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The analysis of symmetric structures using group representation theoryKangwai, Riki Dale January 1998 (has links)
Group Representation Theory is the mathematical language best suited to describing the symmetry properties of a structure, and a structural analysis can utilises Group Representation Theory to provide the most efficient and systematic method of exploiting the full symmetry properties of any symmetric structure. Group Representation Theory methods currently exist for the Stiffness Niethod of structural analysis, where the stiffness matrix of a structure is block-diagonalised into a number of independent submatrices, each of which relates applied loads and displacements with a particular type of symmetry. This dissertation extends the application of Group Representation Theory to the equilibrium and compatibility matrices which are commonly used in the Force Method of structural analysis. Group Representation Theory is used to find symmetry-adapted coordinate systems for both the external vector space which is suitable for representing the loads applied to a structure, and the internal vector space wh",t-k is-suitable for representing the internal forces. Using these symmetry-adapted coordinate systems the equilibrium matrix is block-diagonalised into a number of independent submatrix blocks, thus decomposing the analysis into a number of subproblems which require less computational effort. Each independent equilibrium submatrix block relates applied loads and internal forces with particular symmetry properties, and hence any states of self-stress or inextensional mechanisms in one of these equilibrium submatrix blocks will necessarily have ~rresponding symmetry properties. Thus, a symmetry analysis provides valuable insight into the behaviour of symmetric structures by helping to identify and classif:)'. any states of self-stress .or inextensional mechanisms present in a structure. In certain cases it is also possible for a symmetry analysis to identify when a structure contains a :ijnite rather than infinitesimal mechanism. To do this a symmetry analysis must b~ carried out using the symmetry properties of the inextensional mechanism of interest. If the analysis shows that any states of self-stress which exist in the structure have "lesser" symmetry properties, then the states of self-stress exist independently from the mechanism and cannot prevent its finite motion.
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Hamiltonian cycles in subset and subspace graphs.Ghenciu, Petre Ion 12 1900 (has links)
In this dissertation we study the Hamiltonicity and the uniform-Hamiltonicity of subset graphs, subspace graphs, and their associated bipartite graphs. In 1995 paper "The Subset-Subspace Analogy," Kung states the subspace version of a conjecture. The study of this problem led to a more general class of graphs. Inspired by Clark and Ismail's work in the 1996 paper "Binomial and Q-Binomial Coefficient Inequalities Related to the Hamiltonicity of the Kneser Graphs and their Q-Analogues," we defined subset graphs, subspace graphs, and their associated bipartite graphs. The main emphasis of this dissertation is to describe those graphs and study their Hamiltonicity. The results on subset graphs are presented in Chapter 3, on subset bipartite graphs in Chapter 4, and on subspace graphs and subspace bipartite graphs in Chapter 5. We conclude the dissertation by suggesting some generalizations of our results concerning the panciclicity of the graphs.
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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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Are Artificial Neural Networks the Right Tool for Modelling and Control of Batch and Batch-Like Processes?Mustafa Rashid January 2023 (has links)
The prevalence of batch and batch-like operations, in conjunction with the continued
resurgence of artificial intelligence techniques for clustering and classification applications, has increasingly motivated the exploration of the applicability of deep learning
for modeling and feedback control of batch and batch-like processes. To this end, the
present study seeks to evaluate the viability of artificial intelligence in general, and
neural networks in particular, toward process modeling and control via a case study.
Nonlinear autoregressive with exogeneous input (NARX) networks are evaluated in
comparison with subspace models within the framework of model-based control. A
batch polymethyl methacrylate (PMMA) polymerization process is chosen as a simulation test-bed. Subspace-based state-space models and NARX networks identified
for the process are first compared for their predictive power. The identified models
are then implemented in model predictive control (MPC) to compare the control performance for both modeling approaches. The comparative analysis reveals that the
state-space models performed better than NARX networks in predictive power and
control performance. Moreover, the NARX networks were found to be less versatile
than state-space models in adapting to new process operation. The results of the
study indicate that further research is needed before neural networks may become
readily applicable for the feedback control of batch processes. / Thesis / Master of Applied Science (MASc)
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Recycling Bi-Lanczos Algorithms: BiCG, CGS, and BiCGSTABAhuja, Kapil 21 September 2009 (has links)
Engineering problems frequently require solving a sequence of dual linear systems. This paper introduces recycling BiCG, that recycles the Krylov subspace from one pair of linear systems to the next pair. Augmented bi-Lanczos algorithm and modified two-term recurrence are developed for using the recycle space. Recycle space is built from the approximate invariant subspace corresponding to eigenvalues close to the origin. Recycling approach is extended to the CGS and the BiCGSTAB algorithms. Experiments on a convection-diffusion problem give promising results. / Master of Science
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Effect of different platforms on coupling compensation matrices in AOA estimation algorithms using small size UCAGhazaany, Tahereh S., Zhu, Shaozhen (Sharon), Jones, Steven M.R., Abd-Alhameed, Raed, Noras, James M., Van Buren, T., Marker, S. January 2014 (has links)
No / In this paper the sensitivity of the decoupling matrix used for mutual coupling compensation in small size uniform circular arrays has been studied. The compensation matrix is calculated using the receiving mode technique for a 5-element uniform circular array and applied to two groups of direction finding algorithms, namely phase comparison-based (interferometry) and subspace-based algorithms. In the tracking application considered the receiver array is deployed on a car roof or aircraft, so the geometry of the platform influences the compensation results. In this work, the effect of different ground plane geometries in terms of the standard deviation of angular error for each estimation algorithm using simulation results is investigated. The results show that the calibration conditions used to determine the compensation matrix affect the AOA estimation accuracy.
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Analysis and Implementation Considerations of Krylov Subspace Methods on Modern Heterogeneous Computing ArchitecturesHiggins, Andrew, 0009-0007-5527-9263 05 1900 (has links)
Krylov subspace methods are the state-of-the-art iterative algorithms for solving large, sparse systems of equations, which are ubiquitous throughout scientific computing. Even with Krylov methods, these problems are often infeasible to solve on standard workstation computers and must be solved instead on supercomputers. Most modern supercomputers fall into the category of “heterogeneous architectures”, typically meaning a combination of CPU and GPU processors. Thus, development and analysis of Krylov subspace methods on these heterogeneous architectures is of fundamental importance to modern scientific computing.
This dissertation focuses on how this relates to several specific problems. Thefirst analyzes the performance of block GMRES (BGMRES) compared to GMRES for linear systems with multiple right hand sides (RHS) on both CPUs and GPUs, and modelling when BGMRES is most advantageous over GMRES on the
GPU. On CPUs, the current paradigm is that if one wishes to solve a system of equations with multiple RHS, BGMRES can indeed outperform GMRES, but not always. Our original goal was to see if there are some cases for which BGMRES
is slower in execution time on the CPU than GMRES on the CPU, while on the GPU, the reverse holds. This is true, and we generally observe much faster execution times and larger improvements in the case of BGMRES on the GPU. We
also observe that for any fixed matrix, when the number of RHS increase, there is a point in which the improvements start to decrease and eventually any advantage of the (unrestarted) block method is lost. We present a new computational model which helps us explain why this is so. The significance of this analysis is that it first demonstrates increased potential of block Krylov methods on heterogeneous architectures than on previously studied CPU-only machines. Moreover, the theoretical runtime model can be used to identify an optimal partitioning strategy of the RHS
for solving systems with many RHS.
The second problem studies the s-step GMRES method, which is an implementation of GMRES that attains high performance on modern heterogeneous machines by generating s Krylov basis vectors per iteration, and then orthogonalizing the vectors in a block-wise fashion. The use of s-step GMRES is currently limited because the algorithm is prone to numerical instabilities, partially due to breakdowns in a tall-and-skinny QR subroutine. Further, a conservatively small step size must be used in practice, limiting the algorithm’s performance. To address these issues, first a novel randomized tall-and-skinny QR factorization is presented that is significantly more stable than the current practical algorithms without sacrificing performance on GPUs. Then, a novel two-stage block orthogonalization scheme is introduced that significantly improves the performance of the s-step GMRES algorithm when small step sizes are used. These contributions help make s-step GMRES a more practical method in heterogeneous, and therefore exascale, environments. / Mathematics
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Another Slice of Multivariate Dimension ReductionEkblad, Carl January 2022 (has links)
This thesis presents some methods of multivariate dimension reduction, with emphasis on methods guided by the work of R.A. Fisher. Some of the methods presented can be traced back to the 20th century, while some are much more recent. For the more recent methods, additional attention will paid to the foundational underpinnings. The presentation for each of the methods contains a brief introduction of its general philosophy, accompanied by some theorems and ends with the connection to the work of Fisher. / Den här kandidatuppsatsen presenterar ett antal metoder för dimensionsreducering, där betoning läggs på metoder some följer teori utvecklad av R.A. Fisher. En del av metoderna som presenteras utvecklades redan på tidigt 1900-tal, medan andra är utvecklade i närtid. För metoderna utvecklade i närtid, så kommer större vikt läggas vid den grundläggande teorin för metoden. Presentationen av varje metod består av en kortare beskrivning, följt av satser och slutligen beskrivs dess koppling to Fishers teorier.
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On the generalization of subspace detection in unordered multidimensional data / Sobre a generalização da detecção de subespaços em dados multidimensionais não ordenadosFernandes, Leandro Augusto Frata January 2010 (has links)
Este trabalho apresenta uma solução geral para a detecção de alinhamentos de dados em conjuntos multidimensionais não ordenados e ruidosos. Nesta abordagem, o tipo requerido de alinhamento de dados pode ser uma forma geométrica (e.g., linha reta, plano, círculo, esfera, seção cônica, entre outras) ou qualquer estrutura, com dimensionalidade arbitrária, que possa ser caracterizada por um subespaço linear. A detecção é realizada por meio de um procedimento composto por três etapas. Na etapa de inicialização, um espaço de parâmetros com p (n − p) dimensões é definido de modo que cada ponto neste espaço represente uma instância do alinhamento requerido, descrito por um subespaço p-dimensional em um domínio n-dimensional. Em seguida, uma grade de acumuladores é criada como sendo a representação discreta do espaço de parâmetros. Na segunda etapa do procedimento, cada elemento no conjunto de dados de entrada (também um subespaço no domínio n-dimensional) é mapeado para o espaço de parâmetros como os pontos (no espaço de parâmetros) representando os subespaços requeridos que contém ou que estão contidos no elemento de entrada. À medida que os elementos de entrada são mapeados, as células do acumulador relacionadas com o mapeamento são incrementadas pelo valor de importância do elemento mapeado. A etapa final do procedimento recupera os subespaços p-dimensionais que melhor se ajustam aos dados de entrada como sendo os máximos locais na grade de acumuladores. A parametrização proposta é independente das propriedades geométricas dos alinhamentos a serem detectados. Além disso, o procedimento de mapeamento é independente do tipo de dado de entrada e é capaz de se adaptar a elementos com dimensionalidades arbitrárias. Essas características permitem a utilização da técnica (sem a necessidade de modificações) como uma ferramenta para a detecção de padrões em uma grande quantidade de aplicações. Por conta de sua natureza geral, otimizações desenvolvidas para a abordagem proposta beneficiam, de forma imediata, todos os casos de detecção. Neste trabalho eu demonstro uma implementação em software da técnica proposta e mostro que ela pode ser aplicada tanto em casos simples de detecção, quanto na detecção concorrente de tipos diferentes de alinhamentos, com diferentes interpretações geométricas e em conjuntos de dados compostos por vários tipos de elementos. Esta dissertação também apresenta uma extensão do esquema de detecção para dados de entrada com distribuição Gaussiana de incerteza. A extensão proposta produz distribuições de valores mais suaves na grade de acumuladores e faz com que a técnica fique mais robusta à detecção de subespaços espúrios. / This dissertation presents a generalized closed-form framework for detecting data alignments in large unordered noisy multidimensional datasets. In this approach, the intended type of data alignment may be a geometric shape (e.g., straight line, plane, circle, sphere, conic section, among others) or any other structure, with arbitrary dimensionality that can be characterized by a linear subspace. The detection is performed using a three-step process. In the initialization, a p (n − p)-dimensional parameter space is defined in such a way that each point in this space represents an instance of the intended alignment described by a p-dimensional subspace in some n-dimensional domain. In turn, an accumulator array is created as the discrete representation of the parameter space. In the second step each input entry (also a subspace in the n-dimensional domain) is mapped to the parameter space as the set of points representing the intended p-dimensional subspaces that contain or are contained by the entry. As the input entries are mapped, the bins of the accumulator related to such a mapping are incremented by the importance value of the entry. The subsequent and final step retrieves the p-dimensional subspaces that best fit input data as the local maxima in the accumulator array. The proposed parameterization is independent of the geometric properties of the alignments to be detected. Also, the mapping procedure is independent of the type of input data and automatically adapts to entries of arbitrary dimensionality. This allows application of the proposed approach (without changes) in a broad range of applications as a pattern detection tool. Given its general nature, optimizations developed for the proposed framework immediately benefit all the detection cases. I demonstrate a software implementation of the proposed technique and show that it can be applied in simple detection cases as well as in concurrent detection of multiple kinds of alignments with different geometric interpretations, in datasets containing multiple types of data. This dissertation also presents an extension of the general detection scheme to data with Gaussian-distributed uncertainty. The proposed extension produces smoother distributions of values in the accumulator array and makes the framework more robust to the detection of spurious subspaces.
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On the generalization of subspace detection in unordered multidimensional data / Sobre a generalização da detecção de subespaços em dados multidimensionais não ordenadosFernandes, Leandro Augusto Frata January 2010 (has links)
Este trabalho apresenta uma solução geral para a detecção de alinhamentos de dados em conjuntos multidimensionais não ordenados e ruidosos. Nesta abordagem, o tipo requerido de alinhamento de dados pode ser uma forma geométrica (e.g., linha reta, plano, círculo, esfera, seção cônica, entre outras) ou qualquer estrutura, com dimensionalidade arbitrária, que possa ser caracterizada por um subespaço linear. A detecção é realizada por meio de um procedimento composto por três etapas. Na etapa de inicialização, um espaço de parâmetros com p (n − p) dimensões é definido de modo que cada ponto neste espaço represente uma instância do alinhamento requerido, descrito por um subespaço p-dimensional em um domínio n-dimensional. Em seguida, uma grade de acumuladores é criada como sendo a representação discreta do espaço de parâmetros. Na segunda etapa do procedimento, cada elemento no conjunto de dados de entrada (também um subespaço no domínio n-dimensional) é mapeado para o espaço de parâmetros como os pontos (no espaço de parâmetros) representando os subespaços requeridos que contém ou que estão contidos no elemento de entrada. À medida que os elementos de entrada são mapeados, as células do acumulador relacionadas com o mapeamento são incrementadas pelo valor de importância do elemento mapeado. A etapa final do procedimento recupera os subespaços p-dimensionais que melhor se ajustam aos dados de entrada como sendo os máximos locais na grade de acumuladores. A parametrização proposta é independente das propriedades geométricas dos alinhamentos a serem detectados. Além disso, o procedimento de mapeamento é independente do tipo de dado de entrada e é capaz de se adaptar a elementos com dimensionalidades arbitrárias. Essas características permitem a utilização da técnica (sem a necessidade de modificações) como uma ferramenta para a detecção de padrões em uma grande quantidade de aplicações. Por conta de sua natureza geral, otimizações desenvolvidas para a abordagem proposta beneficiam, de forma imediata, todos os casos de detecção. Neste trabalho eu demonstro uma implementação em software da técnica proposta e mostro que ela pode ser aplicada tanto em casos simples de detecção, quanto na detecção concorrente de tipos diferentes de alinhamentos, com diferentes interpretações geométricas e em conjuntos de dados compostos por vários tipos de elementos. Esta dissertação também apresenta uma extensão do esquema de detecção para dados de entrada com distribuição Gaussiana de incerteza. A extensão proposta produz distribuições de valores mais suaves na grade de acumuladores e faz com que a técnica fique mais robusta à detecção de subespaços espúrios. / This dissertation presents a generalized closed-form framework for detecting data alignments in large unordered noisy multidimensional datasets. In this approach, the intended type of data alignment may be a geometric shape (e.g., straight line, plane, circle, sphere, conic section, among others) or any other structure, with arbitrary dimensionality that can be characterized by a linear subspace. The detection is performed using a three-step process. In the initialization, a p (n − p)-dimensional parameter space is defined in such a way that each point in this space represents an instance of the intended alignment described by a p-dimensional subspace in some n-dimensional domain. In turn, an accumulator array is created as the discrete representation of the parameter space. In the second step each input entry (also a subspace in the n-dimensional domain) is mapped to the parameter space as the set of points representing the intended p-dimensional subspaces that contain or are contained by the entry. As the input entries are mapped, the bins of the accumulator related to such a mapping are incremented by the importance value of the entry. The subsequent and final step retrieves the p-dimensional subspaces that best fit input data as the local maxima in the accumulator array. The proposed parameterization is independent of the geometric properties of the alignments to be detected. Also, the mapping procedure is independent of the type of input data and automatically adapts to entries of arbitrary dimensionality. This allows application of the proposed approach (without changes) in a broad range of applications as a pattern detection tool. Given its general nature, optimizations developed for the proposed framework immediately benefit all the detection cases. I demonstrate a software implementation of the proposed technique and show that it can be applied in simple detection cases as well as in concurrent detection of multiple kinds of alignments with different geometric interpretations, in datasets containing multiple types of data. This dissertation also presents an extension of the general detection scheme to data with Gaussian-distributed uncertainty. The proposed extension produces smoother distributions of values in the accumulator array and makes the framework more robust to the detection of spurious subspaces.
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