A Consistent and Categorical Axiomatization of Differentiation Arithmetic Applicable to First and Higher Order Derivatives

Dawood, Hend; Megahed, Nefertiti

doi:10.5281/zenodo.3479546

Cited by 1 publication

(15 citation statements)

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“…The expressions ‘automatic differentiation’, ‘auto-differentiation’, ‘computational differentiation’, ‘algorithmic differentiation’, and ‘differentiation arithmetic’ are in the just acceptation synonyms. They refer to a subtle and central tool to automatize the simultaneous computation of the numerical values of arbitrarily complex functions and their derivatives with no need for the symbolic representation of the derivative, only the function rule or an algorithm thereof is required ( Dawood & Megahed, 2019 ). Auto-differentiation is thus neither numeric nor symbolic , nor is it a combination of both.…”

Section: Introductionmentioning

confidence: 99%

“…Auto-differentiation is thus neither numeric nor symbolic , nor is it a combination of both. It is also preferable to ordinary numerical methods: In contrast to the more traditional numerical methods based on finite differences, auto-differentiation is ‘in theory’ exact, and in comparison to symbolic algorithms, it is computationally inexpensive ( Dawood & Megahed, 2019 ). The literature on algorithmic differentiation is immense and very diversified.…”

Section: Introductionmentioning

confidence: 99%

“…For further reading, (see, e.g. , Corliss & Rall, 1996 ; Dawood, 2014 , Dawood & Megahed, 2019 ; Griewank & Walther, 2008 ; Moore, 1979 , Neidinger, 2010 , and Mitchell, 1991 ). Currently, for its efficiency and accuracy in computing first and higher order derivatives, auto-differentiation is a celebrated technique with diverse applications in scientific computing and mathematics.…”

Section: Introductionmentioning

confidence: 99%

“…, Rump, 1999 , Chevillard, Joldes & Lauter, 2010 , and Dawood, 2020 ). In practice, there are two types ( modes ) of algorithmic differentiation: a forward-type and a reversed-type ( Dawood & Megahed, 2019 ). Presently, the two types are highly correlated and complementary and both have a wide variety of applications in, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…, non-linear optimization, sensitivity analysis, robotics, machine learning, computer graphics, and computer vision (see, e.g. , Abdelhafez, Schuster & Koch, 2019 ; Dawood, 2014 , Dawood & Megahed, 2019 ; Fries, 2019 , Sommer, Pradalier & Furgale, 2016 , and Tingelstad & Egeland, 2017 ).…”

Section: Introductionmentioning

confidence: 99%

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Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation

Dawood

Megahed

2023

PeerJ Computer Science

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Acquiring reliable knowledge amidst uncertainty is a topical issue of modern science. Interval mathematics has proved to be of central importance in coping with uncertainty and imprecision. Algorithmic differentiation, being superior to both numeric and symbolic differentiation, is nowadays one of the most celebrated techniques in the field of computational mathematics. In this connexion, laying out a concrete theory of interval differentiation arithmetic, combining subtlety of ordinary algorithmic differentiation with power and reliability of interval mathematics, can extend real differentiation arithmetic so markedly both in method and objective, and can so far surpass it in power as well as applicability. This article is intended to lay out a systematic theory of dyadic interval differentiation numbers that wholly addresses first and higher order automatic derivatives under uncertainty. We begin by axiomatizing a differential interval algebra and then we present the notion of an interval extension of a family of real functions, together with some analytic notions of interval functions. Next, we put forward an axiomatic theory of interval differentiation arithmetic, as a two-sorted extension of the theory of a differential interval algebra, and provide the proofs for its categoricity and consistency. Thereupon, we investigate the ensuing structure and show that it constitutes a multiplicatively non-associative S-semiring in which multiplication is subalternative and flexible. Finally, we show how to computationally realize interval automatic differentiation. Many examples are given, illustrating automatic differentiation of interval functions and families of real functions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%