A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function... Given a nonempty set of functions $$\begin{gathered} F = \{ f:[a,b] \to \mathbb{R}: \hfill \\ \hfill \\ {\text{ }}f(x_i ) \in w_i ,i = 0, \ldots ,n,{\text{ and}} \hfill \\ \hfill \\ {\text{ }}f(x) - f(y) \in d_i (x - y){\text{ }}\forall x,y \in [x_{i - 1} ,x_i ],{\text{ }}i = 1, \ldots ,n\} , \hfill \\ \end{gathered}$$ where a = x 0 < ... < x n = b are known nodes and w i , i = 0,...,n, d i , i = 1,..., n, known compact intervals, the main aim of the present paper is to show that the functions $$\underline f :x \mapsto \min \{ f(x):f \in F\} ,{\text{ }}x \in [a,b],$$ and $$\overline f :x \mapsto \max \{ f(x):f \in F\} ,{\text{ }}x \in [a,b],$$ exist, are in F, and are easily computable. This is achieved essentially by giving simple formulas for computing two vectors $$\tilde l,\tilde u \in \mathbb{R}^{n + 1}$$ with the properties $$\begin{gathered} \bullet {\text{ }}\tilde l \leqslant \tilde u{\text{ implies}} \hfill \\ \hfill \\ {\text{ }}\tilde l,\tilde u \in T{\text{ : = \{ }}\xi {\text{ = (}}\xi _0 , \ldots ,\xi _n )^T \in \mathbb{R}^{n + 1} : \hfill \\ \hfill \\ {\text{ }}\xi _i \in w_i ,{\text{ }}i = 0, \ldots ,n,{\text{ and}} \hfill \\ \end{gathered}$$ $$\tilde l,\tilde u$$ ] is the interval hull of (the tolerance polyhedron) T; • $${\tilde l}$$ ≤ ū iff T ≠ 0 iff F ≠ 0. $$\underline f ,\overline f$$ , can serve for solving the following problem: Assume that μ is a monotonically increasing functional on the set of Lipschitz-continuous functions f : [a,b] → R (e.g. μ(f) = ∫ a b f(x) dx or μ(f) = min f([a,b]) or μ(f) = max f([a,b])), and that the available information about a function g : [a,b] → R is "g ∈ F," then the problem is to find the best possible interval inclusion of μ(g). Obviously, this inclusion is given by the interval [μ( $$\underline f$$ ,μ( $$\overline f$$ )]. Complete formulas for computing this interval are given for the case μ(f) = ∫ a b f(x) dx. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 1999 by Kluwer Academic Publishers
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1009928406426
Publisher site
See Article on Publisher Site

Abstract

Given a nonempty set of functions $$\begin{gathered} F = \{ f:[a,b] \to \mathbb{R}: \hfill \\ \hfill \\ {\text{ }}f(x_i ) \in w_i ,i = 0, \ldots ,n,{\text{ and}} \hfill \\ \hfill \\ {\text{ }}f(x) - f(y) \in d_i (x - y){\text{ }}\forall x,y \in [x_{i - 1} ,x_i ],{\text{ }}i = 1, \ldots ,n\} , \hfill \\ \end{gathered}$$ where a = x 0 < ... < x n = b are known nodes and w i , i = 0,...,n, d i , i = 1,..., n, known compact intervals, the main aim of the present paper is to show that the functions $$\underline f :x \mapsto \min \{ f(x):f \in F\} ,{\text{ }}x \in [a,b],$$ and $$\overline f :x \mapsto \max \{ f(x):f \in F\} ,{\text{ }}x \in [a,b],$$ exist, are in F, and are easily computable. This is achieved essentially by giving simple formulas for computing two vectors $$\tilde l,\tilde u \in \mathbb{R}^{n + 1}$$ with the properties $$\begin{gathered} \bullet {\text{ }}\tilde l \leqslant \tilde u{\text{ implies}} \hfill \\ \hfill \\ {\text{ }}\tilde l,\tilde u \in T{\text{ : = \{ }}\xi {\text{ = (}}\xi _0 , \ldots ,\xi _n )^T \in \mathbb{R}^{n + 1} : \hfill \\ \hfill \\ {\text{ }}\xi _i \in w_i ,{\text{ }}i = 0, \ldots ,n,{\text{ and}} \hfill \\ \end{gathered}$$ $$\tilde l,\tilde u$$ ] is the interval hull of (the tolerance polyhedron) T; • $${\tilde l}$$ ≤ ū iff T ≠ 0 iff F ≠ 0. $$\underline f ,\overline f$$ , can serve for solving the following problem: Assume that μ is a monotonically increasing functional on the set of Lipschitz-continuous functions f : [a,b] → R (e.g. μ(f) = ∫ a b f(x) dx or μ(f) = min f([a,b]) or μ(f) = max f([a,b])), and that the available information about a function g : [a,b] → R is "g ∈ F," then the problem is to find the best possible interval inclusion of μ(g). Obviously, this inclusion is given by the interval [μ( $$\underline f$$ ,μ( $$\overline f$$ )]. Complete formulas for computing this interval are given for the case μ(f) = ∫ a b f(x) dx.

Journal

Reliable ComputingSpringer Journals

Published: Oct 22, 2004

References

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