Browsing by Author "Figueroa, Johanna"

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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 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.