Área de Matemática Aplicada
http://hdl.handle.net/10016/19123
Sat, 04 Jul 2020 18:34:24 GMT2020-07-04T18:34:24ZRobust multifrequency imaging with MUSIC
http://hdl.handle.net/10016/28337
Robust multifrequency imaging with MUSIC
Moscoso Castro, Miguel Ángel; Novikov, Alexei; Papanicolaou, George; Tsogka, Chrysoula
In this paper, we study the MUltiple SIgnal Classification (MUSIC) algorithm often used to image small targets when multiple measurement vectors are available. We show that this algorithm may be used when the imaging problem can be cast as a linear system that admits a special factorization. We discuss several active array imaging configurations where this factorization is exact, as well as other configurations where the factorization only holds approximately and, hence, the results provided by MUSIC deteriorate. We give special attention to the most general setting where an active array with an arbitrary number of transmitters and receivers uses signals of multiple frequencies to image the targets. This setting provides all the possible diversity of information that can be obtained from the illuminations. We give a theorem that shows that MUSIC is robust with respect to additive noise provided that the targets are well separated. The theorem also shows the relevance of using appropriate sets of controlled parameters, such as excitations, to form the images with MUSIC robustly. We present numerical experiments that support our theoretical results.
Tue, 04 Dec 2018 00:00:00 GMThttp://hdl.handle.net/10016/283372018-12-04T00:00:00ZQuantitative subsurface imaging in strongly scattering media
http://hdl.handle.net/10016/28336
Quantitative subsurface imaging in strongly scattering media
González Rodríguez, Pedro; Moscoso Castro, Miguel Ángel; Kim, Arnold; Tsogka, Chrysoula
We present a method to obtain quantitatively accurate images of small obstacles or inhomogeneities situated near the surface of a strongly scattering medium. The method uses time-resolved measurements of backscattered light to form the images. Using the asymptotic solution of the radiative transfer equation for this problem, we determine that the key information content in measurements is modeled by a diffusion approximation that is valid for small source-detector distances, and shallow penetration depths. We simplify this model further by linearizing the effect of the inhomogeneities about the known background optical properties using the Born approximation. The resulting model is used in a two-stage imaging algorithm. First, the spatial location of the inhomogeneities are determined using a modification of the multiple signal classification (MUSIC) method. Using those results, we then determine the quantitative values of the inhomogeneities through a least-squares approximation. We find that this two-stage method is most effective for reconstructing a sequence of one-dimensional images along the penetration depth corresponding to none source-detector separations rather than simultaneously using measurements over several source-detector distances. This method is limited to penetration depths and distances between boundary measurements on the order of the scattering mean-free path. (C) 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Mon, 15 Oct 2018 00:00:00 GMThttp://hdl.handle.net/10016/283362018-10-15T00:00:00ZA closed-form formula for the RBF-based approximation of the Laplace-Beltrami operator
http://hdl.handle.net/10016/28335
A closed-form formula for the RBF-based approximation of the Laplace-Beltrami operator
Álvarez Román, Juan Diego; González Rodríguez, Pedro; Moscoso Castro, Miguel Ángel
In this paper we present a method that uses radial basis functions to approximatethe Laplace&-Beltrami operator that allows to solve numerically diffusion (and reaction&-diffusion) equations on smooth, closed surfaces embedded in R3. The novelty of the methodis in a closed-form formula for the Laplace&-Beltrami operator derived in the paper, whichinvolve the normal vector and the curvature at a set of points on the surface of interest.An advantage of the proposed method is that it does not rely on the explicit knowledgeof the surface, which can be simply defined by a set of scattered nodes. In that case, thesurface is represented by a level set function from which we can compute the needed normalvectors and the curvature. The formula for the Laplace&-Beltrami operator is exact for radialbasis functions and it also depends on the first and second derivatives of these functionsat the scattered nodes that define the surface. We analyze the converge of the method andwe present numerical simulations that show its performance. We include an application thatarises in cardiology.
Tue, 29 May 2018 00:00:00 GMThttp://hdl.handle.net/10016/283352018-05-29T00:00:00ZMaximum Entropy Closure of Balance Equations for Miniband Semiconductor Superlattices
http://hdl.handle.net/10016/27909
Maximum Entropy Closure of Balance Equations for Miniband Semiconductor Superlattices
López Bonilla, Luis Francisco; Carretero Cerrajero, Manuel
Charge transport in nanosized electronic systems is described by semiclassical or quantum kinetic equations that are often costly to solve numerically and difficult to reduce systematically to macroscopic balance equations for densities, currents, temperatures and other moments of macroscopic variables. The maximum entropy principle can be used to close the system of equations for the moments but its accuracy or range of validity are not always clear. In this paper, we compare numerical solutions of balance equations for nonlinear electron transport in semiconductor superlattices. The equations have been obtained from Boltzmann-Poisson kinetic equations very far from equilibrium for strong fields, either by the maximum entropy principle or by a systematic Chapman-Enskog perturbation procedure. Both approaches produce the same current-voltage characteristic curve for uniform fields. When the superlattices are DC voltage biased in a region where there are stable time periodic solutions corresponding to recycling and motion of electric field pulses, the differences between the numerical solutions produced by numerically solving both types of balance equations are smaller than the expansion parameter used in the perturbation procedure. These results and possible new research venues are discussed.
Thu, 14 Jul 2016 00:00:00 GMThttp://hdl.handle.net/10016/279092016-07-14T00:00:00Z