Relativistic effects are extremely important for heavy atoms and heavy atom containing molecules. Therefore, a relativistic treatment is needed when calculating molecular properties of these species. The fully- relativistic Dirac treatment involves electronic and positronic wavefunctions and a very large basis set is required. This leads to calculations that are too costly and time-consuming for larger molecules. The Zeroth-Order Regular Approximation (ZORA) is an approximation to the Dirac approach, which only deals with the electronic wavefunction. However, unfortunately this method is plagued by the gauge-dependence problem. The gauge-independent ZORA (ZORA-GI) and strictly atomic ZORA approaches provide solutions to this problem.In this work, the ZORA-GI and strictly atomic ZORA codes have been successfully implemented into the Gaussian 09 program. They have been used to calculate the bond lengths, harmonic vibrational frequencies and dissociation energies of the I2, Au2 and Pt2 diatomic molecules. The results show good agreement with experiment and previous theoretical studies. The non-relativistic, ZORA-GI, strictly atomic ZORA and pseudopotential approximations have been used to investigate the electronic structure of the actinide monoxides, AnO, and actinide monoxide cations, AnO+ (An = Th – Cm). It was found that the ground state configurations were dependent on the relativistic approximation chosen. The bond lengths, harmonic vibrational frequencies and dissociation energies were also calculated, with the ZORA methods generally outperforming the pseudopotential approximation. The first theoretical g-tensor study of the organouranium(V) complexes [U(C7H7)2]-, [U(η8-C8H8)(NEt2)(THF)]+, [U(η5-C5H5)(NMe2)3(THF)]+, [U(η8-C8H8)(NEt2)3], [U(η5-C5H5)2(NEt2)2]+ and [U(η8-C8H8)(η5-C5H5)(NEt2)2] has been carried out. It was demonstrated that the choice of density functional affects the way in which the g-tensor axes are assigned. The ground state spin density and SOMO are also sensitive to the choice of density functional. It is these factors that determine the value of the g-tensor.

The aim of this thesis is to systematically study the statistical guarantee for two representative non-convex optimization problems arsing in the statistics community. The first one is the high-dimensional Gaussian mixture model, which is motivated by the estimation of multiple graphical models arising from heterogeneous observations. The second one is the low-rank tensor estimation model, which is motivated by high-dimensional interaction model. Both optimal statistical rates and numerical comparisons are studied in depth. In the first part of my thesis, we consider joint estimation of multiple graphical models arising from heterogeneous and high-dimensional observations. Unlike most previous approaches which assume that the cluster structure is given in advance, an appealing feature of our method is to learn cluster structure while estimating heterogeneous graphical models. This is achieved via a high dimensional version of Expectation Conditional Maximization (ECM) algorithm. A joint graphical lasso penalty is imposed on the conditional maximization step to extract both homogeneity and heterogeneity components across all clusters. Our algorithm is computationally efficient due to fast sparse learning routines and can be implemented without unsupervised learning knowledge. The superior performance of our method is demonstrated by extensive experiments and its application to a Glioblastoma cancer dataset reveals some new insights in understanding the Glioblastoma cancer. In theory, a non-asymptotic error bound is established for the output directly from our high dimensional ECM algorithm, and it consists of two quantities: statistical error (statistical accuracy) and optimization error (computational complexity). Such a result gives a theoretical guideline in terminating our ECM iterations. In the second part of my thesis, we propose a general framework for sparse and low-rank tensor estimation from cubic sketchings. A two-stage non-convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent, which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability. The non-asymptotic analysis sheds light on an interplay between optimization error and statistical error. The proposed procedure is shown to be rate-optimal under certain conditions. As a technical by-product, novel high-order concentration inequalities are derived for studying high-moment sub-Gaussian tensors. An interesting tensor formulation illustrates the potential application to high-order interaction pursuit in high-dimensional linear regression

Background. Major depressive disorder (MDD) is a leading cause of death and disability worldwide, and many individuals with MDD will experience treatment-resistant depression (TRD). TRD can lead to the development of suicidal ideation and behaviours, and up to 30% of people with refractory depression will attempt suicide at some point in their life. A neurobiological understanding of suicide is lacking, and neuroimaging markers of illness may elucidate the relationship between suicidal ideation and attempt. Diffusion tensor imaging (DTI) is a particularly sensitive neuroimaging modality that quantifies the microstructural integrity of white matter tracts, which may be useful in the investigation of psychiatric disease. The source of white matter changes may be further elucidated using free water imaging to isolate signal specific to the fibre tract and quantify the fractional volume of the free water compartment. Methodology. For this study, data were obtained from N=36 outpatients with TRD (n=20 suicide ideators and n=16 suicide attempters). Clinical characteristics of the patient sample were examined using clinician-rated and self-report questionnaires of depression and suicidal ideation severity. Whole-brain analysis of DTI data was conducted using tract-based spatial statistics (TBSS) via FMRIB Software Library (FSL) to identify between-group differences in white matter microstructure between suicide ideators and attempters. Free water imaging correction was applied through estimation of a constrained bi-tensor model via an in house MatLab-based script developed at Harvard University. Between-group differences of suicide ideators versus attempters were identified at a family-wise error (FWE) corrected significance threshold of p≤0.05. Subsequent exploratory analyses were performed at an uncorrected significance threshold of p≤0.01. Results. Suicide attempters had greater family history of suicide attempt, higher self-reported suicidal ideation severity, and were more likely to have received overnight treatment in a psychiatric facility in the past. TBSS revealed elevated mean diffusivity (MD), axial diffusivity (AD) and free water (FW) in suicide attempters compared to suicide ideators (thresholded p=<0.05, family-wise error corrected). Subsequent exploratory analyses revealed reduced fractional anisotropy (FA) and elevated radial diffusivity (RD) in fronto-thalamo-limbic white matter tracts of suicide attempters (thresholded p=<0.01, uncorrected). Free water correction appeared to increase detection of fractional anisotropy changes and suppress spurious differences in axial and radial diffusivity. Conclusion. The identification of significantly altered diffusion metrics in suicide attempters compared to suicide ideators suggests white matter pathology in TRD and suicide attempt. The effect of free water correction on diffusion metrics and the elevation of free water itself provide evidence toward the source of anisotropic changes. Future investigations to explore the combined impact of these measures in suicide and depression are recommended.

Indiana University-Purdue University Indianapolis (IUPUI) / Knowledge base completion is an important task that allows scientists to reason over knowledge bases and discover new facts. In this thesis, a patient-centric knowledge base is designed and constructed using medical entities and relations extracted from the health forum r/cancer. The knowledge base stores information in binary relation triplets. It is enhanced with an is-a relation that is able to represent the hierarchical relationship between different medical entities. An enhanced Neural Tensor Network that utilizes the frequency of occurrence of relation triplets in the dataset is then developed to infer new facts from the enhanced knowledge base. The results show that when the enhanced inference model uses the enhanced knowledge base, a higher accuracy (73.2 %) and recall@10 (35.4%) are obtained. In addition, this thesis describes a methodology for knowledge base and associated inference model design that can be applied to other chronic diseases.

HCP materials are exceedingly being used as alloys and composites in several high strength light weight applications such as aerospace and aeronautical structures, deep sea maritime applications, and as biocompatible materials. To understand the deformation of HCP materials, reliable tools and techniques are required. One such technique is the Elasto-Plastic Self Consistency (EPSC) model. ESPC models use Eshelby’s Inclusion Theory as their basic formulation to model the strain experienced by a grain within a strained material sample. One of the oldest approximations (or models) used to model the grain’s strain within a strained sample is the Taylor’s Assumption (TA). TA assumes that each grain is strained to the same average value. EPSC models are different from the TA model since each grain modelled by the EPSC model would be strained to a different value. This is possible and obtained by solving an infinite domain boundary value problem. This key advantage of the EPSC model can therefore predict localized weak spots within material samples.EPSC models use the concept of eigen strain where the inhomogeneous grain is replaced with an equivalent inclusion. The technique proposed in this research is used to simulate uniaxial tension of rolled textured Magnesium. The number of deformation modes used in this research is seven. Both slipping systems and twinning systems are included in the simulation. The hardening phenomenon is described as a function of self-hardening as well as latent-hardening. As stated in (S. Kweon, 2020), modelling the interactive hardening requires a more robust numerical iterative technique. An improved robust iterative numerical technique is explained in (Daniel Raja, 2021) and (Soondo Kweon D. S., 2021). This research implements the equivalent inclusion theory in combination with the numerical iterative technique developed in the aforementioned papers.The report begins with the need for this research and advocates for the same. Then, the conceptional theories and the imaginary thought experiment performed by John D. Eshelby is presented. The concept of “Eigen Strain” which serves as the base work needed to understand and formulate the Equivalent Inclusion Theory is described in detail. The Equivalent Inclusion is then presented and developed. The concept of Green’s Function is presented and explained. These concepts serve as the building block for the derivation and calculation of the Eshelby Tensor which relates the concepts of eigen strain and constrained strain. The report concludes the theory section with the amalgamation of the ideas of the Green’s Function and Eigen Strain to develop the Eshelby Tensor for an Isotropic material as well as Anisotropic materials. In the following section, the unit cell accompanied with the deformation modes within the unit cell of an HCP material that are used in these simulations are presented. Following unit cell model, the crystal plasticity model which includes plastic deformation, hardening laws, and elastic deformation is elaborated. The results obtained from the simulation are presented and salient features are highlighted that are observed in the results. Lastly, the report concludes by pointing out key “take aways” from this research and identifies possible avenues for future research.Additionally, ten appendices are included towards the end of this report to enhance understanding of complicated derivations and solutions. Lastly, the author’s vita is included at the end of the report.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20921号 / 理博第4373号 / 新制||理||1628(附属図書館) / 京都大学大学院理学研究科地球惑星科学専攻 / (主査)教授 大志万 直人, 准教授 吉村 令慧, 教授 中西 一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

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This thesis studies data-parallelism in tensor assignments. Building on an existent domain specific language for tensor calculations developed at the Chair of Compiler Construction, an extension is proposed to detect so called compatible statements, which describe when a statement is data-parallel. Using a type system, the correctness is shown and a conjecture about the optimality is proposed. By applying the extension, two optimizations for exploiting the data-parallelism are described. These optimizations reduce the memory usage for computation, therefore reducing cache misses and improving runtime. The speedup which can be gained mostly depends on the complexity of the kernel and the size of the tensors. For simple kernels like multiplication of a vector with a scalar or elementwise multiplication of two vectors, a speedup of up to 2x can be achieved. For more complex kernels like a kernel containing matrix-matrix multiplication, the speed difference is a few percent. Additionally, a kernel called interpolation consisting of incompatible statements is analysed to check whether a similar optimization can be applied. The result is that while the optimization does not result in a speedup, similar improvements might be possible. Finally, in order to gain a better understanding of what effect the optimizations might have, different kernels are analysed regarding the data size at which parallelism makes sense.:1 Introduction 1.1 Parallelization 1.2 Existing DSLs and compilers 2 Background 2.1 Tensors and tensor operations 2.2 A language for tensor manipulation 3 Compatible statements 3.1 Detecting compatible statements 4 Extension of the DSL 5 Correctness of the extension 6 Performance evaluation 6.1 Copy vs. in-place (avoid-copy) 6.2 Other variable vs. in-place (reduce-cache-miss) 6.2.1 Explanation of the optimization 6.2.2 Measuring the impact 6.3 Memory reusing for incompatible statements 7 Evaluation of data sizes for parallelization 8 Summary 9 Outlook Appendices

A material model capable of reproducing the anisotropic behavior of polycrystalline materials will prove to be useful in simulations in which directional properties are of key importance. The primary contributor to anisotropic behavior in polycrystalline materials is the development of texture through the rotation and alignment of slip systems due to plastic deformation. A large concentration of aligned slip systems will influence the glide of dislocations in the respective global deformation direction resulting in a directionally dependent flow stress. The Evolving Microstructural Model of Inelasticity (EMMI) is modified to account for evolving anisotropy due to the development of texture. Texture is characterized via a second order orientation tensor and is incorporated into EMMI through various modifications to the EMMI equations based on physical assumptions. Evolving anisotropy is captured via a static yield surface through a modification to the flow rule based on the assumption loading is entirely elastic within the yield surface. A separate modification to EMMI captures evolving anisotropy through an apparent yield surface via a modification to the EMMI internal state variable evolution equations. The apparent yield surface is the result of a smaller yield surface translating through stress space and assumes the state of the material is disturbed at stresses much lower than indicated by experimental yield surfaces.

X-ray dark-field imaging allows to resolve the conflict between the demand for centimeter scaled fields of view and the spatial resolution required for the characterization of fibrous materials structured on the micrometer scale. It draws on the ability of X-ray Talbot interferometers to provide full field images of a sample's ultra small angle scattering properties, bridging a gap of multiple orders of magnitude between the imaging resolution and the contrasted structure scale. The correspondence between shape anisotropy and oriented scattering thereby allows to infer orientations within a sample's microstructure below the imaging resolution. First demonstrations have shown the general feasibility of doing so in a tomographic fashion, based on various heuristic signal models and reconstruction approaches. Here, both a verified model of the signal anisotropy and a reconstruction technique practicable for general imaging geometries and large tensor valued volumes is developed based on in-depth reviews of dark-field imaging and tomographic reconstruction techniques. To this end, a wide interdisciplinary field of imaging and reconstruction methodologies is revisited. To begin with, a novel introduction to the mathematical description of perspective projections provides essential insights into the relations between the tangible real space properties of cone beam imaging geometries and their technically relevant description in terms of homogeneous coordinates and projection matrices. Based on these fundamentals, a novel auto-calibration approach is developed, facilitating the practical determination of perspective imaging geometries with minimal experimental constraints. A corresponding generalized formulation of the widely employed Feldkamp algorithm is given, allowing fast and flexible volume reconstructions from arbitrary tomographic imaging geometries. Iterative reconstruction techniques are likewise introduced for general projection geometries, with a particular focus on the efficient evaluation of the forward problem associated with tomographic imaging. A highly performant 3D generalization of Joseph's classic linearly interpolating ray casting algorithm is developed to this end and compared to typical alternatives. With regard to the anisotropic imaging modality required for tensor tomography, X-ray dark-field contrast is extensively reviewed. Previous literature is brought into a joint context and nomenclature and supplemented by original work completing a consistent picture of the theory of dark-field origination. Key results are explicitly validated by experimental data with a special focus on tomography as well as the properties of anisotropic fibrous scatterers. In order to address the pronounced susceptibility of interferometric images to subtle mechanical imprecisions, an efficient optimization based evaluation strategy for the raw data provided by Talbot interferometers is developed. Finally, the fitness of linear tensor models with respect to the derived anisotropy properties of dark-field contrast is evaluated, and an iterative scheme for the reconstruction of tensor valued volumes from projection images is proposed. The derived methods are efficiently implemented and applied to fiber reinforced plastic samples, imaged at the ID19 imaging beamline of the European Synchrotron Radiation Facility. The results represent unprecedented demonstrations of X-ray dark-field tensor tomography at a field of view of 3-4cm, revealing local fiber orientations of both complex shaped and low-contrast samples at a spatial resolution of 0.1mm in 3D. The results are confirmed by an independent micro CT based fiber analysis. / Die Röntgen-Dunkelfeld-Bildgung vermag den Widerspruch zwischen dem Bedarf nach großen Sichtfeldern im Zentimeterbereich und der nötigen Bildauflösung zur Charakterisierung von Fasermaterialien mit Strukturgrößen im Mikrometerbereich aufzulösen. Sie bedient sich dafür der Eigenschaft von Röntgen-Talbot-Interferometern, Ultrakleinwinkelstreueigenschaften einer Probe vollflächig abzubilden, womit eine Lücke von mehreren Größenordnung zwischen der Bildauflösung und der konstrastgebenden Strukturgröße überbrückt werden kann. Der Zusammenhang zwischen Strukturanisotropie und gerichteter Streuung ermöglicht dabei Rückschlüsse auf die Orientierung der Mikrostruktur einer Probe unterhalb der Bildauflösung. Erste Demonstrationen haben, basiered auf verschiedenen heuristischen Signalmodellen und Rekonstruktrionsansätzen, die grundsätzliche Erweiterbarkeit auf die Volumen-Bildgebung gezeigt. In der vorliegenden Arbeit wird, aufbauend auf einer umfassenden Analyse der Dunkelfeld-Bildgebung und tomographischer Rekonstruktionsmethoden, sowohl ein verifiziertes Modell der Signalanisotropie als auch eine Rekonstruktionstechnik entwickelt, die für große tensorwertige Volumina und allgemeine Abbildungsgeometrien praktikabel ist. In diesem Sinne wird ein weites interdisziplinäres Feld von Bildgebungs- und Rekonstruktionsmethoden aufgearbeitet. Zunächst werden anhand einer neuen Einführung in die mathematische Beschreibung perspektivischer Projektionen essenzielle Einsichten in die Zusammenhänge zwischen der greifbaren Realraum-Darstellung der Kegelstrahl-Geometrie und ihrer technisch relevanten Beschreibung mittels homogener Koordinaten und Projektionsmatrizen gegeben. Aufbauend auf diesen Grundlagen wird eine neue Methode zur Auto-Kalibration entwickelt, die die praktische Bestimmung von perspektivischen Abbildungsgeometrien unter minimalen Anforderungen an die experimentelle Ausführung ermöglicht. Passend dazu wird eine verallgemeinerte Formulierung des weit verbreiteten Feldkamp-Algorithmus gegeben, um eine schnelle und flexible Volumenrekonstruktion aus beliebigen tomographischen Bildgebungsgeometrien zu ermöglichen. Iterative Rekonstruktionsverfahren werden ebenfalls für allgemeine Aufnahmegeometrien eingeführt, wobei ein Schwerpunkt auf der effizienten Berechnung des mit der tomographischen Bildgebung assoziierten Vorwärtsproblems liegt. Zu diesem Zweck wird eine hochperformante 3D-Erweiterung des klassischen, linear interpolierenden Linienintegrationsalgorithmus von Joseph entwickelt und mit typischen Alternativen verglichen. In Bezug auf die anisotrope Bildmodalität, die die Grundlage der Tensortomographie bildet, wird der Röntgen-Dunkelfeld-Kontrast umfassend besprochen. Die vorhandende Literatur wird dazu in einen gemeinsamen Kontext und eine gemeinsame Nomenklatur gebracht und mit neuen Überlegungen zu einer konsistenten Darstellung der Theorie zur Dunkelfeldsignalentstehung vervollständigt. Zentrale Ergebnisse werden dabei explizit anhand experimenteller Daten verifiziert, wobei besonders die Tomographie und die Eigenschaften anisotroper, faseriger Streuer im Vordergrund stehen. Um die ausgeprägte Empfindlichkeit interferometrischer Bilder auf feinste mechanische Instabilitäten zu kompensieren, wird ein effizientes Optimierungsverfahren zur Auswertung der Rohdaten aus Talbot-Interferometern entwickelt. Schließlich wird die Anwendbarkeit von linearen Tensor-Modellen in Bezug auf die hergeleiteten Anisotropie-Eigenschaften des Dunkelfeld-Kontrastes diskutiert, und ein iteratives Verfahren für die Rekonstruktion tensorwertiger Volumen aus Projektionsbildern vorgeschlagen. Die entwickelten Methoden werden effizient implementiert und auf Proben aus faserverstärktem Kunstoff angewandt, die dafür an der Bildgebungs-Strahllinie ID19 des Europäischen Synchrotrons ESRF abgebildet wurden. Die Ergebnisse stellen eine bisher einmalige Demonstration von Röntgen-Dunkelfeld-Tensor-Tomographie mit einem Sichtfeld von 3-4cm dar, wobei lokale Faserorientierung sowohl für komplex geformte als auch kontrastarme Objekte mit einer räumlichen Auflösung von 0.1mm in 3D dargestellt werden kann. Ein unabhängiger Vergleich mit Mikro-CT basierter Faser-Analyse bestätigt die Ergebnisse.