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Verfahren zur schnellen Lösung von grossen Gleichungssystemen in der MomentenmethodeAstner, Miguel January 2009 (has links)
Zugl.: Hamburg, Techn. Univ., Diss., 2009
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Migration preconditioning with curvelets.Moghaddam, Peyman P., Herrmann, Felix J. January 2004 (has links)
In this paper, the property of Curvelet transforms for preconditioning the migration and normal operators is investigated. These operators belong to the class of Fourier integral operators and pseudo-differential operators, respectively. The effect of this preconditioner is shown in term of improvement of sparsity, convergence rate, number of iteration for the Krylov-subspace solver and clustering of singular(eigen) values. The migration operator, which we employed in this work is the common-offset Kirchoff-Born migration.
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Hybrid Sn/Diffusion and Sn/P3 Neutronics CalculationsManolov, Sergiy 02 October 2013 (has links)
In this thesis we investigate coupling and preconditioning techniques for 19D hybrid neutronics calculations. Each problem is represented by two spatial regions with Sn in one region and either Diffusion (P1) or P3 in the other region. For each of these two cases we define one coupling scheme and two different preconditioned systems. These systems are solved with both fixed9point iteration and the GMRES Krylov method. The solution techniques are compared in terms of iteration count and computational cost. Preconditioning with a global diffusion operator is found to be very effective for the most difficult problems.
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Krylov and Finite State Projection methods for simulating stochastic biochemical kinetics via the Chemical Master EquationShevarl MacNamara Unknown Date (has links)
Computational and mathematical models of cellular processes promise great benets in important elds such as molecular biology and medicine. Increasingly, researchers are incorporating the fundamentally discrete and stochastic nature of biochemical processes into the mathematical models that are intended to represent them. This has led to the formulation of models for genetic networks as continuous-time, discrete state, Markov processes, giving rise to the so-called Chemical Master Equation (CME), which is a discrete, partial dierential equation, that governs the evolution of the associated probability distribution function (PDF). While promising many insights, the CME is computationally challenging, especially as the dimension of the model grows. In this thesis, novel methods are developed for computing the PDF of the Master Equation. The problems associated with the high-dimensional nature of the Chemical Master Equation are addressed by adapting Krylov methods, in combination with Finite State Projection methods, to derive algorithms well-suited to the Master Equation. Variations of the approach that incorporate the Strang splitting and a stochastic analogue of the total quasi-steady-state approximation are also derived for chemical systems with disparate rates. Monte Carlo approaches, such as the Stochastic Simulation Algorithm, that simulate trajectories of the process governed by the CME have been a very popular approach and we compare these with the PDF approaches developed in this thesis. The thesis concludes with a discussion of various implementation issues along with numerical results for important applications in systems biology, including the gene toggle, the Goldbeter-Koshland switch and the Mitogen-Activated Protein Kinase Cascade.
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Krylov and Finite State Projection methods for simulating stochastic biochemical kinetics via the Chemical Master EquationShevarl MacNamara Unknown Date (has links)
Computational and mathematical models of cellular processes promise great benets in important elds such as molecular biology and medicine. Increasingly, researchers are incorporating the fundamentally discrete and stochastic nature of biochemical processes into the mathematical models that are intended to represent them. This has led to the formulation of models for genetic networks as continuous-time, discrete state, Markov processes, giving rise to the so-called Chemical Master Equation (CME), which is a discrete, partial dierential equation, that governs the evolution of the associated probability distribution function (PDF). While promising many insights, the CME is computationally challenging, especially as the dimension of the model grows. In this thesis, novel methods are developed for computing the PDF of the Master Equation. The problems associated with the high-dimensional nature of the Chemical Master Equation are addressed by adapting Krylov methods, in combination with Finite State Projection methods, to derive algorithms well-suited to the Master Equation. Variations of the approach that incorporate the Strang splitting and a stochastic analogue of the total quasi-steady-state approximation are also derived for chemical systems with disparate rates. Monte Carlo approaches, such as the Stochastic Simulation Algorithm, that simulate trajectories of the process governed by the CME have been a very popular approach and we compare these with the PDF approaches developed in this thesis. The thesis concludes with a discussion of various implementation issues along with numerical results for important applications in systems biology, including the gene toggle, the Goldbeter-Koshland switch and the Mitogen-Activated Protein Kinase Cascade.
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Linear estimation and detection in Krylov subspaces : with 11 tables /Dietl, Guido K. E. January 2007 (has links) (PDF)
Techn. Univ., Diss.--München, 2006.
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A stable cubically convergent GR algorithm and Krylov subspace methods for non-hermitian matrix eigenvalue problemsZiegler, Markus. Unknown Date (has links) (PDF)
University, Diss., 2001--Tübingen.
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Avaliação dos algoritmos de Picard-Krylov e Newton-Krylov na solução da equação de Richards / Evaluation of algorithms of Picard-Krylov and Newton-Krylov in solution of Richards equationMarcelo Xavier Guterres 13 December 2013 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A engenharia geotécnica é uma das grandes áreas da engenharia civil que estuda a
interação entre as construções realizadas pelo homem ou de fenômenos naturais com o ambiente
geológico, que na grande maioria das vezes trata-se de solos parcialmente saturados.
Neste sentido, o desempenho de obras como estabilização, contenção de barragens, muros
de contenção, fundações e estradas estão condicionados a uma correta predição do fluxo de
água no interior dos solos. Porém, como a área das regiões a serem estudas com relação à
predição do fluxo de água são comumente da ordem de quilômetros quadrados, as soluções
dos modelos matemáticos exigem malhas computacionais de grandes proporções, ocasionando
sérias limitações associadas aos requisitos de memória computacional e tempo de
processamento. A fim de contornar estas limitações, métodos numéricos eficientes devem
ser empregados na solução do problema em análise. Portanto, métodos iterativos para
solução de sistemas não lineares e lineares esparsos de grande porte devem ser utilizados
neste tipo de aplicação. Em suma, visto a relevância do tema, esta pesquisa aproximou
uma solução para a equação diferencial parcial de Richards pelo método dos volumes finitos
em duas dimensões, empregando o método de Picard e Newton com maior eficiência
computacional. Para tanto, foram utilizadas técnicas iterativas de resolução de sistemas
lineares baseados no espaço de Krylov com matrizes pré-condicionadoras com a biblioteca
numérica Portable, Extensible Toolkit for Scientific Computation (PETSc). Os resultados
indicam que quando se resolve a equação de Richards considerando-se o método de
PICARD-KRYLOV, não importando o modelo de avaliação do solo, a melhor combinação
para resolução dos sistemas lineares é o método dos gradientes biconjugados estabilizado
mais o pré-condicionador SOR. Por outro lado, quando se utiliza as equações de van
Genuchten deve ser optar pela combinação do método dos gradientes conjugados em conjunto
com pré-condicionador SOR. Quando se adota o método de NEWTON-KRYLOV,
o método gradientes biconjugados estabilizado é o mais eficiente na resolução do sistema
linear do passo de Newton, com relação ao pré-condicionador deve-se dar preferência ao
bloco Jacobi. Por fim, há evidências que apontam que o método PICARD-KRYLOV
pode ser mais vantajoso que o método de NEWTON-KRYLOV, quando empregados na
resolução da equação diferencial parcial de Richards. / Geotechnical Engineering is the area of Civil Engineering that studies the interaction
between constructions carried out by man or natural phenomena with geological
environment, which most of times is partially saturated soil. In this sense, work developing
as stabilization, dam containing, retaining walls, foundations and highways are conditioned
to a right prediction of water flow into the soil. However, considering the water flow,
the studied region areas are commonly on the order of square kilometers, mathematical
models solutions require computational meshes of large proportions, causing serious limitations
linked to computational memory requirements and processing time. In order to
overcome these limitations, efficient numerical methods must be used in the solution of
the considered problem. Hence iterative methods for solving nonlinear and large sparse
linear systems must be used in this type of application. In short, this study approached
a solution to the Richard partial differential equation by the two dimensions finite volume
method, bringing Picard and Newton method with greater efficiency. Linear system
resolution iterative techniques based on Krylov space with pre-conditioners matrix were
used. Portable Extensible Toolkit for Scientific Computation (PETSc) numerical library
was a tool used during the task. The results indicate when a Richards equation is solved
considering thr PICARD-KRYLOV method, no matter the soil evaluation model, the
best combination for solving linear systems is the stabilized double gradient method and
the SOR preconditioning. On the other hand, when the van Genuchten equations are
used the gradients methods with the SOR preconditioning must be chosen. Adopting
the NEWTON-KRYLOV method, the stabilized double gradient method is more efficient
in soling Newton linear system, in relation to the preconditioning it must be giving
preference to the Jacob block. Finally, there are strong indications that the PICARDKRYLOV
method can be more effective than the NEWTON-KRYLOV one, when used
for solving Richards partial differential equation.
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Avaliação dos algoritmos de Picard-Krylov e Newton-Krylov na solução da equação de Richards / Evaluation of algorithms of Picard-Krylov and Newton-Krylov in solution of Richards equationMarcelo Xavier Guterres 13 December 2013 (has links)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / A engenharia geotécnica é uma das grandes áreas da engenharia civil que estuda a
interação entre as construções realizadas pelo homem ou de fenômenos naturais com o ambiente
geológico, que na grande maioria das vezes trata-se de solos parcialmente saturados.
Neste sentido, o desempenho de obras como estabilização, contenção de barragens, muros
de contenção, fundações e estradas estão condicionados a uma correta predição do fluxo de
água no interior dos solos. Porém, como a área das regiões a serem estudas com relação à
predição do fluxo de água são comumente da ordem de quilômetros quadrados, as soluções
dos modelos matemáticos exigem malhas computacionais de grandes proporções, ocasionando
sérias limitações associadas aos requisitos de memória computacional e tempo de
processamento. A fim de contornar estas limitações, métodos numéricos eficientes devem
ser empregados na solução do problema em análise. Portanto, métodos iterativos para
solução de sistemas não lineares e lineares esparsos de grande porte devem ser utilizados
neste tipo de aplicação. Em suma, visto a relevância do tema, esta pesquisa aproximou
uma solução para a equação diferencial parcial de Richards pelo método dos volumes finitos
em duas dimensões, empregando o método de Picard e Newton com maior eficiência
computacional. Para tanto, foram utilizadas técnicas iterativas de resolução de sistemas
lineares baseados no espaço de Krylov com matrizes pré-condicionadoras com a biblioteca
numérica Portable, Extensible Toolkit for Scientific Computation (PETSc). Os resultados
indicam que quando se resolve a equação de Richards considerando-se o método de
PICARD-KRYLOV, não importando o modelo de avaliação do solo, a melhor combinação
para resolução dos sistemas lineares é o método dos gradientes biconjugados estabilizado
mais o pré-condicionador SOR. Por outro lado, quando se utiliza as equações de van
Genuchten deve ser optar pela combinação do método dos gradientes conjugados em conjunto
com pré-condicionador SOR. Quando se adota o método de NEWTON-KRYLOV,
o método gradientes biconjugados estabilizado é o mais eficiente na resolução do sistema
linear do passo de Newton, com relação ao pré-condicionador deve-se dar preferência ao
bloco Jacobi. Por fim, há evidências que apontam que o método PICARD-KRYLOV
pode ser mais vantajoso que o método de NEWTON-KRYLOV, quando empregados na
resolução da equação diferencial parcial de Richards. / Geotechnical Engineering is the area of Civil Engineering that studies the interaction
between constructions carried out by man or natural phenomena with geological
environment, which most of times is partially saturated soil. In this sense, work developing
as stabilization, dam containing, retaining walls, foundations and highways are conditioned
to a right prediction of water flow into the soil. However, considering the water flow,
the studied region areas are commonly on the order of square kilometers, mathematical
models solutions require computational meshes of large proportions, causing serious limitations
linked to computational memory requirements and processing time. In order to
overcome these limitations, efficient numerical methods must be used in the solution of
the considered problem. Hence iterative methods for solving nonlinear and large sparse
linear systems must be used in this type of application. In short, this study approached
a solution to the Richard partial differential equation by the two dimensions finite volume
method, bringing Picard and Newton method with greater efficiency. Linear system
resolution iterative techniques based on Krylov space with pre-conditioners matrix were
used. Portable Extensible Toolkit for Scientific Computation (PETSc) numerical library
was a tool used during the task. The results indicate when a Richards equation is solved
considering thr PICARD-KRYLOV method, no matter the soil evaluation model, the
best combination for solving linear systems is the stabilized double gradient method and
the SOR preconditioning. On the other hand, when the van Genuchten equations are
used the gradients methods with the SOR preconditioning must be chosen. Adopting
the NEWTON-KRYLOV method, the stabilized double gradient method is more efficient
in soling Newton linear system, in relation to the preconditioning it must be giving
preference to the Jacob block. Finally, there are strong indications that the PICARDKRYLOV
method can be more effective than the NEWTON-KRYLOV one, when used
for solving Richards partial differential equation.
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Teoretické otázky popisu chování krylovovských metod / Teoretické otázky popisu chování krylovovských metodStrnad, Otto January 2011 (has links)
The presented thesis is focused on the GMRES convergence analysis. The basic principles of CG, MINRES and GMRES are briefly explained. The thesis summarizes some known convergence results of these methods. The known characterizations of the matrices and the right hand sides gen- erating the same Krylov residual spaces are summarized. Connections and the differences between the different points of view on GMRES convergence analysis are shown. We expect that if the convergence curve of GMRES applied to the nonnormal matrix and the right hand side seems to be de- termined by the eigenvalues of the matrix then exists a matrix that is close to normal and has the same spectrum as the matrix and for the right hand side has the same GMRES convergence curve (We assume that the initial approximation 0 = 0). Several numerical experiments are done to examine this assumption. This thesis describes an unpublished result of Gérard Meu- rant which is the formula for the norm of the -th error of GMRES applied to the matrix and right hand side and its derivation. The upper estimate of the -th GMRES error is derived. This estimate is minimized via spectrum.
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