We study and analyze a nonmonotone globally convergent method for minimization on closed sets. This method is based on the ideas from trust-region and Levenberg-Marquardt methods. Thus, the subproblems consists in minimizing a quadratic model of the objective function subject to the constraint set. We incorporate concepts of bidiagonalization and calculation of the SVD "with inaccuracy'' to improve the performance of the algorithm, since the solution of the subproblem by traditional techniques, which is required in each iteration, is computationally expensive. Other feasible methods are mentioned, including a curvilinear search algorithm and a minimization along geodesics algorithm. Finally, we illustrate the numerical performance of the methods when applied to the Orthogonal Procrustes Problem.<object id="c828aa19-8948-aba9-58e8-19af48b78cb9" width="0" height="0" type="application/gas-events-bb"></object>
In this paper, we consider the numerical treatment of an inverse acoustic scattering problem that involves an impenetrable obstacle embedded in a layered medium. We begin by employing a modified version of the well known factorization method, in which a computationally effective numerical scheme for the reconstruction of the shape of the scatterer is presented. This is possible, due to a mixed reciprocity principle, which renders the computation of the Green function at the background medium unnecessary. Moreover, to further refine our inversion algorithm, an efficient Tikhonov parameter choice technique, called Improved Maximum Product Criterion (IMPC) is exploited. Our regularization parameter is computed via a fast iterative algorithm which requires no a priori knowledge of the noise level in the far-field data. Finally, the effectiveness of IMPC is illustrated with various numerical examples.
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