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Applications of fractional calculus in scientific computing

Grant number: 18/06037-7
Support type:Scholarships in Brazil - Scientific Initiation
Effective date (Start): July 01, 2018
Effective date (End): June 30, 2019
Field of knowledge:Physical Sciences and Mathematics - Computer Science - Computational Mathematics
Principal researcher:Eliana Contharteze Grigoletto
Grantee:Samuel Ferreira Batista
Home Institution: Faculdade de Ciências Agronômicas (FCA). Universidade Estadual Paulista (UNESP). Campus de Botucatu. Botucatu , SP, Brazil

Abstract

We often have to resort to numerical optimization because the most parameter estimation problems in machine learning cannot be solved in closed form. In this project, we intend provide a differentiated approach to describe some, of the common optimization techniques used in machine learning, through the inclusion of fractional derivative, an important tool of fractional calculus, which has shown to be an very promising area of research for modeling of some physical systems and also in the field of neural networks. We propose to analyze the convergence ability of the fractional order gradient method, using different fractional derivatives, and compare the results with the convergence ability of the classical gradient method. Furthermore, we can investigate the efficiency of the generalized Taylor's formula, whose applications include approximation of functions and solutions of fractional differential equations, we can yet discuss the fractional differentiability of a class of functions associated with eigenvalues and eigenvectors of symmetric matrices, and propose algorithms, with the inclusion of fractional derivatives, to solve, approximately, the Lp-norm minimization problems for both super-Gaussian (1

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