Nature-inspired algorithms for solving some hard numerical problems
Autor(a) principal: | |
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Data de Publicação: | 2020 |
Tipo de documento: | Dissertação |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.13/3010 |
Resumo: | Optimisation is a branch of mathematics that was developed to find the optimal solutions, among all the possible ones, for a given problem. Applications of optimisation techniques are currently employed in engineering, computing, and industrial problems. Therefore, optimisation is a very active research area, leading to the publication of a large number of methods to solve specific problems to its optimality. This dissertation focuses on the adaptation of two nature inspired algorithms that, based on optimisation techniques, are able to compute approximations for zeros of polynomials and roots of non-linear equations and systems of non-linear equations. Although many iterative methods for finding all the roots of a given function already exist, they usually require: (a) repeated deflations, that can lead to very inaccurate results due to the problem of accumulating rounding errors, (b) good initial approximations to the roots for the algorithm converge, or (c) the computation of first or second order derivatives, which besides being computationally intensive, it is not always possible. The drawbacks previously mentioned served as motivation for the use of Particle Swarm Optimisation (PSO) and Artificial Neural Networks (ANNs) for root-finding, since they are known, respectively, for their ability to explore high-dimensional spaces (not requiring good initial approximations) and for their capability to model complex problems. Besides that, both methods do not need repeated deflations, nor derivative information. The algorithms were described throughout this document and tested using a test suite of hard numerical problems in science and engineering. Results, in turn, were compared with several results available on the literature and with the well-known Durand–Kerner method, depicting that both algorithms are effective to solve the numerical problems considered. |
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Nature-inspired algorithms for solving some hard numerical problemsOtimizaçãoOtimização por enxame de partículasRedes neurais artificiaisRaízesPolinómiosEquações não linearesOptimisationParticle swarm optimisationArtificial neural networksRootsPolynomialsNon-linear equationsMathematics, Statistics and Applications.Faculdade de Ciências Exatas e da EngenhariaDomínio/Área Científica::Ciências Naturais::MatemáticasOptimisation is a branch of mathematics that was developed to find the optimal solutions, among all the possible ones, for a given problem. Applications of optimisation techniques are currently employed in engineering, computing, and industrial problems. Therefore, optimisation is a very active research area, leading to the publication of a large number of methods to solve specific problems to its optimality. This dissertation focuses on the adaptation of two nature inspired algorithms that, based on optimisation techniques, are able to compute approximations for zeros of polynomials and roots of non-linear equations and systems of non-linear equations. Although many iterative methods for finding all the roots of a given function already exist, they usually require: (a) repeated deflations, that can lead to very inaccurate results due to the problem of accumulating rounding errors, (b) good initial approximations to the roots for the algorithm converge, or (c) the computation of first or second order derivatives, which besides being computationally intensive, it is not always possible. The drawbacks previously mentioned served as motivation for the use of Particle Swarm Optimisation (PSO) and Artificial Neural Networks (ANNs) for root-finding, since they are known, respectively, for their ability to explore high-dimensional spaces (not requiring good initial approximations) and for their capability to model complex problems. Besides that, both methods do not need repeated deflations, nor derivative information. The algorithms were described throughout this document and tested using a test suite of hard numerical problems in science and engineering. Results, in turn, were compared with several results available on the literature and with the well-known Durand–Kerner method, depicting that both algorithms are effective to solve the numerical problems considered.A Optimização é um ramo da matemática desenvolvido para encontrar as soluções óptimas, de entre todas as possíveis, para um determinado problema. Actualmente, são várias as técnicas de optimização aplicadas a problemas de engenharia, de informática e da indústria. Dada a grande panóplia de aplicações, existem inúmeros trabalhos publicados que propõem métodos para resolver, de forma óptima, problemas específicos. Esta dissertação foca-se na adaptação de dois algoritmos inspirados na natureza que, tendo como base técnicas de optimização, são capazes de calcular aproximações para zeros de polinómios e raízes de equações não lineares e sistemas de equações não lineares. Embora já existam muitos métodos iterativos para encontrar todas as raízes ou zeros de uma função, eles usualmente exigem: (a) deflações repetidas, que podem levar a resultados muito inexactos, devido ao problema da acumulação de erros de arredondamento a cada iteração; (b) boas aproximações iniciais para as raízes para o algoritmo convergir, ou (c) o cálculo de derivadas de primeira ou de segunda ordem que, além de ser computacionalmente intensivo, para muitas funções é impossível de se calcular. Estas desvantagens motivaram o uso da Optimização por Enxame de Partículas (PSO) e de Redes Neurais Artificiais (RNAs) para o cálculo de raízes. Estas técnicas são conhecidas, respectivamente, pela sua capacidade de explorar espaços de dimensão superior (não exigindo boas aproximações iniciais) e pela sua capacidade de modelar problemas complexos. Além disto, tais técnicas não necessitam de deflações repetidas, nem do cálculo de derivadas. Ao longo deste documento, os algoritmos são descritos e testados, usando um conjunto de problemas numéricos com aplicações nas ciências e na engenharia. Os resultados foram comparados com outros disponíveis na literatura e com o método de Durand–Kerner, e sugerem que ambos os algoritmos são capazes de resolver os problemas numéricos considerados.Lopes, Luiz Carlos GuerreiroDias, Fernando Manuel Rosmaninho Morgado FerrãoDigitUMaFreitas, Diogo Nuno Teixeira2021-10-02T00:30:15Z2020-10-022020-10-02T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttp://hdl.handle.net/10400.13/3010TID:202538583enginfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-06-11T03:30:12Zoai:digituma.uma.pt:10400.13/3010Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T15:05:51.224314Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Nature-inspired algorithms for solving some hard numerical problems |
title |
Nature-inspired algorithms for solving some hard numerical problems |
spellingShingle |
Nature-inspired algorithms for solving some hard numerical problems Freitas, Diogo Nuno Teixeira Otimização Otimização por enxame de partículas Redes neurais artificiais Raízes Polinómios Equações não lineares Optimisation Particle swarm optimisation Artificial neural networks Roots Polynomials Non-linear equations Mathematics, Statistics and Applications . Faculdade de Ciências Exatas e da Engenharia Domínio/Área Científica::Ciências Naturais::Matemáticas |
title_short |
Nature-inspired algorithms for solving some hard numerical problems |
title_full |
Nature-inspired algorithms for solving some hard numerical problems |
title_fullStr |
Nature-inspired algorithms for solving some hard numerical problems |
title_full_unstemmed |
Nature-inspired algorithms for solving some hard numerical problems |
title_sort |
Nature-inspired algorithms for solving some hard numerical problems |
author |
Freitas, Diogo Nuno Teixeira |
author_facet |
Freitas, Diogo Nuno Teixeira |
author_role |
author |
dc.contributor.none.fl_str_mv |
Lopes, Luiz Carlos Guerreiro Dias, Fernando Manuel Rosmaninho Morgado Ferrão DigitUMa |
dc.contributor.author.fl_str_mv |
Freitas, Diogo Nuno Teixeira |
dc.subject.por.fl_str_mv |
Otimização Otimização por enxame de partículas Redes neurais artificiais Raízes Polinómios Equações não lineares Optimisation Particle swarm optimisation Artificial neural networks Roots Polynomials Non-linear equations Mathematics, Statistics and Applications . Faculdade de Ciências Exatas e da Engenharia Domínio/Área Científica::Ciências Naturais::Matemáticas |
topic |
Otimização Otimização por enxame de partículas Redes neurais artificiais Raízes Polinómios Equações não lineares Optimisation Particle swarm optimisation Artificial neural networks Roots Polynomials Non-linear equations Mathematics, Statistics and Applications . Faculdade de Ciências Exatas e da Engenharia Domínio/Área Científica::Ciências Naturais::Matemáticas |
description |
Optimisation is a branch of mathematics that was developed to find the optimal solutions, among all the possible ones, for a given problem. Applications of optimisation techniques are currently employed in engineering, computing, and industrial problems. Therefore, optimisation is a very active research area, leading to the publication of a large number of methods to solve specific problems to its optimality. This dissertation focuses on the adaptation of two nature inspired algorithms that, based on optimisation techniques, are able to compute approximations for zeros of polynomials and roots of non-linear equations and systems of non-linear equations. Although many iterative methods for finding all the roots of a given function already exist, they usually require: (a) repeated deflations, that can lead to very inaccurate results due to the problem of accumulating rounding errors, (b) good initial approximations to the roots for the algorithm converge, or (c) the computation of first or second order derivatives, which besides being computationally intensive, it is not always possible. The drawbacks previously mentioned served as motivation for the use of Particle Swarm Optimisation (PSO) and Artificial Neural Networks (ANNs) for root-finding, since they are known, respectively, for their ability to explore high-dimensional spaces (not requiring good initial approximations) and for their capability to model complex problems. Besides that, both methods do not need repeated deflations, nor derivative information. The algorithms were described throughout this document and tested using a test suite of hard numerical problems in science and engineering. Results, in turn, were compared with several results available on the literature and with the well-known Durand–Kerner method, depicting that both algorithms are effective to solve the numerical problems considered. |
publishDate |
2020 |
dc.date.none.fl_str_mv |
2020-10-02 2020-10-02T00:00:00Z 2021-10-02T00:30:15Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10400.13/3010 TID:202538583 |
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http://hdl.handle.net/10400.13/3010 |
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TID:202538583 |
dc.language.iso.fl_str_mv |
eng |
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eng |
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info:eu-repo/semantics/openAccess |
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openAccess |
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application/pdf |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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