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Automatic differentiation of uncertainties: an interval computational differentiation for first and higher
Hend Dawood1, Nefertiti Megahed1
1Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
Peerj. Computer Science
|June 22, 2023
Summary
This study introduces a systematic theory for dyadic interval differentiation numbers, enabling automatic differentiation under uncertainty. This approach combines interval mathematics and algorithmic differentiation for enhanced computational reliability.
Area of Science:
- Computational Mathematics
- Interval Analysis
Background:
- Acquiring reliable knowledge under uncertainty is crucial in modern science.
- Interval mathematics effectively handles uncertainty and imprecision.
- Algorithmic differentiation offers advantages over numeric and symbolic methods.
Purpose of the Study:
- To develop a systematic theory of dyadic interval differentiation numbers.
- To address first and higher-order automatic derivatives under uncertainty.
- To extend real differentiation arithmetic using interval mathematics and algorithmic differentiation.
Main Methods:
- Axiomatization of a differential interval algebra.
- Definition of interval extensions for real functions and interval functions.
- Development of an axiomatic theory for interval differentiation arithmetic.
- Proofs of categoricity and consistency for the proposed theory.
Main Results:
- The developed structure forms a multiplicatively non-associative S-semiring.
- Multiplication within this structure is shown to be subalternative and flexible.
- Computational realization of interval automatic differentiation is demonstrated.
Conclusions:
- The proposed theory provides a robust framework for automatic differentiation under uncertainty.
- This method enhances the power and applicability of differentiation arithmetic.
- The study illustrates practical applications through examples of interval functions and real function families.
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