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The number of high-dimensional datasets recording multiple aspects of a single phenomenon is ever increasing in many areas of science. This is accompanied by a fundamental need for mathematical frameworks that can compare data tabulated as multiple large-scale matrices of di erent numbers of rows. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. This thesis addresses this limitation and de fines a higher-order GSVD (HO GSVD) of N > 2 datasets, that provides a mathematical framework that can compare multiple high-dimensional datasets tabulated as large-scale matrices of different numbers of rows. / text

Made available in DSpace on 2015-03-03T15:32:43Z (GMT). No. of bitstreams: 1 IvanildoFP_DISSERT.pdf: 6193808 bytes, checksum: 2b4b204c68da306ef20f2a99dc91d9c9 (MD5) Previous issue date: 2013-09-20 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In general, an inverse problem corresponds to find a value of an element x in a suitable vector space, given a vector y measuring it, in some sense. When we discretize the problem, it usually boils down to solve an equation system f(x) = y, where f : U Rm ! Rn represents the step function in any domain U of the appropriate Rm. As a general rule, we arrive to an ill-posed problem. The resolution of inverse problems has been widely researched along the last decades, because many problems in science and industry consist in determining unknowns that we try to know, by observing its effects under certain indirect measures. Our general subject of this dissertation is the choice of Tykhonov?s regulaziration parameter of a poorly conditioned linear problem, as we are going to discuss on chapter 1 of this dissertation, focusing on the three most popular methods in nowadays literature of the area. Our more specific focus in this dissertation consists in the simulations reported on chapter 2, aiming to compare the performance of the three methods in the recuperation of images measured with the Radon transform, perturbed by the addition of gaussian i.i.d. noise. We choosed a difference operator as regularizer of the problem. The contribution we try to make, in this dissertation, mainly consists on the discussion of numerical simulations we execute, as is exposed in Chapter 2. We understand that the meaning of this dissertation lays much more on the questions which it raises than on saying something definitive about the subject. Partly, for beeing based on numerical experiments with no new mathematical results associated to it, partly for being about numerical experiments made with a single operator. On the other hand, we got some observations which seemed to us interesting on the simulations performed, considered the literature of the area. In special, we highlight observations we resume, at the conclusion of this work, about the different vocations of methods like GCV and L-curve and, also, about the optimal parameters tendency observed in the L-curve method of grouping themselves in a small gap, strongly correlated with the behavior of the generalized singular value decomposition curve of the involved operators, under reasonably broad regularity conditions in the images to be recovered / Problemas inversos, usualmente recaem em resolver alguma equa??o do tipo f(x) = b, onde cada equa??o fi(x) = bi pode ser pensada como uma medida de um dado x a ser recuperado. Usualmente s?o mal postos, no sentido de corresponderem a equa??es que podem n?o ter solu??o exata, podem ainda ter muitas solu??es, ou ainda, o que ? o mais comum, ter solu??es muito inst?veis a ru?dos na obten??o de b. H? v?rias formas de regularizar a obten??o de solu??es de tais problemas e a mais popular seria a de Tykhonov, que corresponde a: Minimizar ||f(x) b||2 + l ||L(x x0) ||2 (I) A regulariza??o pretendida corresponde a se escolher o operador l, de tal forma que o problema I tenha solu??es est?veis com perturba??es em b e que aproximem solu??es do problema de m?nimos quadrados usual, no caso de se fazer l 0. O primeiro termo de (I) representa o ajuste aos dados e o segundo termo penaliza a solu??o de forma a regularizar o problema e produzir uma solu??o est?vel a ru?dos. Se l = 0, isto significa que estamos procurando uma solu??o de quadrados m?nimos para o problema, o que usualmente ? insuficiente para problemas mal postos. O termo de regulariza??o adicionado introduz um vi?s na solu??o ao penalizar o ajuste com um termo adicional. Se L for a identidade, por exemplo, isto significa que estamos apostando que a solu??o estaria relativamente pr?xima de x0. Se L for o operador gradiente, estamos apostando que a solu??o x ? razoavelmente suave. Nas aplica??es, L usualmente ? escolhido como um operador adaptado ao problema estudado e de forma se valer de informa??es a priori dispon?veis sobre as solu??es procuradas. A escolha do par?metro l > 0 ? crucial neste m?todos, pelo fato que se l ? excessivo, isto tende a enfraquecer excessivamente o ajuste aos dados, induzindo um ajuste da solu??o ? x0. Se l for pequeno demais a regulariza??o pretendida acaba n?o acontecendo e a solu??o do problema (I) usualmente acaba ficando muito inst?vel e contaminada por ru?dos. H? v?rias t?cnicas dispon?veis na literatura para tal escolha, sobretudo se f ? uma fun??o linear f(x) = Ax. O objetivo da disserta??o ? o de estudar algumas destas t?cnicas de ajuste do par?metro l no caso de operadores discretizados, vale dizer, x no Rn. Em especial, destacamos os m?todos de ajuste do par?metro l reconhecidos na literatura como L-curve, GCV e m?todo da discrep?ncia, e objetiva-se comparar estes m?todos em testes feitos com a transformada de Radon e tendo como regularizador um operador de derivada de primeira ordem. Os resultados dos testes realizados revelam pontos interessantes na rela??o entre os diferentes estimadores para o par?metro de regulariza??o e que sugerem um aprofundamento te?rico al?m do escopo desta disserta??o

<p dir="ltr">Acoustic holography is a powerful tool in the visualization of sound fields and sound sources. It provides engineers and researchers clear insights into sound fields as well as their sound sources. Some widely-used methods include Nearfield Acoustical Holography (NAH), Statistically Optimized Nearfield Acoustic Holography (SONAH) and the Equivalent Source Method (ESM). SONAH and ESM were developed specifically to tackle the intrinsic deficiency of the Fourier-based NAH which requires that the sound field fall to negligible levels at the edges of the measurement aperture, a requirement rarely met in practice. Besides the aforementioned methods, the Inverse Boundary Element Method (IBEM) can be used, given sufficient measurements and computational resources. As useful as they are in visualizing the sound field, none of these methods can provide direct guidance on potential design modifications of the observed structure in order to unequivocally reduce sound power radiation. Acoustic radiation mode analysis has previously been primarily associated with active noise control applications. Since the radiation modes radiate sound power independently, it is only necessary to modify the surface vibration patterns so that they do not couple well with the radiation modes in order to guarantee a reduction of the radiated sound power. Since the radiation modes are orthogonal and complete, they can be used as the basis functions through which the source surface vibration can be described. Therefore, an acoustic holography method based on the acoustic radiation modes will enable the sound power ranking of the modal components of the surface vibration pattern, and in turn, point out the component(s) which should be targeted in order to reduce the overall sound power. However, use of the acoustic radiation modes in the inverse procedure comes with a price: the detailed geometry of the object to be measured must be obtained, thus enabling the calculation of acoustic radiation modes and the modal pressures. But this is not an issue for original equipment manufacturers given that almost all prototypes are now designed with CAD, as is the case with the engine example to be described next.</p><p dir="ltr">In modern engine design, downsizing and reducing weight while still providing an increased amount of power has been a general trend in recent decades. Traditionally, an engine design with superior NVH performance usually comes with a heavier, thus sturdier structure. Therefore, modern engine design requires that NVH be considered in the very early design stage to avoid modifications of engine structures at the last minute, when very few changes can be made. NVH design optimization of engine components has become more practical due to the development of computer software and hardware. However, there is still a need for smarter algorithms to draw a direct relationship between the design and the radiated sound power. At the moment, techniques based on modal acoustic transfer vectors (MATVs) have gained popularity in design optimization for their good performance in sound pressure prediction. Since MATVs are derived based on structural modes, they are not independent with respect to radiated sound power. In contrast, as noted, acoustic radiation modes are an orthogonal set of velocity distributions on the structure’s surface that contribute to the radiated sound power independently. As a result, it is beneficial to describe structural vibration in terms of acoustic radiation modes in order to identify the velocity distributions that contribute the majority of the radiated sound power. Measures can then be taken to modify the identified vibration patterns to reduce their magnitudes, which will in turn result in an unequivocal reduction of the radiated sound power. A workflow of the structural optimization procedure is proposed in this dissertation.</p><p dir="ltr">While acoustic radiation modes have great efficiencies in describing radiated acoustic power, the computation of acoustic radiation modes can be time consuming. In the last chapter of this thesis, a novel way of calculating acoustic radiation modes is proposed, which differs from the traditional singular value decomposition of the power radiation resistance matrix, and which is more efficient than previously proposed procedures. </p><p><br></p>

Indiana University-Purdue University Indianapolis (IUPUI) / Magnetic resonance (MR) spectroscopy is a neuroimaging technique. It is widely used to quantify the concentration of important metabolites in a brain tissue. Imbalance in concentration of brain metabolites has been found to be associated with development of neurological impairment. There has been increasing trend of using MR spectroscopy as a diagnosis tool for neurological disorders. We established statistical methodology to analyze data obtained from the MR spectroscopy in the context of the HIV associated neurological disorder. First, we have developed novel methodology to study the association of marker of neurological disorder with MR spectrum from brain and how this association evolves with time. The entire problem fits into the framework of scalar-on-function regression model with individual spectrum being the functional predictor. We have extended one of the existing cross-sectional scalar-on-function regression techniques to longitudinal set-up. Advantage of proposed method includes: 1) ability to model flexible time-varying association between response and functional predictor and (2) ability to incorporate prior information. Second part of research attempts to study the influence of the clinical and demographic factors on the progression of brain metabolites over time. In order to understand the influence of these factors in fully non-parametric way, we proposed LongCART algorithm to construct regression tree with longitudinal data. Such a regression tree helps to identify smaller subpopulations (characterized by baseline factors) with differential longitudinal profile and hence helps us to identify influence of baseline factors. Advantage of LongCART algorithm includes: (1) it maintains of type-I error in determining best split, (2) substantially reduces computation time and (2) applicable even observations are taken at subject-specific time-points. Finally, we carried out an in-depth analysis of longitudinal changes in the brain metabolite concentrations in three brain regions, namely, white matter, gray matter and basal ganglia in chronically infected HIV patients enrolled in HIV Neuroimaging Consortium study. We studied the influence of important baseline factors (clinical and demographic) on these longitudinal profiles of brain metabolites using LongCART algorithm in order to identify subgroup of patients at higher risk of neurological impairment. / Partial research support was provided by the National Institutes of Health grants U01-MH083545, R01-CA126205 and U01-CA086368