In the last few years, several techniques to fill holes of a given surface by means of minimal energy surfaces have been proposed. In all cases, the filling patches are obtained by minimizing an "energy functional" defined in a vector space of spline functions over the Powell-Sabin triangulation associated to a Delta1-type triangulation of a given domain D. The energy functional and the space of spline functions are defined in order to the filling patch fulfills certain geometric features. In this work we present, for the first time, a general framework to include most of techniques above referred. Under this general new frame, we review the main filling-holes techniques developed until now, we give their main characteristics, the computation aspects as well as some graphical examples.

We consider the numerical computation of matrix functions f(x) via matrix ODE integration. The solution is modeled as an asymptotic steady state of a proper differential system. The framework we propose, allows to define flows of sparse matrices leading to sparse approximations to f(x). We discuss of this approach giving stability and approximation results in a general case. We apply our method to the factorization of matrices (LU, Cholesky) as well as the computation of the square root. Numerical illustrations are presented.

Inexact restoration (IR) is a well established technique for continuous minimization problems with constraints that can be applied to constrained optimization problems with specific structures. When some variables are restricted to be integer, an IR strategy seems to be appropriate. The IR strategy employs a restoration procedure in which one solves a standard nonlinear programming problem and an optimization procedure in which the constraints are linearized and techniques for mixed-integer (linear or quadratic) programming can be employed.

The Modified Spectral Projected Subgradient (MSPS) was proposed to solve Langrangen Dual Problems, and its convergence was shown when the momentum term was zero. The MSPS uses a momentum term in order to speed up its convergence. The momentum term is built on the multiplication of a momentum parameter and the direction of the previous iterate. In this work, we show convergence when the momentum parameter is a non-zero constant. We also propose heuristics to choose the momentum parameter intended to avoid the Zigzagging Phenomenon of Kind I. This phenomenon is present in the MSPS when at an iterate the subgradient forms an obtuse angle with the previous direction. We identify and diminish the Zigzagging Phenomenon of Kind I on Setcovering problems, and compare our numerical results to those of the original MSPS algorithm.

We present a derivative-free algorithm for solving bound constrained systems of nonlinear monotone equations. The algorithm generates feasible iterates using in a systematic way the residual as search direction and a suitable step-length closely related to the Barzilai-Borwein choice. A convergence analysis is described. We also present one application in solving problems related with the study of reaction-diffusion processes that can be described by nonlinear partial differential equations of elliptic type. Numerical experiences are included to highlight the efficacy of proposed algorithm.

This work addresses a spectral correction for the Gauss-Newton model in the solution of nonlinear least-squares problems within a globally convergent algorithmic framework. The nonmonotone line search of Zhang and Hager is the chosen globalization tool. We show that the search directions obtained from the corrected Gauss-Newton model satisfy the conditions that ensure the global convergence under such a line search scheme. A numerical study assesses the impact of using the spectral correction for solving two sets of test problems from the literature.