Interval extensions and interval iterations

Computing ◽  
1980 ◽  
Vol 24 (2-3) ◽  
pp. 119-129 ◽  
Author(s):  
R. Krawczyk
Keyword(s):  
Interval Extensions ◽  
Interval Iterations

A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

2016 ◽  
Vol 26 (13) ◽  
pp. 1650226 ◽  
Author(s):  
Eduardo M. A. M. Mendes ◽  
Erivelton G. Nepomuceno
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Equations Of Motion ◽  
Original System ◽  
Glass System ◽  
Largest Lyapunov Exponent ◽  
Simple Method ◽  
Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Hénon, Lorenz and Rössler equations and the Mackey–Glass system.

Interval iterations for including a set of solutions

Computing ◽  
1984 ◽  
Vol 32 (1) ◽  
pp. 13-31 ◽  
Author(s):  
R. Krawczyk
Keyword(s):  
Interval Iterations

scholarly journals Two stochastic filters and their interval extensions

2016 ◽  
Vol 49 (5) ◽  
pp. 49-54
Author(s):  
Tran Tuan Anh ◽  
Françoise Le Gall ◽  
Carine Jauberthie ◽  
Louise Travé-Massuyès
Keyword(s):  
Interval Extensions

Application of interval iterations to the entrainment problem in respiratory physiology

2009 ◽  
Vol 367 (1908) ◽  
pp. 4717-4739 ◽  
Author(s):  
Jacques Demongeot ◽  
Jules Waku
Keyword(s):  
Population Dynamics ◽  
Limit Cycles ◽  
Numerical Results ◽  
Respiratory Physiology ◽  
New Type ◽  
Classical Point ◽  
Interval Iterations ◽  
Asymptotic Behaviours

We present here some theoretical and numerical results about interval iterations. We consider first an application of the interval iterations theory to the problem of entrainment in respiratory physiology for which the classical point iterations theory fails. Then, after a brief review of some of the main aspects of point iterations, we explain what is meant by the term ‘interval iterations’. It consists essentially in replacing in the point iterations the function to iterate by a set-valued map. We present both theoretical and numerical aspects of this new type of iterations and we observe the dynamical behaviours encountered, such as fixed intervals and interval limit cycles. The comparison between point and interval iterations is carried out with respect to a parameter ε , which determines the thickness of a neighbourhood around the function to iterate. We will finally focus our attention on the Verhulst and Ricker functions largely used in population dynamics, which exhibit various asymptotic behaviours.

Interval Extensions of Signed Distance Functions: iSDF-reps and Reliable Membership Classification

2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Duane Storti ◽  
Chris Finley ◽  
Mark Ganter
Keyword(s):  
Distance Function ◽  
Distance Functions ◽  
Upper And Lower Bounds ◽  
Uniform Grid ◽  
Signed Distance Function ◽  
Computationally Efficient ◽  
Signed Distance ◽  
Interval Extension ◽  
Interval Extensions ◽  
Signed Distance Functions

This paper considers the problem of inferring the geometry of an object from values of the signed distance sampled on a uniform grid. The problem is motivated by the desire to effectively and efficiently model objects obtained by 3D imaging technology such as magnetic resonance, computed tomography, and positron emission tomography. Techniques recently developed for automated segmentation convert intensity to signed distance, and the voxel structure imposes the uniform sampling grid. The specification of the signed distance function (SDF) throughout the ambient space would provide an implicit and function-based representation (f-rep) model that uniquely specifies the object, and we refer to this particular f-rep as the signed distance function representation (SDF-rep). However, a set of uniformly sampled signed distance values may uniquely determine neither the distance function nor the shape of the object. Here, we employ essential properties of the signed distance to construct the upper and lower bounds on the allowed variation in signed distance, which combine to produce interval-valued extensions of the signed distance function. We employ an interval extension of the signed distance function as an interval SDF-rep that defines the range of object geometries that are consistent with the sampled SDF data. The particular interval extensions considered include a tight global extension and more computationally efficient local extensions that provide useful criteria for root exclusion/isolation. To illustrate a useful application of the interval bounds, we present a reliable approach to top-down octree membership classification for uniform samplings of signed distance functions.

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