Thesis (M.Oc.E.)--Oregon State University, 2011. / Printout. Includes bibliographical references (leaf 70). Also available on the World Wide Web.

Orientador: Diego Sebastian Ledesma / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T14:31:48Z (GMT). No. of bitstreams: 1 YalandaMuelas_YamitYesid_M.pdf: 7930295 bytes, checksum: 93c5826f6c164fed94abacbb42db5256 (MD5) Previous issue date: 2012 / Resumo: Nesta dissertação apresentamos generalizações de três resultados muito conhecidos em geometria Riemanniana: o Teorema de Myers, o Teorema de Bochner e o Teorema de decomposição de Cheeger-Gromoll. Em particular veremos que fazendo uma pequena modificação sobre os requisitos destes teoremas no que se refere ao tensor de Ricci, os resultados permanecem inalterados / Abstract: In this dissertation we present generalizations of three well-known results in Riemannian geometry: The Myers's theorem, Bochner's theorem and the Cheeger-Gromoll splitting theorem. In particular, we will prove that making a small modification of the requirements of these theorems related to the Ricci tensor, the results remain unchanged / Mestrado / Matematica / Mestre em Matemática

The interdisciplinary field of machine learning studies algorithms in which functionality is dependent on data sets. This data is often treated as a matrix, and a variety of mathematical methods have been developed to glean information from this data structure such as matrix decomposition. The Laplacian matrix, for example, is commonly used to reconstruct networks, and the eigenpairs of this matrix are used in matrix decomposition. Moreover, concepts such as SVD matrix factorization are closely connected to manifold learning, a subfield of machine learning that assumes the observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. Since many data sets have natural higher dimensions, tensor methods are being developed to deal with big data more efficiently. This thesis builds on these ideas by exploring how matrix methods can be extended to data presented as tensors rather than simply as ordinary vectors.

This thesis consists of four chapters. In Chapter 1, we focus on a class of transportation inequalities known as the transportation-information inequalities. These inequalities bound optimal transportation costs in terms of relative Fisher information, and are known to characterize certain concentration properties of Markov processes around their invariant measures. We provide a characterization of the quadratic transportation-information inequality in terms of a dimension-free concentration property for i.i.d. copies of the underlying Markov process, identifying the precise high-dimensional concentration property encoded by this inequality. We also illustrate how this result is an instance of a general convex-analytic tensorization principle. In Chapter 2, we study the entropic gradient flow property of McKean--Vlasov diffusions via a stochastic analysis approach. We formulate a trajectorial version of the relative entropy dissipation identity for these interacting diffusions, which describes the rate of relative entropy dissipation along every path of the diffusive motion. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality. In Chapter 3, we further extend the trajectorial approach to a class of degenerate diffusion equations that includes the porous medium equation. These equations are posed on a bounded domain and are subject to no-flux boundary conditions, so that their corresponding probabilistic representations are stochastic differential equations with normal reflection on the boundary. Our stochastic analysis approach again leads to a new derivation of the Wasserstein gradient flow property for these nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context. Finally, in Chapter 4, we turn our attention to mean field approximation -- a method widely used to study the behavior of large stochastic systems of interacting particles. We propose a new approach to deriving quantitative mean field approximations for any strongly log-concave probability measure. Our framework is inspired by the recent theory of nonlinear large deviations, for which we offer an efficient non-asymptotic perspective in log-concave settings based on functional inequalities. We discuss three implications, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems.

This dissertation seeks to describe advancements made in computational methods for multi-messenger astrophysics (MMA) using gravitational waves GW and neutrinos during Advanced LIGO (aLIGO)’s first through third observing runs (O1-O3) and, looking forward, to describe novel computational techniques suited to the challenges of both the burgeoning MMA field and high-performance computing as a whole. The first two chapters provide an overview of MMA as it pertains to gravitational wave/high energy neutrino (GWHEN) searches, including a summary of expected astrophysical sources as well as GW, neutrino, and gamma-ray detectors used in their detection. These are followed in the third chapter by an in-depth discussion of LIGO’s timing system, particularly the diagnostic subsystem, describing both its role in MMA searches and the author’s contributions to the system itself. The fourth chapter provides a detailed description of the Low-Latency Algorithm for Multi-messenger Astrophysics (LLAMA), the GWHEN pipeline developed by the author and used in O2 and O3. Relevant past multi-messenger searches are described first, followed by the O2 and O3 analysis methods, the pipeline’s performance, scientific results, and finally, an in-depth account of the library’s structure and functionality. In particular, the author’s high-performance multi-order coordinates (MOC) HEALPix image analysis library, HPMOC, is described. HPMOC increases performance of HEALPix image manipulations by several orders of magnitude vs. naive single-resolution approaches while presenting a simple high-level interface and should prove useful for diverse future MMA searches. The performance improvements it provides for LLAMA are also covered. The final chapter of this dissertation builds on the approaches taken in developing HPMOC, presenting several novel methods for efficiently storing and analyzing large data sets, with applications to MMA and other data-intensive fields. A family of depth-first multi-resolution ordering of HEALPix images — DEPTH9, DEPTH19, and DEPTH40 — is defined, along with algorithms and use cases where it can improve on current approaches, including high-speed streaming calculations suitable for serverless compute or FPGAs. For performance-constrained analyses on HEALPix data (e.g. image analysis in multi-messenger search pipelines) using SIMD processors, breadth-first data structures can provide short-circuiting calculations in a data-parallel way on compressed data; a simple compression method is described with application to further improving LLAMA performance. A new storage scheme and associated algorithms for efficiently compressing and contracting tensors of varying sparsity is presented; these demuxed tensors (D-Tensors) have equivalent asymptotic time and space complexity to optimal representations of both dense and sparse matrices, and could be used as a universal drop-in replacement to reduce code complexity and developer effort while improving performance of existing non-optimized numerical code. Finally, the big bucket hash table (B-Table), a novel type of hash table making guarantees on data layout (vs. load factor), is described, along with optimizations it allows for (like hardware acceleration, online rebuilds, and hard realtime applications) that are not possible with existing hash table approaches. These innovations are presented in the hope that some will prove useful for improving future MMA searches and other data-intensive applications.

Selection of an appropriate structure for storage and analysis of complex datasets is a vital but often overlooked decision in the design of data mining and machine learning experiments. Most present techniques impose a matrix structure on the dataset, with rows representing observations and columns representing features. While this assumption is reasonable when features are scalar and do not exhibit co-dependence, the matrix data model becomes inappropriate when dependencies between non-target features must be modeled in parallel, or when features naturally take the form of higher-order multilinear structures. Such datasets particularly abound in functional medical imaging modalities, such as fMRI, where accurate integration of both spatial and temporal information is critical. Although necessary to take full advantage of the high-order structure of these datasets and built on well-studied mathematical tools, tensor analysis methodologies have only recently entered widespread use in the data mining community and remain relatively absent from the literature within the biomedical domain. Furthermore, naive tensor approaches suffer from fundamental efficiency problems which limit their practical use in large-scale high-order mining and do not capture local neighborhoods necessary for accurate spatiotemporal analysis. To address these issues, a comprehensive framework based on wavelet analysis, tensor decomposition, and the WaveCluster algorithm is proposed for addressing the problems of preprocessing, classification, clustering, compression, feature extraction, and latent concept discovery on large-scale high-order datasets, with a particular emphasis on applications in computer-assisted diagnosis. Our framework is evaluated on a 9.3 GB fMRI motor task dataset of both high dimensionality and high order, performing favorably against traditional voxelwise and spectral methods of analysis, discovering latent concepts suggestive of subject handedness, and reducing space and time complexities by up to two orders of magnitude. Novel wavelet and tensor tools are derived in the course of this work, including a novel formulation of an r-dimensional wavelet transform in terms of elementary tensor operations and an enhanced WaveCluster algorithm capable of clustering real-valued as well as binary data. Sparseness-exploiting properties are demonstrated and variations of core algorithms for specialized tasks such as image segmentation are presented. / Computer and Information Science

We use molecular mechanics (MM) simulations with the tight-binding (TB) potential to study local and global instabilities in initially defect-free finite specimens of gold crystals deformed in shear, simple shear, tension/compression, simple tension/compression, and triaxial tension/compression. The criteria used to delineate local instabilities in a system include the following: (i) a second order spatial derivative of the displacement field having large values relative to its average value in the body, (ii) the minimum eigenvalue of the Hessian of the potential energy of an atom becoming nonpositive, (iii) and structural changes represented by a high value of the common neighborhood parameter. A specimen becomes globally unstable when its potential energy decreases significantly with a small increase in its deformations. It is found that the three criteria for local instability are satisfied essentially simultaneously at the same atomic position. Deformations of a specimen are quite different when it is deformed with some bounding surfaces free from external forces as opposed to essential boundary conditions prescribed on all bounding surfaces. It is found that the initial unloaded configuration (or the reference configuration) of the minimum potential energy has significant in-plane stresses on the bounding surfaces and nonzero normal stresses at interior points. In tensile/compressive deformations of a rectangular prismatic nanobar the yield stress defined as the average axial stress when the average axial stress vs. the average axial strain curve exhibits a sharp discontinuity depends upon the specimen size; a similar result holds for simulations of shear deformations. Specimens deformed with essential boundary conditions on all bounding surfaces experience instabilities at a higher value of the average strain than identical specimens deformed similarly but with one or more pairs of opposite bounding surfaces traction free. For the former set of deformations, the response of a specimen prior to the onset of instability is the same as that of a hyperelastic body with the strain energy derived from the TB potential and deformations obeying the Cauchy-Born rule. Specimens with some traction free bounding surfaces experience local instabilities prior to the onset of a global instability but the two instabilities occur simultaneously in specimens with essential boundary conditions prescribed on all bounding surfaces. It is believed that because of residual stresses in the reference configuration, the average axial stress at yield in compression is nearly one-half of that in tension. / Ph. D.

The results of a systematic 35Cl, 81Br, and 127I SSNMR spectroscopic study of a series of halogen-substituted anilinium halide salts are presented. Solid-state NMR of these nuclides, bromine-/81 and iodine-127 in particular, is not well established. Twenty-one compounds thought to exhibit halogen bonding were prepared based on modified literature procedures, and two crystal structures were solved. Experiments show that collection of SSNMR spectra of the anions is feasible, though ultrahigh magnetic fields (21.1 T) and variable offset data acquisition were found to be essential. Electric field gradient and chemical shift tensors are measured experimentally for all 21 compounds, significantly expanding the body of data for the quadrupolar halogen nuclei. Quadrupolar coupling constants for chlorine-35 ranged from 2.12 to 6.04 MHz, for bromine-81 from 12.3 to 45.3 MHz, and for iodine-127 from 57.50 to 152.50 MHz. Gauge-including projector-augmented wave density functional theory (GIPAW-DFT) calculations were used to provide insight as to how the NMR parameters vary with local environment and long-range crystal packing. Overall, calculations reproduced the experimental trends in quadrupolar coupling constants and chemical shift tensor span (Ω) but failed to provide quantitative agreement within experimental error. Experimental and computational data were analyzed in order to provide insight into how halogen bonding influences NMR parameters. Several trends were elucidated from this study, including an inverse correlation between Ω and the length of the shortest halogen-halide contact (d). In selected bromine compounds, for example, Ω (81Br) was measured to increase from 120 to 240 ppm as d decreased from 3.838 to 3.443 Å. In summary, this study has demonstrated the feasibility and utility of quadrupolar halogen SSNMR, and that these techniques may prove useful in characterizing halogen bonding interactions in solids.

This thesis examines the advantages to nonlinear least-squares (NLS) fitting of diffusion-weighted MRI data over the commonly used linear least-squares (LLS) approach. A modified fitting algorithm is proposed which accounts for the positive bias experienced in magnitude images at low SNR. For b-values in the clinical range (~1000 s/mm2), the increase in precision of FA and fibre orientation estimates is almost negligible, except at very high anisotropy. The optimal b-value for estimating tensor parameters was slightly higher for NLS. The primary advantage to NLS was improved performance at high b-values, for which complex fibre architectures were more easily resolved. This was demonstrated using a model-selection classifier based on higher-order diffusion models. Using a b-value of 3000 s/mm2 and magnitude-corrected NLS fitting, at least 10% of voxels in the brain exhibited diffusion profiles which could not be represented by the tensor model.

This thesis examines the advantages to nonlinear least-squares (NLS) fitting of diffusion-weighted MRI data over the commonly used linear least-squares (LLS) approach. A modified fitting algorithm is proposed which accounts for the positive bias experienced in magnitude images at low SNR. For b-values in the clinical range (~1000 s/mm2), the increase in precision of FA and fibre orientation estimates is almost negligible, except at very high anisotropy. The optimal b-value for estimating tensor parameters was slightly higher for NLS. The primary advantage to NLS was improved performance at high b-values, for which complex fibre architectures were more easily resolved. This was demonstrated using a model-selection classifier based on higher-order diffusion models. Using a b-value of 3000 s/mm2 and magnitude-corrected NLS fitting, at least 10% of voxels in the brain exhibited diffusion profiles which could not be represented by the tensor model.