An interval extension based on occurrence (original) (raw)
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An Interval Extension Based on Occurrence Grouping: Method and Properties
HAL (Le Centre pour la Communication Scientifique Directe), 2011
In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper (interval) image than the natural interval extension does. This paper presents a so-called "occurrence grouping" interval extension [ f ] og of a function f. When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f og that is monotonic w.r.t. two subsets x a and x b of the occurrences of x: f og is increasing w.r.t. x a and decreasing w.r.t. x b. [ f ] og is the interval extension by monotonicity of f og and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based overestimate of the image diameter of [ f ] og. Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.
An interval extension based on occurrence grouping
Computing, 2011
In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper image than the natural interval extension does. This paper presents a so-called "occurrence grouping" interval extension [ f ] og of a function f. When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f og that is monotonic w.r.t. two subsets x a and x b of the occurrences of x: f og is increasing w.r.t. x a and decreasing w.r.t. x b. [ f ] og is the interval extension by monotonicity of f og and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based overestimate of the image diameter of [ f ] og. Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.
Power and beauty of interval methods
Interval calculus is a relatively new branch of mathematics. Initially understood as a set of tools to assess the quality of numerical calculations (rigorous control of rounding errors), it became a discipline in its own rights today. Interval methods are usefull whenever we have to deal with uncertainties, which can be rigorously bounded. Fuzzy sets, rough sets and probability calculus can perform similar tasks, yet only the interval methods are able to (dis)prove, with mathematical rigor, the (non)existence of desired solution(s). Known are several problems, not presented here, which cannot be effectively solved by any other means. This paper presents basic notions and main ideas of interval calculus and two examples of useful algorithms.
Interval Computations: Introduction, Uses, and Resources
1996
Interval analysis is a broad Þeld in which rigorous mathematics is associated with with scientiÞc computing. A number,of researchers worldwide have produced a voluminous literature on the subject. This article introduces interval arithmetic and its interaction with established mathematical theory. The article provides pointers to traditional literature collections, as well as electronic resources. Some successful scientiÞc and engineering applications
A class of problems that can be solved using interval algorithms
2011
The paper discusses several theoretical and implementational problems of interval branch-and-bound methods. A trial to define a class of problems that can be solved with such methods is done. Features and variants of the method are presented. Useful data structures and shared-memory parallelization issues are considered.
Interval-Arithmetic-Oriented Interval Computing Technique for Global Optimization
Firstly, a brief survey of the existing works on comparing and ranking any two interval numbers on the real line is presented, and then pointing out the drawbacks of these definitions, a new approach is proposed in the context of decision maker’s (optimistic and pessimistic) point of view. Secondly, an interval technique is proposed to solve unconstrained multimodal optimization problems with continuous variables. In this proposed method, the search region is divided into two equal subregions successively and in each subregion, the lower and upper bounds of the objective function are computed with the help of interval arithmetic. Then, by comparing these two interval objective values and considering the subregion containing the better objective value, the global optimal value of the objective function or close to it is obtained. Finally, the proposed method is applied to solve several number of test problems of global optimization with lower as well as higher dimension and is compared with the existing methods with respect to the number of function evaluations.
arXiv (Cornell University), 2020
The interval numbers is the set of compact intervals of R with addition and multiplication operation, which are very useful for solving calculations where there are intervals of error or uncertainty, however, it lacks an algebraic structure with an inverse element, both additive and multiplicative This fundamental disadvantage results in overestimation of solutions in an interval equation or also overestimation of the image of a function over square regions. In this article we will present an original solution, through a morphism that preserves both the addiction and the multiplication between the space of the interval numbers to the space of square diagonal matrices.
Interval arithmetic: From principles to implementation
Journal of the ACM, 2001
We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication, and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard's specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems, we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally, we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed.
An analysis of slow convergence in interval propagation
2007
When performing interval propagation on integer variables with a large range, slow-convergence phenomena are often observed: it becomes difficult to reach the fixpoint of the propagation. This problem is practically important, as it hinders the use of propagation techniques for problems with large numerical ranges, and notably problems arising in program verification.