A. P. Calderón published by the Brazilian Mathematical Society (SBM) in ATAS of SBM (Rio de Janeiro), pp. 65-73, 1980. The original paper had no abstract, so this reprint to be truthful to the original work is published with no abstract.

Resumo em Inglês:

In the present work we propose the use of weighted Bregman distances in the construction of regularization terms for the Tikhonov functional applied for the formulation and solution of the inverse problem of photoacoustic spectroscopy. Test case results demonstrate that better estimates were obtained for the simultaneous estimation of the thermal diffusivity and optical absorption coefficient using, as synthetic experimental data, the information on both the amplitude and phase-lag of the temperature at the interface sample-gas between the material under analysis and the air chamber of the closed photoacoustic cell.

Resumo em Inglês:

This paper reviews four variants of global Carleman weights that are especially adapted to some singular controllability and inverse problems in partial differential equations. These variants arise when studying: i) one measurement stationary source inverse problems for the heat equation with discontinuous coefficients, ii) one measurement stationary potential inverse problems for the heat equation with discontinuous coefficients, iii) null controllability for fluid-structure problems in mobile domains and iv) recovering coefficients from locally supported boundary observations for the wave equation. In all the case we explain how to explicitly construct the Carleman weight.

Resumo em Inglês:

We investigate the problem of identifying discontinuous doping profiles in semiconductor devices from data obtained by the stationary voltage-current (VC) map. The related inverse problem correspond to the inverse problem for the Dirichlet-to-Neumann (DN) map with partial data.

Resumo em Inglês:

We describe the continuous and discrete mathematical models for Emission Computed Tomography (ECT) and the need for attenuation correction. Then we analyse the problem of retrieving the attenuation directly from the emission data, nonuniqueness and its consequences. We present the existing and new approaches for solving the problem. Methods are compared and illustrated by numerical simulations. The presentation focuses on Positron Emission Tomography, but we also discuss the extensions to Single Photon Emission Computed Tomography.

Resumo em Inglês:

First we prove a Carleman estimate for a hyperbolic integro-differential equation. Next we apply such a result to identify a spatially dependent function in a source term by an (additional) single measurement on the boundary.

Resumo em Inglês:

The aim of this article is to propose a new method for the inpainting problem. Inpainting is the problem of filling-in holes in images. We consider in this article the crack localization problem, which can be solved using the Dirichlet to Neumann approach and the topological gradient. In a similar way, we can define a Dirichlet and a Neumann inpainting problem. We then define a cost function measuring the discrepancy between the two corresponding solutions. The minimization is done using the topological asymptotic analysis, and is performed in only one iteration. The optimal solution provides the best localization of the missing edges, and it is then easy to inpaint the holes.

Resumo em Inglês:

We consider the three dimensional electromagnetic inverse scattering problem of determining information about a buried coated object from a knowledge of the electric and magnetic fields measured on the surface of the earth corresponding to time harmonic electric dipoles as incident fields. We assume that the buried object is a perfect conductor that is (possibly) partially coated by a thin dielectric layer. No a priori assumption is made on the extent of the coating, i.e. the object can be fully coated, partially coated or not coated at all. We present an algorithm based on the linear sampling method and reciprocity gap functional for reconstructing the shape of the scattering obstacle together with an estimate of the surface impedance of the coating.

Resumo em Inglês:

We consider electromagnetic evasion-interrogation games wherein the evader can use ferroelectric material coatings to attempt to avoid detection while the interrogator can manipulate the interrogating frequencies (wave numbers) and angles of incidence of the interrogating inputs to enhance detection and identification. The resulting problems are formulated as two player games in which one player wishes to minimize the reflected signal while the other wishes to maximize it. Simple deterministic strategies are easily defeated and hence the players must introduce uncertainty to disguise their intentions and confuse their opponent. Mathematically, the resulting game is carried out over spaces of probability measures which in many cases are appropriately metrized using the Prohorov metric.

Resumo em Inglês:

Tsallis' non-extensive entropy is used as a regularization operator. The parameter ''q'' (non-extensivity parameter) has a central role in the Tsallis' thermostatiscs formalism. Here, several values of q are investigated in inverse problems, using q < 1 and q > 1. Two standard regularization techniques are recovered for special q-values: (i) q = 2 is the well known Tikhonov regularization; (ii) q = 1 is the standard Boltzmann-Gibbs-Shannon formulation for entropy. The regularization feature is illustrated in an inverse test problem: the estimation of initial condition in heat conduction problem. Two methods are studied for determining the regularization parameter, the maximum curvature for the L-curve, and the Morozov's discrepancy principle. The new regularization of higher order is applied to the retrieval of the atmospheric vertical profile from satellite data.

Resumo em Inglês:

In this paper we consider the inverse problem of recovering the collision kernel for the time dependent linear Boltzmann equation via a finite number of boundary measurements. We prove that this kernel can be uniquely determined by at most k measurements, provided that it belongs to a finite k-dimensional vector space.

Resumo em Inglês:

In this paper, we derive a sharp observability inequality for Kirchhoff plate equations with lower order terms. More precisely, for any T > 0 and suitable boundary observation domains (satisfying the geometric conditions that the multiplier method imposes), we prove an estimate with an explicit observability constant for Kirchhoff plate systems with an arbitrary finite number of components and in any space dimension with lower order bounded potentials.