Probabilist Set Inversion using Pseudo-Intervals Arithmetic

Abstract

In this paper, we present how to use an interval arithmetics framework based on free algebra construction, in order to build better defined inclusion function for interval semi-group and for its associated vector space. One introduces the ψ-algorithm, which performs set inversion of functions and exhibits some numerical examples developped with the python programming langage.

Linear Algebra and Differential Calculus in Pseudo-Intervals Vector Space

“…Moore [20,21,22,29,30] as the first mathematician who has proposed a rigorous framework for interval computations, the famous Archimedes from Syracuse (287-212 b.C) proposed 23 centuries before a two-sides bounding of π: 3 + Interval arithmetic, and interval analysis (IA) have been introduced as a computing framework which allows to perform analysis by computing mathematic bounds. Its extensions of the areas in applied and computational mathematics are important: non-linear problems, partial differential equations, inverse problems, global optimization and set inversion [1,13,14,20,21,22,30,32]. It finds a large place of applications in controllability, automation, robotics, embedded systems, biomedical, haptic interfaces, form optimization, analysis of architecture plans, etc.…”

“…Some approaches using boolean inclusion Scientific COnsulting for Research & Engineering, (SCORE), Kingersheim, Alsace, France. E-mail: kenoufi@s-core.fr tests, series or limited expansions of the natural function where the derivatives are computed at a certain point of the intervals, have been developed to circumvent this problem [1,13,14]. Nevertheless, those transfers from real functions to functions defined on intervals are not systematic and not given by a formal process.…”

“…This article reminds first in Section 2 the definition and characteristics of the intervals semigroup IR and the construction of its associated vector space IR, and how to get an associative and distributive arithmetic of intervals, called pseudo-intervals arithmetic, by embedding the vector space into a free algebra with dimension 4, 5, or 7 [1,5,13]. A systematic way to build inclusion functions, i.e.…”

“…In Section 3, one shows that IR can be endowed with a Banach space topology and allows differential calculus. Since a new algorithm for set inversion within pseudo-intervals framework has already been presented in previous papers [1,13], only examples of interval matrix eigenmode calculation, inversion, and minimization of interval functions are exhibited in each section. We have choosen python programming langage [28] to implement pseudo-intervals arithmetic.…”

“…In this paper one proposes to use a new approach of interval arithmetic, the so-called pseudointervals [1,5,13]. It uses a construction which is more canonical and based on the semi-group completion into the group, and it allows to build a Banach vector space.…”