Nonlinear thresholding of multiresolution decompositions adapted to the presence of discontinuities

International audienceA new nonlinear representation of multiresolution decompositions and new thresholding adapted to the presence of discontinuities are presented and analyzed. They are based on a nonlinear modification of the multiresolution details coming from an initial (linear or nonlinear) scheme and on a data dependent thresholding. Stability results are derived. Numerical advantages are demonstrated on various numerical experiments

Primera generalización de la Extrapolación Recíproca

Partiremos de una alternativa no lineal a la extrapolación clásica; la extrapolación recíproca. Revisaremos la primera generalización del método de Richardson y realizaremos una generalización de la extrapolación recíproca. Por último, veremos como esta nueva generalización tiene un mejor comportamiento en problemas donde la región de estabilidad es demasiado pequeña

Some Remarks on a Variational Method for Stiff Differential Equations

We have recently proposed a variational framework for the approximation of systems of differential equations. We associated, in a natural way, with the original problem, a certain error functional. The discretization is based on standard descent schemes, and we can use a variable-step implementation. The minimization problem has a unique solution, and the approach has a global convergence. The use of our error-functional strategy was considered by other authors, but using a completely different way to derive the discretization. Their technique was based on the use of an integral form of the Euler equation for a related optimal control problem, combined with an adapted version of the shooting method, and the cyclic coordinate descent method. In this note, we illustrate and compare our strategy to theirs from a numerical point of view

Error bounds for a class of subdivision schemes based on the two-scale refinement equation

Subdivision schemes are iterative procedures to construct curves and constitute fundamental tools in Computer Aided Design. Starting with an initial control polygon, a subdivision scheme refines the computed values at the previous step according to some basic rules. The scheme is said to be convergent if there exists a limit curve. The computed values define a control polygon in each step. This paper is devoted to estimate error bounds between the “ideal” limit curve and the control polygon defined after k subdivision stages. In particular, a stop criteria of convergence is obtained. The considered refinement rules in the paper are widely used in practice and are associated to the well known two-scale refinement equation including as particular examples Daubechies’ schemes. Companies such as Pixar have made subdivision schemes the basic tool for much of their computer graphicsmodelling software.Research supported in part by MTM2007-62945

On a Variational Method for Stiff Differential Equations Arising from Chemistry Kinetics

For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test its numerical performance through some experiments. The main advantage with respect to other implicit methods is that our approach has a global convergence. The other approaches need to ensure convergence of the iterative scheme used to approximate the associated nonlinear equations that appear for the implicitness. Notice that these iterative methods, for these nonlinear equations, have bounded basins of attraction

¿Mentimos a nuestros hijos cuando les decimos que 1+1 son 2?

La gente cree que la forma de contar sigue unas reglas predeterminadas por la aritmética convencional y que no existen ni pueden existir otro tipo de aritméticas. El objetivo de este artículo es mostrar la existencia de aritméticas distintas de la usual que dan respuesta a problemas reales en los que las reglas cotidianas entran en contradicciones o paradojas. En muchas ocasiones, tenemos que utilizar diferentes reglas para contar y esto es un signo de la existencia de distintas aritméticas. A estas aritméticas las llamaremos no diofantinas, en honor a Diophantus cuyas contribuciones a la aritmética clásica fueron fundamentales