scholarly journals Error bounds in stochastic-geometric normal approximation

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Mathew Penrose ◽  
Tom Rosoman
Keyword(s):  
Poisson Process ◽  
Error Bounds ◽  
Normal Approximation ◽  
Approximation Error ◽  
Geometric Graphs ◽  
International Audience

International audience We provide normal approximation error bounds for sums of the form $\sum_x \xi_x$, indexed by the points $x$ of a Poisson process (not necessarily homogeneous) in the unit $d$-cube, with each term $\xi_x$ determined by the configuration of Poisson points near to $x$ in some sense. We consider geometric graphs and coverage processes as examples of our general results.

scholarly journals Nonnormal Small Jump Approximation of Infinitely Divisible Distributions

2014 ◽  
Vol 46 (04) ◽  
pp. 963-984
Author(s):  
Zhiyi Chi
Keyword(s):  
Computational Complexity ◽  
Total Variation ◽  
Random Sampling ◽  
Normal Approximation ◽  
Approximation Error ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Normal Distributions ◽  
Jump Size ◽  
Variation Distance

We study a type of nonnormal small jump approximation of infinitely divisible distributions. By incorporating compound Poisson, gamma, and normal distributions, the approximation has a higher order of cumulant matching than its normal counterpart, and, hence, in many cases a higher rate of approximation error decay as the cutoff for the jump size tends to 0. The parameters of the approximation are easy to fix, and its random sampling has the same order of computational complexity as the normal approximation. An error bound of the approximation in terms of the total variation distance is derived. Simulations empirically show that the nonnormal approximation can have a significantly smaller error than its normal counterpart.

scholarly journals Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Oswin Aichholzer ◽  
Sergio Cabello ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Thomas Hackl ◽  
...  
Keyword(s):  
Extremal Problem ◽  
General Position ◽  
Geometric Graph ◽  
Perfect Matchings ◽  
Geometric Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Edge Removal ◽  
International Audience ◽  
Point Set

Graphs and Algorithms International audience A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

scholarly journals R-adaptation par l'estimateur d'erreur hiérarchique

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Abdellah Alla ◽  
Zoubida Mghazli ◽  
Michel Fortin ◽  
Frédéric Hecht
Keyword(s):  
Numerical Test ◽  
Approximation Error ◽  
Finite Element Discretization ◽  
Optimal Position ◽  
Element Discretization ◽  
Adaptation Procedure ◽  
International Audience ◽  
Éléments Finis ◽  
Node Displacement ◽  
Nous Utilisons

International audience The aim of this work is to devise a method to determine the optimal position of the nodes in a finite element discretization for a boundary value problem. The node displacement procedure (also called R-adaptation) is a crucial step in a global mesh adaptation procedure. In the present approch, we determine the nodal position by minimizing the approximation error. This error is evaluated using a hierarchical estimator. A numerical test is presented. L'objectif de ce travail est de déterminer la meilleure position des noeuds d'un maillage, utilisé lors de la discrétisation d'un problème aux limites par la méthode des éléments finis. La procédure de déplacement des noeuds (appelé aussi R-adaptation) est une étape importante dans la stratégie globale d'adaptation de maillage. La position optimale des noeuds est déterminée en minimisant l'erreur d'approximation. Pour évaluer cette erreur nous utilisons l'estimateur d'erreur hiérarchique. Un test numérique est présenté.

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