Browsing by Author "Cores, Débora"

Now showing 1 - 7 of 7
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    A convex optimization approach for solving large scale linear systems
    (Bulletin of Computational Applied Mathematics, 2017-01) Cores, Débora; Figueroa, Johanna
    The well-known Conjugate Gradient (CG) method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the co- efficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the characteristics of the coefficient matrix. For finding the global minimizers, of this new convex function, any low-cost iterative opti- mization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG) method, which allow us to extend this optimization approach for solving consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex con- straints and with and without preconditioning strategies. Our numerical results indicate that the new scheme outperforms state-of-the-art iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefi-nite, and also for solving linear feasibility problems.
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    A deterministic optimization approach for solving the rainfall disaggregation problem
    (Bulletin of Computational Applied Mathematics, 2015-10-03) Cores, Débora; Guenni, Lelys; Torres, Lisbeth
    One of the main problems in hydrology is the time scale of the historical rainfall data, available from many meteorological data bases. Most of the rainfall data is given at a time scale coarser than the one needed for many applications in hydrology and environmental sciences, as the estimation of spatially continuous rainfall at finer time scales, for drainage systems design and extreme rainfall analysis. A method to disaggregate monthly rainfall to daily or finer temporal scale is very important in many applications. Many authors have addressed this problem by using some stochastic methods including several stochastic rainfall models. The lowering resolution methods must be low-cost and low-storage since the amount of rainfall data is large. The purpose of this work is to formulate this problem as a constrained optimization problem and solve it with a low-cost and low-storage deterministic optimization method. We modify the objective function proposed by Guenni and Bárdossy for solving the disaggregation rainfall problem and we use the low-cost spectral projected gradient (SPG) method. In contrast with the stochastic method, a deterministic approach will take into account important information, as for example the gradient of the objective func- tion. The proposed method was applied to a data set from a rainfall network of the central plains of Venezuela, in which rainfall is highly seasonal and data avail- ability at a daily time scale or even higher temporal resolution is very limited. The numerical results show that the SPG method for solving the disaggregation rainfall problem avoids daily precipitations outliers that might occur as an artifact of the simulation procedure and accurately reproduces the probability distribution. Also, the proposed model and methodology outperforms the one proposed by Guenni and Bárdossy (2002) in the sense that it reduces the absolute error value for the statistical properties from the observed data.
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    A Generalized Two Point Ellipsoidal Anisotropic Ray Tracing for Converted Waves
    (Optimization and Engineering, 2007-12) Cores, Débora; Loreto, Milagros C.
    Rocks can be anisotropic due to a variety of reasons. When estimating rock velocities from seismic data, failure to introduce anisotropy into earth models could generate distortions in the final images that can have enormous economic impact. To estimate anisotropic earth velocities by tomographic methods, it is necessary to trace rays or to solve the wave equation in models where anisotropy has been properly considered. Thus, in this work we present a 3-D generalized ellipsoidal travel time formulation that allow us to trace rays in an anisotropic medium. We propose to trace rays in anisotropic media by solving a set of nonlinear optimization problems, where the group velocities for P and S wave propagation modes are 3-D ellipsoidal approximations that have been recently obtained. Moreover, we prove that this 3-D ellipsoidal anisotropic ray tracing formulation is a convex nonlinear optimization problem, and therefore any solution of the problem is a global minimum. Each optimization problem is solved by the global spectral gradient method, which requires first order information and has low computation and low storage requirements. Our approach for tracing rays in anisotropic media is a generalization in the sense that handles titled axis of symmetry and, close to the axis of symmetry, it is an accu-rate formulation for 2-D transversely isotropic media and 3-D orthorhombic media, depending on the input parameters. Moreover, this formulation gives the exact ray trajectories in 2-D and 3-D homogeneous isotropic media. The simplicity of the formulation and the low computational cost of the optimization method allow us to present a variety of numerical results that illustrate the behavior and computational advantages of the approach, and the difficulties when working in anisotropic media.
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    A low-cost optimization approach for solving minimum norm linear systems and linear least-squares problems
    (Journal of Computational Mathematics, 2024) Cores, Débora; Figueroa, Johanna
    Recently, the authors proposed a low-cost approach, named OPALS (Optimization Approach for Linear Systems) for solving any kind of a consistent linear system regarding the structure, characteristics, and dimension of the coe cient matrix A. The results obtained by this approach for matrices with no structure and with inde nite symmetric part were encouraging when compare with other recent and well-known techniques. In this work, we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem (LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative low-cost gradient-type method, avoiding the construction of the matrices ATA or AAT , and taking full advantage of the structure and form of the gradient of the proposed nonlinear objective function in the gradient direction. The combination of those conditions together with the choice of the initial iterate allow us to produce a novel and e cient low-cost numerical scheme for solving both problems. Moreover, the scheme presented in this work can also be used and extended for the weighted minimum norm linear systems and minimum norm linear least-squares problems. We include encouraging numerical results to illustrate the practical behavior of the proposed schemes.
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    Ajuste de modelo por optimización no lineal usados en la segmentación de estructuras vasculares en imágenes de tomografía computarizada
    (Memorias CIMENICS 2010, 2010-10) Landaeta, Luis; La Cruz, Alexandra; Cores, Débora
    Las imágenes de Tomografia Computarizada (TC) son las más utilizadas para el diagnóstico y evaluación de enfermedades vasculares de las extremidades inferiores. Una de las técnicas más usadas y ampliamente aceptadas por expertos radiólogos para la visualización de estas imágenes, es la técnica de Reformación de Curva Planar (Curve Planar Reformation-CPR), la cual utiliza las líneas centrales de las arterias. Dicha línea central debe describir el camino central de las arterias de manera correcta. Actualmente se utiliza un método de aproximación de líneas centrales bastante preciso, el cual trabaja tantos en segmentos arteriales sanos como en los no sanos (obstruidos, calcificados, etc). Sin embargo cuando se encuentra con una bifurcación falla en la determinación correcta de la línea central, por lo que el proceso de segmentación de líneas centrales en las bifurcaciones debe hacerse manualmente. Asumiendo que la imagen del corte transversal en una bifurcación se asemeja más a una spira de Perseus (spiric of Perseus) o sección spírica (spiric section), la cual es un caso particular de una sección tórica (toric section). Este trabajo consiste en la construcción de un modelo geométrico que permita estimar los parámetros de una sección espírica y los valores de la densidad media, tanto de las arterias como del fondo, que mejor se aproxime a la imagen del corte transversal de una bifurcación en una arteria, utilizando un método de optimización no lineal conocido como el método de la región de confianza de Newton. Esto nos da una aproximación más precisa del camino central de las arterias, incluyendo las bifurcaciones.
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    Nonsmooth spectral gradient methods for unconstrained optimization
    (Euro Journal on Computational Optimization, 2017) Loreto, Milagros; Aponte, Hugo; Cores, Débora; Raydan, Marcos
    To solve nonsmooth unconstrained minimization problems, we combine the spectral choice of step length with two well-established subdifferential-type schemes: the gradient sampling method and the simplex gradient method. We focus on the interesting case in which the objective function is continuously differentiable almost everywhere, and it is of- ten not differentiable at minimizers. In the case of the gradient sampling method, we also present a simple differentiability test that allows us to use the exact gradient direction as frequently as possible, and to build a stochastic subdifferential direction only if the test fails. The proposed spectral gradient sampling method is combined with a monotone line search globalization strategy. On the other hand, the simplex gradient method is a direct search method that only requires function evaluations to build an approximation to the gradient direction. In this case, the proposed spectral simplex gradient method is combined with a suitable nonmonotone line search strategy. For both scenarios, we present preliminary nu- merical results on a set of nonsmooth test functions. These numerical results indicate that using a spectral step length can improve the practical performance of both methods.
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    On the use of the Spectral Projected Gradient method for Support Vector Machines
    (Computational and Applied Mathematics, 2009) Cores, Débora; Escalante, René; González-Lima, María; Jiménez, Oswaldo
    In this work we study how to solve the SVM optimization problem by using the Spectral Projected Gradient (SPG) method with three different strategies for computing the projection onto the constrained set. One of the strategies is based on Dykstra’s alternating projection algorithm since there is not a mathematical equation for the projection onto the whole constrained set but the projection on each restriction is easy to compute with exact formulations. We present another strategy based on the Karush-Kunh-Tucker optimality conditions, we call it the Projected-KKT algorithm. We compare these strategies with a third one proposed by Dai and Fletcher. The three schemes are low computational cost and their use within the SPG algorithm leads to a solution of the SVM problem. We study the computational performance of the three strategies when solving randomly generated as well as real life SVM problems. The numerical results show that Projected-KKT is competitive in general with the Dai and Fletcher algorithm, and it is more efficient for some specific problems. They both outperform Dykstra’s algorithm in all the tests.