Miguel A. Fortes

On Chebyshev-type integral quasi-interpolation operators

Spline quasi-interpolants on the real line are approximating splines to given functions with optimal approximation orders. They are called integral quasi-interpolants if the coefficients in the spline series are linear combinations of weighted mean values of the function to be approximated. This paper is devoted to the construction of new integral quasi-interpolants with compactly supported piecewise polynomial weights. The basic idea consists of minimizing an expression appearing in an estimate for the quasi-interpolation error. It depends on how well the quasi-interpolation operator approximates the first non-reproduced monomial. Explicit solutions as well as some numerical tests in the B-spline case are given.

Preconditioned conjugate gradient method for finding minimal energy surfaces on Powell–Sabin triangulations

Abstract We present an iterative proposal, based on the preconditioned conjugate gradient method, to solve the linear system associated to the problem of approximating a data set by a minimal energy surface constructed through a Powell–Sabin finite element over a Δ 1 -type triangulation defined on a polygonal domain. These approximation problems give rise to symmetric, banded, and positive definite matrices with a very special block structure that depends on the basis functions of the associated vector space, and also on the numeration of the nodes in the triangulation. In practice, the associated sparse matrices are large and ill-conditioned. The special structure of these matrices allows us to adapt and explore several known preconditioned strategies to improve the performance of the conjugate gradient method. We adapt and explore five different preconditioning strategies, including some well-known direct and also some recent inverse strategies. Special attention is paid to the delicate and difficult task of choosing the related parameters in each case. The quality of each preconditioner is evaluated by observing the clustering of the preconditioned matrix eigenvalues, and the obtained reduction in number of iterations. We report on seven different surfaces, and our results indicate that the best preconditioning strategies for this application are the ones based on incomplete factorizations. Nevertheless, from a computational-cost point of view, all the explored strategies are competitive.

A hole filling method for surfaces by using C1-Powell-Sabin splines. Estimation of the smoothing parameters

Abstract: Let D@?R^2 be a polygonal domain, H be a subdomain of D and f@?C^1(D@?-H@?). In this paper we propose a method to reconstruct f over the whole D@? using a technique based on the minimization of an energy functional J. More precisely, we construct a new C^1-Powell-Sabin spline function f^* over D@? that approximates f outside H, and fills the hole of f inside H. The resulting filling patch strongly depends on the values of two smoothing parameters involved in the functional J. We give a criteria to select optimum values of the parameters and we present some graphical and numerical examples.

Multiresolution analysis for minimal energy C r -surfaces on Powell-Sabin type meshes

This paper is intended to provide a multiresolution analysis (MRA) scheme to obtain a sequence of Cr-spline surfaces over a Powell-Sabin triangulation of a polygonal domain approximating a Lagrangian data set and minimizing a certain “energy functional”. We define certain non separable scaling and wavelet functions in bidimensional domains, and we give the decomposition and reconstruction formulas in the framework of lifting schemes. Two important applications of the theory are given: In the first one we develop an algorithm for noise reduction of signals. The second one is related to the localization of the regions where the energy of a given function is mostly concentrated. Some numerical and graphical examples for different test functions and resolution levels are given.

Filling holes with geometric and volumetric constraints

Abstract We propose and analyze different methods to reconstruct a function that is defined outside a sub-domain (hole) of a given domain. The reconstructed function is a smooth Powell–Sabin spline that is defined also inside this hole, filling then this lack of information, and, at the same time, fulfills certain global geometric considerations and other local volume constraints on the hole. We give several examples and we include a technique to estimate the volume of the function inside the hole by using just the data of the function where it is known, that is, outside the hole.

Inverse-free recursive multiresolution algorithms for a data approximation problem

We present inverse-free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization. During the multiresolution process a linear system needs to be solved at each different resolution level, which can be solved with direct or iterative methods. Numerical results are reported, using the sparse Cholesky factorization, for two applications: one concerning the localization of regions in which the energy of a given surface is mostly concentrated, and another one regarding noise reduction of a given dataset. In addition, for large-scale data approximation problems that require a very fine resolution, we discuss the use of the Preconditioned Conjugate Gradient (PCG) iterative method coupled with a specialized monolithic preconditioner, for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels.

Original article: A hole filling method for explicit and parametric surfaces by using C1-Powell Sabin splines

In this work we develop a method to fill a hole in a surface, either explicit or in parametric form, or just in a set of three dimensional scattered data. We will construct a new surface which is very close to the original one where it is known and that fills the hole in a homogeneous way, in such a way that the final reconstruction is of class C^1. We give results which prove the existence and uniqueness of solution of the proposed method, and we present several graphical examples which show the efficiency of the theory developed.

Approximation of patches by Cr-finite elements of Powell–Sabin type

In this work we develop a method to extend a function that is defined in a finite set of disjoint patches to a bigger domain containing all of them. The way to extend the function is by minimizing an energy functional which controls the closeness of the extended function to the original one over the patches, as well as the smoothness of the final reconstructed function. We show the existence and uniqueness of solution of this problem and we give a convergence result as well as several graphical and numerical examples.