Browsing by Author "Rojas, M."

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    Accelerating the LSTRS Algorithm
    (2009-07) Lampe, J.; Rojas, M.; Sorensen, D.C.; Voss, H.
    In a recent paper [Rojas, Santos, Sorensen: ACM ToMS 34 (2008), Article 11] an efficient method for solvingthe Large-Scale Trust-Region Subproblem was suggested which is based on recasting it in terms of a parameter dependent eigenvalue problem and adjusting the parameter iteratively. The essential work at each iteration is the solution of an eigenvalue problem for the smallest eigenvalue of the Hessian matrix (or two smallest eigenvalues in the potential hard case) and associated eigenvector(s). Replacing the implicitly restarted Lanczos method in the original paper with the Nonlinear Arnoldi method makes it possible to recycle most of the work from previous iterations which can substantially accelerate LSTRS.
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    Efficient Numerical Methods for Least-Norm Regularization
    (2010-03) Sorensen, D.C.; Rojas, M.
    The problem min ||x||, s.t. ||b-Ax||≤ ε arises in the regularization of discrete forms of ill-posed problems when an estimate of the noise level in the data is available. After deriving necessary and sufficient optimality conditions for this problem, we propose two different classes of algorithms: a factorization-based algorithm for small to medium problems, and matrix-free iterations for the large-scale case. Numerical results illustrating the performance of the methods demonstrate that both classes of algorithms are efficient, robust, and accurate. An interesting feature of our formulation is that there is no situation corresponding to the so-called hard case that occurs in the standard trust-region subproblem. Neither small singular values nor vanishing coefficients present any difficulty to solving the relevant secular equations.