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Rydberg, David. "GPU Predictor-Corrector Interior Point Method for Large-Scale Linear Programming." Thesis, KTH, Numerisk analys, NA, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-168672.
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This master’s thesis concerns the implementation of a GPUaccelerated version of Mehrotra’s predictor-corrector interior point algorithm for large-scale linear programming (LP). The implementations are tested on LP problems arising in the financial industry, where there is high demand for faster LP solvers. The algorithm was implemented in C++, MATLAB and CUDA, using double precision for numerical stability. A performance comparison showed that the algorithm can be accelerated from 2x to 6x using an Nvidia GTX Titan Black GPU compared to using only an Intel Xeon E5-2630v2 CPU. The amount of memory on the GPU restricts the size of problems that can be solved, but all tested problems that are small enough to fit on the GPU could be accelerated.Detta masterexamensarbete behandlar implementeringen av en grafikkortsaccelererad inrepunktsmetod av predictor-corrector-typ för storskalig linjärprogrammering (LP). Implementeringarna testas på LP-problem som uppkommer i finansbranschen, där det finns ett stort behov av allt snabbare LP-lösare. Algoritmen implementeras i C++, MATLAB och CUDA, och dubbelprecision används för numerisk stabilitet. En prestandajämförelse visade att algoritmen kan accelereras 2x till 6x genom att använda ett Nvidia GTX Titan Black jämfört med att bara använda en Intel Xeon E5-2630v2. Mängden minne på grafikkortet begränsar problemstorleken, men alla testade problem som får plats i grafikkortsminnet kunde accelereras.
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Haasler, Isabel. "A predictor corrector method for a Finite ElementMethod for the variable density Navier-Stokes equations." Thesis, KTH, Numerisk analys, NA, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-214837.
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Constructing Finite Element Methods for the incompressible Navier-Stokes equations that are computationally feasible requires the use of pressure segregation methods such as the predictor corrector scheme. From an algebraic point of view these methods correspond to a preconditioned Richardson iteration on the pressure Schur complement. In this work the theory is extended to a variable density model. We suggest a modified preconditioner and investigate its performance experimentally.Konstruktion av Finita Elementmetoder för de inkompressibla Navier-Stokes ekvationer som är beräkningsmässigt genomförbara kräver användning av trycksegregeringsmetoder såsom prediktorkorrigeringsschemat. Från en algebraisk synvinkel motsvarar dessa metoder en förkonditionerad Richardsoniteration på tryck-Schur-komplementet. I detta arbete utvidgas teorin till en variabel-densitet-modell. Vi föreslår en modifierad preconditioner och undersöker prestanda experimentellt.
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Carvalho, Lilian Milena Ramos. ""Métodos de pontos interiores aplicados ao pré-despacho de um sistema hidroelétrico usando o princípio de mínimo esforço - comparação com o modelo de fluxo em redes"." Universidade de São Paulo, 2005. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-06072006-153856/.
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Neste trabalho, os métodos de pontos interiores primal-dual e preditor corretor são estudados e desenvolvidos para o problema de minimização de custos na geração e perdas na transmissão do pré-despacho DC (fluxo de carga em corrente contínua) de um sistema de potência hidroelétrico, com base no modelo de fluxo em redes e no princípio do mínimo esforço. A estrutura matricial, resultante da simplificação do problema proposto pela inclusão do princípio do mínimo esforço, é estudada visando implementações eficientes.In this work, the primal-dual and predictor corrector interior points methods are studied and developed for the predispatch DC problem that minimizes generation and transmission losses on hydroelectric power systems, on the basis of the network flow model and the minimum effort principle. The matrix structure, resulting of the simplification of the problem considered by inclusion of the minimum effort principle, is studied aiming efficient implementations. A disturbed primal-dual method is considered on the basis of a heuristic definition that determine the choice of the disturbance parameter. This method showed to be efficient in practice and converged in fewer iterations when compare with an existing implementation of the network flow model.
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Shirabayashi, Wesley Vagner Ines. "Estrategias de segunda ordem para problemas de complementaridade." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306537.
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Orientadores: Sandra Augusta Santos, Roberto AndreaniTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Neste trabalho reformulamos o problema de complementaridade não linear generalizado (GNCP) em cones poliedrais como um sistema não linear com restrição de não negatividade em algumas variáveis, e trabalhamos na resolução de tal reformulação por meio de estratégias de pontos interiores. Em particular, definimos dois algoritmos e provamos a convergência local de tais algoritmos sob hipóteses usuais. O primeiro algoritmo é baseado no método de Newton, e o segundo, no método tensorial de Chebyshev. O algoritmo baseado no método de Chebyshev pode ser visto como um método do tipo preditor-corretor. Tal algoritmo, quando aplicado a problemas em que as funções envolvidas são afins, e com escolhas adequadas dos parâmetros, torna-se o bem conhecido algoritmo preditor-corretor de Mehrotra. Também apresentamos resultados numéricos que ilustram a competitividade de ambas as propostas.
Abstract: In this work we reformulate the generalized nonlinear complementarity problem (GNCP) in polyhedral cones as a nonlinear system with nonnegativity in some variables and propose the resolution of such reformulation through interior-point methods. In particular we define two algorithms and prove the local convergence of these algorithms under standard assumptions. The first algorithm is based on Newton's method and the second, on the Chebyshev's tensorial method. The algorithm based on Chebyshev's method may be considered a predictor-corrector one. Such algorithm, when applied to problems for which the functions are affine, and the parameters are properly chosen, turns into the well-known Mehrotra's predictor corrector algorithm. We also present numerical results that illustrate the competitiveness of both proposals.
Doutorado
Otimização
Doutor em Matemática Aplicada
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Mariano, Cristiane Regina. "Estudo e análise do desempenho do método barreira modificada." Universidade de São Paulo, 2006. http://www.teses.usp.br/teses/disponiveis/18/18154/tde-15022007-163305/.
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Este trabalho tem por objetivo estudar e analisar a influência do parâmetro de barreira e de seu fator de correção no processo de convergência dos métodos de pontos interiores primal-dual, primal-dual barreira modificada e primal-dual barreira modificada com as técnicas preditor-corretor e Newton composto. A grande motivação para o desenvolvimento desta pesquisa está relacionada com a busca de métodos eficientes para resolver problemas de otimização de programação não-linear, existentes na área de engenharia elétrica mais especificamente na operação de sistemas elétricos de potência. Esses métodos foram aplicados a um problema de programação não-linear e aos sistemas elétricos de três e de trinta barras para analisar a sensibilidade em relação ao parâmetro de barreira e ao seu fator de correção.This work has for objective to study and to analyze the influence of the barrier parameter and its correction factor in the convergence process of the methods primal-dual interior point, primal-dual modified barrier and primal-dual barrier modified with the techniques predictor-corrector and composed Newton. The great motivation for the development of this research is related with the search of efficient methods to solve nonlinear programming optimization problems, existent in the area of electric engineering more specifically in the operation of power systems. Those methods were applied to a nonlinear programming problem and the electric systems of three and thirty buses to analyze the sensibility in relation to the barrier parameter and its correction factor.
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Santos, Claudia Augusta dos. "Métodos numéricos para o retoque digital /." São José do Rio Preto : [s.n.], 2005. http://hdl.handle.net/11449/94281.
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Orientador: Maurílio BoaventuraBanca: Antonio Castelo Filho
Banca: Heloisa Helena Marino Silva
Resumo: O objetivo deste trabalho þe aplicar Mþetodos Numþericos de ordem de precisão mais alta ao problema de Retoque Digital, visando melhorar a qualidade da aproximação quando comparada com o Método de Euler, que þe geralmente utilizado para esse tipo de problema. Para testar a eficiência de tais métodos, utilizamos três modelos de Retoque Digital: o modelo proposto por Bertalmþýo, Sapiro, Ballester e Caselles (BSBC), o modelo de Rudin, Osher e Fatemi conhecido como Variacional Total (TV) e o modelo de Chan e Shen, chamado de Difusão Guiada pela Curvatura (CDD).
Abstract: The purpose of this work is to apply Numerical Methods of higher order to the problem of Digital Inpainting, aiming to improve the quality of the approach when compared with the Eulers Method which is generally used for this kind of problem. To test the e ciency of these methods we use three models of Digital Inpainting: the model considered by Bertalmþýo, Sapiro, Ballester and Caselles (BSBC), the model of Rudin, Osher and Fatemi known as Total Variation (TV) and the model of Chan and Shen, named Curvature Driven Di usion (CDD)
Mestre
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Santos, Claudia Augusta dos [UNESP]. "Métodos numéricos para o retoque digital." Universidade Estadual Paulista (UNESP), 2005. http://hdl.handle.net/11449/94281.
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O objetivo deste trabalho þe aplicar Mþetodos Numþericos de ordem de precisão mais alta ao problema de Retoque Digital, visando melhorar a qualidade da aproximação quando comparada com o Método de Euler, que þe geralmente utilizado para esse tipo de problema. Para testar a eficiência de tais métodos, utilizamos três modelos de Retoque Digital: o modelo proposto por Bertalmþýo, Sapiro, Ballester e Caselles (BSBC), o modelo de Rudin, Osher e Fatemi conhecido como Variacional Total (TV) e o modelo de Chan e Shen, chamado de Difusão Guiada pela Curvatura (CDD).
The purpose of this work is to apply Numerical Methods of higher order to the problem of Digital Inpainting, aiming to improve the quality of the approach when compared with the Euler s Method which is generally used for this kind of problem. To test the e ciency of these methods we use three models of Digital Inpainting: the model considered by Bertalmþýo, Sapiro, Ballester and Caselles (BSBC), the model of Rudin, Osher and Fatemi known as Total Variation (TV) and the model of Chan and Shen, named Curvature Driven Di usion (CDD)
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Yang, Liming. "Parallel block predictor-corrector methods for the numerical solution of ODE's." Thesis, University of Ottawa (Canada), 1993. http://hdl.handle.net/10393/6802.
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Stability and efficiency (i.e. derivative function evaluations per processor) are the two main considerations in deriving good numerical methods for ODE's. The underlying challenge is to increase the stability region while maintaining or even improving efficiency. To achieve this, some extensions of predictor-corrector based methods, which apply a fixed number of corrector iterations, are considered. This thesis studies two particular members of a family of methods called the Parallel Block Predictor-Corrector Family, which are based on these extensions. These two members are called PBPC/2 and PBPC/3. They are characterized by iterated corrector evaluations carried out in two adjacent blocks. Stability properties of these methods are analyzed and compared with some existing block-based parallel predictor-corrector methods. Performance of the PBPC/2 and PBPC/3 methods and these existing block-based parallel predictor-corrector methods is compared using solution formulas which extend over a range of integration orders and which use various number of processors. The results obtained from a stability analysis and from a collection of numerical experiments indicate that the proposed methods provide a potential opportunity to balance stability properties and efficiency in the parallel computer systems.
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Solakoglu, Erhan. "Modélisation des écoulements compressibles turbulents. Interaction onde de choc/couche limite turbulente." Rouen, 1994. http://www.theses.fr/1994ROUES023.
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Le travail présenté dans ce mémoire est consacré à la modélisation de la turbulence, étudiée dans le cadre particulier de la simulation d'interactions onde de choc/couche limite. Les cas-tests étudiés sont: canal transsonique 2D avec bosse à la paroi, écoulement supersonique 2D sur une marche montante et canal transsonique 3D avec bosse en flèche. Les modèles de turbulence utilisés sont: modèle algébrique de Baldwin-Lomax et modèle k-epsilon de Jones-Launder. Une revue bibliographique fait l'objet du premier chapitre. Le deuxième chapitre développe des équations de Navier-Stokes, moyennées au sens de Favre. Le problème de la fermeture des équations moyennées et les différents modèles de turbulence sont présentés dans le troisième chapitre. Dans le quatrième chapitre, la méthode numérique prédicteur-correcteur de Maccormack, implicite et volumes finis, est détaillée. L'exploitation et l'analyse des résultats numériques constituent le cinquième chapitre. Le dernier chapitre est consacré aux conclusions et perspectives
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Chen, Min. "An investigation of the parallel predictor-corrector class of methods for the solution of ode's." Thesis, University of Ottawa (Canada), 1989. http://hdl.handle.net/10393/5993.
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Lörcher, Frieder. "Predictor-corrector discontinuous galerkin schemes for the numerical solution of the compressible navier-stokes equations in complex domains." München Verl. Dr. Hut, 2009. http://d-nb.info/993731309/04.
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Suñagua, Salgado Porfirio 1963. "Uma nova abordagem para encontrar uma base do precondicionador separador para sistemas lineares no método de pontos interiores." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306744.
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Orientador: Aurelio Ribeiro Leite de OliveiraTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Uma abordagem do método preditor-corretor de Mehrotra para resolver problemas de programação linear de grande porte, que utiliza uma classe do precondicionador separador, para resolver sistemas lineares envolvidos por métodos iterativos precondicionados, necessita de uma base que é encontrada por um sofisticado processo de fatoração LU retangular da matriz de restrições. O precondicionador separador tem bom desempenho perto de uma solução ótima, onde as matrizes envolvidas ficam muito mal condicionadas. Neste trabalho, primeiro desenvolvemos o método preditor-corretor com o parâmetro de penalização a fim de reduzir o mau condicionamento da matriz de equações normais. O sucesso desta abordagem é garantido pela demonstração do teorema de convergência de penalização mista com o parâmetro de barreira. Em seguida, implementamos uma nova abordagem para encontrar uma base para o precondicionador separador mediante um processo de fatoração LU retangular padrão aplicada à matriz transposta de restrições escalada. Na maioria das vezes, esta base encontrada é melhor condicionada que a base do método de fatoração retangular anterior. Testes computacionais comprovam uma redução da média do número de iterações do método de gradientes conjugados precondicionado. Também, a eficácia e a robustez da nova abordagem é comprovada por conseguir uma melhor curva de desempenho
Abstract: The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra's predictor-corrector method for large-scale linear programming problems needs to find a base by a sophisticated process based upon applying a rectangular LU factorization. The class of splitting preconditioners works better near a solution of the linear programming problems when the matrices are highly ill-conditioned. In this work, we develop the penalty parameter in Mehrotra's predictor-corrector method in order to reduce the ill-conditioning of the normal equations matrix. The success of this approach is guaranteed by the proof of the theorem of convergence of mixed penalty with the barrier parameter. In addition, we develop and implement a new approach to find a basis for the splitting preconditioner, based upon standard rectangular LU factorization with partial permutation of the transpose of the scaled linear programming constraint matrix. In most cases, the basis is better conditioned than the existing one. Computational tests show a reduction in the average number of iterations of the preconditioned conjugate gradient method. Also, the efficiency and robustness of the new approach is demonstrated by achieving better performance profile
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada
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Sehnalová, Pavla. "Stabilita a konvergence numerických výpočtů." Doctoral thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2011. http://www.nusl.cz/ntk/nusl-261248.
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Tato disertační práce se zabývá analýzou stability a konvergence klasických numerických metod pro řešení obyčejných diferenciálních rovnic. Jsou představeny klasické jednokrokové metody, jako je Eulerova metoda, Runge-Kuttovy metody a nepříliš známá, ale rychlá a přesná metoda Taylorovy řady. V práci uvažujeme zobecnění jednokrokových metod do vícekrokových metod, jako jsou Adamsovy metody, a jejich implementaci ve dvojicích prediktor-korektor. Dále uvádíme generalizaci do vícekrokových metod vyšších derivací, jako jsou např. Obreshkovovy metody. Dvojice prediktor-korektor jsou často implementovány v kombinacích modů, v práci uvažujeme tzv. módy PEC a PECE. Hlavním cílem a přínosem této práce je nová metoda čtvrtého řádu, která se skládá z dvoukrokového prediktoru a jednokrokového korektoru, jejichž formule využívají druhých derivací. V práci je diskutována Nordsieckova reprezentace, algoritmus pro výběr proměnlivého integračního kroku nebo odhad lokálních a globálních chyb. Navržený přístup je vhodně upraven pro použití proměnlivého integračního kroku s přístupe vyšších derivací. Uvádíme srovnání s klasickými metodami a provedené experimenty pro lineární a nelineární problémy.
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Souza, Márcio Augusto da Silva. "Investigação e aplicação de métodos primal - dual pontos interiores em problemas de despacho econômico e ambiental /." Bauru : [s.n.], 2010. http://hdl.handle.net/11449/87187.
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Orientador: Antonio Roberto BalboBanca: Márcia Marcondes Altimari Samed
Banca: Edmea Cassia Baptista
Resumo: Este trabalho visa a investigação e implementação de métodos Primal - Dual Previsor-Corretor de Pontos Interiores com a estratégia de busca unidimensional, e a aplicação destes em problemas de Despacho Econômico e Ambiental. Objetiva-se utilizar estes métodos para determinar soluções aproximadas e consistentes dos problemas causados citados, que forneçam a solução de minimização dos custos dos combustíveis empregados na geração termoelétrica de energia, otimizando um processo de alocação da demanda de energia elétrica entre as unidades geradoras disponíveis, de tal forma que as restrições operacionais sejam atendidas e que o custo de geração é minimizado. Pretende-se também, analisar o problema de Despacho Ambiental com um objetivo único quando se acopla a este o Problema de Despacho Econômico e busca-se, simultaneamente, a minimização dos custos de geração e a redução da emissão de poluentes na natureza. Os métodos foram implementados, testados em Problemas de Despacho Econômico e Ambiental, e o seu desempenho foi comparado com outros métodos já utilizados, cujos resultados são encontrados na literatura
Abstract: This work aims the investigation and implementation of Primal-Dual Predictor-Corrector interior points methods, with the strategy of one-dimensional search, and its application in Economic and Environmental Dispatch Problems. It pretends to use these methods to determine approximate and consistent solutions of the mentioned problems, that provide the solution to minimize the fuel costs used in thermoelectric power generation, optimizing an allocations process of eletric power demand among available generation units, such that the operational constraints are attended and that generation cost is minimized. It too pretends to analyze the Environmental Dispatch Problem with the one objective when it is joined with the Dispatch Problems and it searchs, simultaneously, the minimization of the generation costs and the reduction of emission of the polluants in the nature. The methods were implemented, tested on the Economic and Environemental Dispatch Problems and its performance was compared with others method currently used, whose results are found in the literature
Mestre
15
Souza, Márcio Augusto da Silva [UNESP]. "Investigação e aplicação de métodos primal - dual pontos interiores em problemas de despacho econômico e ambiental." Universidade Estadual Paulista (UNESP), 2010. http://hdl.handle.net/11449/87187.
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Este trabalho visa a investigação e implementação de métodos Primal - Dual Previsor-Corretor de Pontos Interiores com a estratégia de busca unidimensional, e a aplicação destes em problemas de Despacho Econômico e Ambiental. Objetiva-se utilizar estes métodos para determinar soluções aproximadas e consistentes dos problemas causados citados, que forneçam a solução de minimização dos custos dos combustíveis empregados na geração termoelétrica de energia, otimizando um processo de alocação da demanda de energia elétrica entre as unidades geradoras disponíveis, de tal forma que as restrições operacionais sejam atendidas e que o custo de geração é minimizado. Pretende-se também, analisar o problema de Despacho Ambiental com um objetivo único quando se acopla a este o Problema de Despacho Econômico e busca-se, simultaneamente, a minimização dos custos de geração e a redução da emissão de poluentes na natureza. Os métodos foram implementados, testados em Problemas de Despacho Econômico e Ambiental, e o seu desempenho foi comparado com outros métodos já utilizados, cujos resultados são encontrados na literatura
This work aims the investigation and implementation of Primal-Dual Predictor-Corrector interior points methods, with the strategy of one-dimensional search, and its application in Economic and Environmental Dispatch Problems. It pretends to use these methods to determine approximate and consistent solutions of the mentioned problems, that provide the solution to minimize the fuel costs used in thermoelectric power generation, optimizing an allocations process of eletric power demand among available generation units, such that the operational constraints are attended and that generation cost is minimized. It too pretends to analyze the Environmental Dispatch Problem with the one objective when it is joined with the Dispatch Problems and it searchs, simultaneously, the minimization of the generation costs and the reduction of emission of the polluants in the nature. The methods were implemented, tested on the Economic and Environemental Dispatch Problems and its performance was compared with others method currently used, whose results are found in the literature
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張書豪. "Varitions on th predictor-corrector interior point method." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/75398233901035865390.
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碩士國立中正大學
應用數學研究所
87
In chapter 2 of this paper, we will introduce the basic ideas of the interior point algorithms. Consider the problems we would like to solve, and propose how to solve it based on the logarithmic barrier function approach. Then compute some kinds of the search directions and choose the step lengths. Introduce an feasible interior-point algorithm, and its iteration complexity. The last part of chapter 2, we would like to find a suitable parameter for the algorithm by Klee-Minty problems. Then in chapter 3, write down some variate algorithms we used, explain their characteristics and how to complete the computer programs (By FORTRAN 77) for them. Chapter 4 is numerical experiments and conclusions. The first section is about the experiment results for all algorithms. We will record the iteration numbers and CPU time of them. Then is the conclusions.
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cheng, chih-chieh, and 鄭至傑. "Block predictor – corrector method on Python-based distributed computing environment." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/40855202325662510816.
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碩士中華大學
應用統計學系碩士班
99
In this thesis, we study how to apply Python to do parallel computation in a distributed computing structure. MPI-2 (Message Passing Interface) is implemented to be a basic standard on parallel processing to communicate between each CPU inside the structure. MPICH2 (Message Passing Interface Chameleon) is implemented to let individual CPU to do data transmission and reception between other CPUs. While controlling and manipulation of the process is proceed by applying the functions; send、recv、get_size、get_rank, etc, in the package of mpi4py by Python. Numerical experiments are carried on the block method to solve some stiff ordinary differential equations, and their speed up between 1 CPU and two CPU are explored to show positive achievement of the proposed distributed parallel structure.
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Liu, Shu Chen, and 劉淑貞. "The Stable Regions of Multi-Block Predictor-Corrector Method for ODE's." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/75407656621700670775.
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碩士國立中正大學
數學研究所
91
The multi-block predictor-corrector method(BPC) is one of the efficient and accurate method for solving the numerical solution of the ordinary differential equations. By choosing some suitable matrix coefficients in two schemes with repeated self-correcting processes, P(EC)^nE,n=1,2,...,we investigate and compare the corresponging changes of the stable regions. We also vary the coefficients matrix in the local truncation error to study the responses about the stable respectively.
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簡鶴昇. "Predictor-Corrector Method on Adaptive Grids for One-Dimensional Parabolic Problems." Thesis, 1999. http://ndltd.ncl.edu.tw/handle/55145896860794062291.
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碩士國立彰化師範大學
數學系
87
Three issues are investigated for numerical methods of one-dimensional parabolic equations in this thesis. First, we heuristically show that a predictor-corrector method performs much better than explicit difference method by testing on two typical nonlinear examples on uniform grids. Second, we develop an algorithm for adaptive grids which performs well on the testing suite. Third, comparing with standard finite difference methods on the analysis of convergence for linear parabolic equations on nonuniform grids, we have found that the predictor-corrector method relaxes more conditions on its assumption.
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Fang, Huei-Jen, and 方惠真. "On Adaptive Block Predictor-Corrector Methods for ODE's." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/81866989847792491462.
Full textAbstract:
博士國立中正大學
應用數學研究所
89
In this thesis, the multi-block predictor corrector methods (BPC) are studied in solving the initial value problem of ordinary differential equations. In Part I, the qualitative analysis of BPC schemes with self-correction processes, EP(EC)ι, have been provided. Based on the criteria of stability, accuracy and speed-up effect, we have proposed the proper 2-block and 3-block schemes that have been constituted by the fourth order predictor and corrector. In addition to the optimal stability intercept, our results indicate that the shape of the region of absolute stability provide more important character in choosing the proper schemes. In Part II, based on the studies of Stetter and Worland, the self-correction process is replaced by the linear combination (LC). The limiting behavior of LC is studied and it yields the qualitative analysis of the associated BPC schemes. Our simulation results show that, in comparing with EP(EC)ι, the speed-up effect of BPC scheme with LC is significant with the compatible accuracy.
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Chen, Jr-Sheng, and 陳志昇. "On Real Time Simulation of Block Predictor-Corrector Methods." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/53931849885220139321.
Full textAbstract:
碩士國立中正大學
應用數學研究所
90
The BPC methods for solving IVP can be parallelized in parallel machines. However, how complicated computations does the problem needs, the parallel method is worth to implement instead of using sequential methods? In this paper, we would concentrate on 2-block schemes proposed in H. J. Fang [29]. First, we make the comparison of some efficiency with 2-block schemes and RK45 by processing fifteen benchmark problems. Consider the conditions of functional evaluations and maximal error, B2T2 scheme has better parallelism than others. Furthermore, to study the speedup, we perform five test problems with increasing complexity by B2T2 on two different parallel platforms: PC Cluster and IBM SP2. As the given functional computations become more complicated, the parallelism gets more better and PC Cluster would be superior to IBM SP2. But the speedup of performance with 2 processors is still less than 1. With a comparison in the process, it needs at least four messages to be exchanged in one clock time. As estimating the admissible local error control, parallel processing is superior to sequential performing. However, the speedup is still less than 1. When we rewrite the corresponding coefficient matrix of the vector predictor derivative value in the corrector formula of the BPC scheme, that is, to reduce one message-passing, the speedup raises up to 1.2. The superiority of parallelism appears instantly. For the high dimensional BPC schemes, if send and receive message are synchronous in hardware, we predict that it would have better speedup with multiple division of computation.
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陳香妘. "On the Stability of Predictor-Corrector Methods with Linear Combination Update." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/74291506504254891569.
Full textAbstract:
碩士國立中正大學
數學研究所
90
The process of linear combination was originally introduced to extend the absolute stability intercept for the predictor-corrector (PC) methods. In this thesis, we present the limiting behavior of some one-dimensional PC schemes with the linear combination update. By applying the root-locus plot, the corresponding with stability regions of various LC update process are successfully obtained. Especially for the schemes on the parallel clustering computation platform,our simulation results show that ,the suitable linear combination will improve the efficiency in solving the problem with complex eigenvalues.
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Wu, Sunny, and 吳山林. "On the Stability of Block Predictor-Corrector Methods with Self-Correcting Process." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/10368565886066258176.
Full textAbstract:
碩士國立中正大學
數學研究所
90
Abstract In this thesis, we are interested in studying the configuration of stability regions of numerical schemes when solving an initial value problem in ODEs. By applying the predictor-corrector scheme with repeated correcting processes, the variation of stability region could be changed drastically. We propose a delicate algorithm on which the boundary of the stability region could be located efficiently. By using the algorithm, the stability of some block predictor-corrector schemes with repeated self-correction process have been studied. Moreover, some incorrect results of optimal stability intercept in some literatures have also been found. Key words: Block predictor-corrector Methods, Absolute Stability, Self-Correction Process
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Wang, May-Ya, and 王美雅. "On The Stability of Block Predictor-Corrector Methods with Linear Combination Process." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/46091222911320787440.
Full textAbstract:
碩士國立中正大學
應用數學研究所
91
We study the stability and related efficiency on the parallel predictor-corrector schemes in solving the initial value problem of ODE's. On contract to the block predictor-corrector methods (BPC), the given schemes were designated that the predictor and corrector are to be performed on the individual clusters. By applying more delicate locus plot scheme, the stability regions were recomputed for the schemes proposed by Birta and Yang [16] with repeated self-corrections (PBPC/M). Some stability data are revised. In addition to PBPC/M, the linear combination scheme (PBPC(LC)^{K}) was also studied. Our results indicate that PBPC(LC)^{K} gains the advantage in reducing the number of functional evaluation and the data interchanges between computation nodes. Therefore, the computation efficiency may be further improved.
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Wu, Ming-Tsung, and 吳銘宗. "Study of Predictor-Corrector Methods for Optimum Blank Design of Sheet Metal Forming." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/38567339300889385012.
Full textAbstract:
博士淡江大學
機械與機電工程學系博士班
95
This thesis discusses the optimization analysis of the blank geometry profile in square cup and cylindrical cup of the metal forming. Using the explicit dynamic finite element method to carry on the numerical simulation, cooperate with streamline method, true strain method and adaptive network fuzzy inference system(ANFIS) three optimization theory to predict the shape of optimum blank. With this way, can be save a lot of time than using “Try and Error” and repeating experiments. In order to ask the verification of the numerical simulation and optimization theory, that to design the multi-form experiment mould in accordance with the condition and restriction of the multi-formly forming. After it made forming in order to verify the numerical simulation relation between reason the geometry measurement of the blank, the loading of the punch and stroke and the thickness distribute of the blank. Compare via number value between analysis and experimental result, shows that explicit dynamic finite element could correctly simulation of situation during forming process of blank form comparing numerical analysis data and experimental data. The adaptive network fuzzy inference system of the optimization theory is the best to can effective and fast to get the optimization of the blank. Moreover, it can to supply optimization reference when making process and designing mold. Finally, it reduces the research and development of products channel into the technology of CAE/CAD in the micro cylindrical cup, that wish to substitute for testing in fact consuming time with computer simulation. Then probe into comparing to the important parameter of micro forming (size effect and stress-strain curve). Channel the thickness of the blank and grain size into material stress strain equation in size effect, and ask the way of solving to seek the proper conjecture value with number value. In the hope of making can predict stress-strain curve of the different material thickness and different grain size with the material through the single revising type. Than comparison between the error of size effect factor equation and simple stress strain equation form all in permitting range through result, so can provable revise equation dependability. And whether it give in accordance with revising type different material thickness and different grain size definitely estimate it each relevant materials after forming in advance coming, can offer the reference of relevant micro forming.
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Singh, Samar B. "Study of Higher Order Split-Step Methods for Stiff Stochastic Differential Equations." Thesis, 2013. http://etd.iisc.ernet.in/2005/3354.
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Stochastic differential equations(SDEs) play an important role in many branches of engineering and science including economics, finance, chemistry, biology, mechanics etc. SDEs (with m-dimensional Wiener process) arising in many applications do not have explicit solutions, which implies the development of effective numerical methods for such systems. For SDEs, one can classify the numerical methods into three classes: fully implicit methods, semi-implicit methods and explicit methods. In order to solve SDEs, the computation of Newton iteration is necessary for the implicit and semi-implicit methods whereas for the explicit methods we do not need such computation.
In this thesis the common theme is to construct explicit numerical methods with strong order 1.0 and 1.5 for solving Itˆo SDEs. The five-stage Milstein(FSM)methods, split-step forward Milstein(SSFM)methods and M-stage split-step strong Taylor(M-SSST) methods are constructed for solving SDEs. The FSM, SSFM and M-SSST methods are fully explicit methods. It is proved that the FSM and SSFM methods are convergent with strong order 1.0, and M-SSST methods are convergent with strong order 1.5.Stiffness is a very important issue for the numerical treatment of SDEs, similar to the case of deterministic ordinary differential equations. Stochastic stiffness can lead someone to use smaller step-size for the numerical simulation of the SDEs. However, such issues can be handled using numerical methods with better stability properties.
The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the FSM and SSFM methods are larger than the Milstein and three-stage Milstein methods. The M-SSST methods possess large mean square stability region as compared to the order 1.5 strong Itˆo-Taylor method. SDE systems simulated with the FSM, SSFM and M-SSST methods show the computational efficiency of the methods.
In this work, we also consider the problem of computing numerical solutions for stochastic delay differential equations(SDDEs) of Itˆo form with a constant lag in the argument. The fully explicit methods, the predictor-corrector Euler(PCE)methods, are constructed for solving SDDEs. It is proved that the PCE methods are convergent with strong order γ = ½ in the mean-square sense. The conditions under which the PCE methods are MS-stable and GMS-stable are less restrictive as compared to the conditions for the Euler method.
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Govender, Nadrajh. "A brief analysis of certain numerical methods used to solve stochastic differential equations." Diss., 2007. http://hdl.handle.net/2263/26558.
Full textAbstract:
Stochastic differential equations (SDE’s) are used to describe systems which are influenced by randomness. Here, randomness is modelled as some external source interacting with the system, thus ensuring that the stochastic differential equation provides a more realistic mathematical model of the system under investigation than deterministic differential equations. The behaviour of the physical system can often be described by probability and thus understanding the theory of SDE’s requires the familiarity of advanced probability theory and stochastic processes. SDE’s have found applications in chemistry, physical and engineering sciences, microelectronics and economics. But recently, there has been an increase in the use of SDE’s in other areas like social sciences, computational biology and finance. In modern financial practice, asset prices are modelled by means of stochastic processes. Thus, continuous-time stochastic calculus plays a central role in financial modelling. The theory and application of interest rate modelling is one of the most important areas of modern finance. For example, SDE’s are used to price bonds and to explain the term structure of interest rates. Commonly used models include the Cox-Ingersoll-Ross model; the Hull-White model; and Heath-Jarrow-Morton model. Since there has been an expansion in the range and volume of interest rate related products being traded in the international financial markets in the past decade, it has become important for investment banks, other financial institutions, government and corporate treasury offices to require ever more accurate, objective and scientific forms for the pricing, hedging and general risk management of the resulting positions. Similar to ordinary differential equations, many SDE’s that appear in practical applications cannot be solved explicitly and therefore require the use of numerical methods. For example, to price an American put option, one requires the numerical solution of a free-boundary partial differential equation. There are various approaches to solving SDE’s numerically. Monte Carlo methods could be used whereby the physical system is simulated directly using a sequence of random numbers. Another method involves the discretisation of both the time and space variables. However, the most efficient and widely applicable approach to solving SDE’s involves the discretisation of the time variable only and thus generating approximate values of the sample paths at the discretisation times. This paper highlights some of the various numerical methods that can be used to solve stochastic differential equations. These numerical methods are based on the simulation of sample paths of time discrete approximations. It also highlights how these methods can be derived from the Taylor expansion of the SDE, thus providing opportunities to derive more advanced numerical schemes.Dissertation (MSc (Mathematics of Finance))--University of Pretoria, 2007.
Mathematics and Applied Mathematics
MSc
unrestricted