Interval unions

Schichl, Hermann; Domes, Ferenc; Montanher, Tiago; Kofler, Kevin

doi:10.1007/s10543-016-0632-y

Cited by 9 publications

(5 citation statements)

References 15 publications

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“…We propose a gap filling strategy based on the ideas described by [24]. The resulting method is given by 3.…”

Section: Discussionmentioning

confidence: 99%

“…A similar phenomenon was already observed by Hyvönen [11] for the propagation of discontinuous intervals; however, the remedy proposed there-simply to take the interval hull-unnecessarily discards useful information. As a more flexible remedy, [24] introduced the notion of gap filling. In this section we describe a gap filling strategy that (among several strategies tried) proved useful for the interval union Gauss-Seidel procedure.…”

Section: Gap Fillingmentioning

confidence: 99%

“…Treating this using complements of intervals (see e.g, [23]) only postpones the problem a little, while interval union arithmetic, introduced by [24] as arithmetic on finite ordered sets of disjoint closed, possibly unbounded intervals, allow a mathematically and computationally natural approach to this problem. Indeed, the collection of interval unions (treated as closed sets in the obvious way) is closed under set-theoretic addition, subtraction, multiplication, division (after adding end points in case of an unbounded divisor), and all continuous elementary operations.…”

Section: Introductionmentioning

confidence: 99%

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Using interval unions to solve linear systems of equations with uncertainties

et al. 2017

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An interval union is a finite set of closed and disjoint intervals. In this paper we introduce the interval union Gauss-Seidel procedure to rigorously enclose the solution set of linear systems with uncertainties given by intervals or interval unions. We also present the interval union midpoint and Gauss-Jordan preconditioners. The Gauss-Jordan preconditioner is used in a mixed strategy to improve the quality and efficiency of the algorithm. Numerical experiments on interval linear systems generated at random show the capabilities of our approach.

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“…We propose a gap filling strategy based on the ideas described by [24]. The resulting method is given by 3.…”

Section: Discussionmentioning

confidence: 99%

Section: Gap Fillingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Using interval unions to solve linear systems of equations with uncertainties

et al. 2017

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“…Schichl et al [30] introduced an arithmetic of unions of intervals. Such domains can be propagated through the constraints, which leads to more precise domains.…”

Section: Holes and Boundsmentioning

confidence: 99%

Interval Tests and Contractors Based on Optimality Conditions for Bound-Constrained Global Optimization

Granvilliers

2020

Int. J. Artif. Intell. Tools

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We study the problem of finding the global optimum of a nonlinear real function over an interval box by means of complete search techniques, namely interval branch-and-bound algorithms. Such an algorithm typically generates a tree of boxes from the initial box by alternating branching steps and contraction steps in order to remove non optimal sub-boxes. In this paper, we introduce a new contraction method that is designed to handle the boundary of the initial box where a minimizer may not be a stationary point. This method exploits the first-order optimality conditions and we show that it subsumes the classical monotonicity test based on interval arithmetic. A new branch-and-bound algorithm has been implemented in the interval solver Realpaver. An extensive experimental study based on a set of standard benchmarks is presented.

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“…However, thus far, there is no specific SM for DIs that directly assesses their similarity. Instead, DIs are commonly converted into their continuous form (CIs) using some common approaches like interval addition [11], interval union [12], or a 'convexify' function [13] [14] and then the respective CI SM is applied to compute the similarity. However, the 'DI to CI' conversion involves the loss of discontinuity information of the DIs, changing the original meaning of the data, and thus affecting the accuracy of the similarity of the DIs.…”

Section: Introductionmentioning

confidence: 99%

Measuring Similarity Between Discontinuous Intervals - Challenges and Solutions

Kabir

Wagner

Havens

et al. 2019

2019 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

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Discontinuous intervals (DIs) arise in a wide range of contexts, from real world data capture of human opinion to α-cuts of non-convex fuzzy sets. Commonly, for assessing the similarity of DIs, the latter are converted into their continuous form, followed by the application of a continuous interval (CI) compatible similarity measure. While this conversion is efficient, it involves the loss of discontinuity information and thus limits the accuracy of similarity results. Further, most similarity measures including the most popular ones, such as Jaccard and Dice, suffer from aliasing, that is, they are liable to return the same similarity for very different pairs of CIs. To address both of these challenges, this paper proposes a generalized approach for calculating the similarity of DIs which leverages the recently introduced bidirectional subsethood based similarity measure (which avoids aliasing) while accounting for all pairs of the continuous subintervals within the DIs to be compared. We provide detail of the proposed approach and demonstrate its behaviour when applying bidirectional subsethood, Jaccard and Dice as similarity measures, using different pairs of synthetic DIs. The experimental results show that the similarity outputs of the new generalized approach follow intuition for all three similarity measures; however, it is only the proposed integration with the bidirectional subsethood similarity measure which also avoids aliasing for DIs.

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Interval unions

Cited by 9 publications

References 15 publications

Using interval unions to solve linear systems of equations with uncertainties

Using interval unions to solve linear systems of equations with uncertainties

Interval Tests and Contractors Based on Optimality Conditions for Bound-Constrained Global Optimization

Measuring Similarity Between Discontinuous Intervals - Challenges and Solutions

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