Despite the increasing popularity of high-order methods in computational fluid dynamics, their application to practical problems still remains challenging. In order to exploit the advantages of high-order methods with geometrically complex computational domains, coarse curved meshes are necessary, i.e. high-order representations of the geometry. This dissertation presents a strategy for the generation of curved high-order surface meshes. The mesh generation method combines least-squares fitting with energy functionals, which approximate physical bending and stretching energies, in an incremental energy-minimizing fitting strategy. Since the energy weighting is reduced in each increment, the resulting surface representation features high accuracy. Nevertheless, the beneficial influence of the energy-minimization is retained. The presented method aims at enabling the utilization of the superior convergence properties of high-order methods by facilitating the construction of coarser meshes, while ensuring accuracy by allowing an arbitrary choice of geometric approximation order. Results show surface meshes of remarkable quality, even for very coarse meshes representing complex domains, e.g. blood vessels.

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第21374号 / 農博第2298号 / 新制||農||1066(附属図書館) / 学位論文||H30||N5147(農学部図書室) / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 村上 章, 教授 藤原 正幸, 教授 渦岡 良介 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

Ten-millijoule, thirty-five femtosecond, 800 nm (~40 nm bandwidth) laser pulses are used to study high-order harmonic generation in helium- and neon-filled gas cells of various lengths. Harmonic orders in the range of 50 to 100 are investigated. A semi-infinite cell geometry produces brighter harmonics than cells of sub-centimeter length. In the semi-infinite geometry, the gas occupies the region from the focusing lens to a thin exit foil near the laser focus. Counter-propagating light is used to directly probe where the high harmonics are generated within the laser focus and to investigate phase matching. The phase matching under optimized harmonic generation conditions was found to be unexpectedly good with phase zones many millimeters long. Restricting the laser beam with an 8 mm aperture in front of the focusing lens increases the emission of most harmonic orders observed by as much as an order of magnitude. Optimal harmonic generation pressures were found to be about 55 torr in neon and 110 torr in helium. The optimal position of the laser focus was found to be a few millimeters inside the exit foil of the gas cell. Probing with counter-propagating light reveals that in the case of neon the harmonics are generated in the last few millimeters before the exit foil. In helium, the harmonics are produced over a longer distance. Direct measurement shows that the re-absorption limit for mid-range harmonics in neon has been reached.

It is currently a well-established fact that the dynamics of interacting fluid compartments of the central nervous system (CNS) may play a role in the CNS fluid physiology and pathology of a number of neurological disorders, including neurodegenerative diseases associated with accumulation of waste products in the brain. However, the mechanisms and routes of waste clearance from the brain are still unclear. One of the main components of this interacting cerebral fluids dynamics is blood flow. In the last decades, mathematical modeling and fluid dynamics simulations have become a valuable complementary tool to experimental approaches, contributing to a deeper understanding of the circulatory physiology and pathology. However, modeling blood flow in the brain remains a challenging and demanding task, due to the high complexity of cerebral vascular networks and the difficulties that consequently arise to describe and reproduce the blood flow dynamics in these vascular districts. The first part of this work is devoted to the development of efficient numerical strategies for blood flow simulations in complex vascular networks. In cardiovascular modeling, one-dimensional (1D) and lumped-parameter (0D) models of blood flow are nowadays well-established tools to predict flow patterns, pressure wave propagation and average velocities in vascular networks, with a good balance between accuracy and computational cost. Still, the purely 1D modeling of blood flow in complex and large networks can result in computationally expensive simulations, posing the need for extremely efficient numerical methods and solvers. To address these issues, we develop a novel modeling and computational framework to construct hybrid networks of coupled 1D and 0D vessels and to perform computationally efficient and accurate blood flow simulations in such networks. Starting from a 1D model and a family of nonlinear 0D models for blood flow, with either elastic or viscoelastic tube laws, this methodology is based on (i) suitable coupling equations ensuring conservation principles; (ii) efficient numerical methods and numerical coupling strategies to solve 1D, 0D and hybrid junctions of vessels; (iii) model selection criteria to construct hybrid networks, which provide a good trade-off between accuracy in the predicted results and computational cost of the simulations. By applying the proposed hybrid network solver to very complex and large vascular networks, we show how this methodology becomes crucial to gain computational efficiency when solving networks and models where the heterogeneity of spatial and/or temporal scales is relevant, still ensuring a good level of accuracy in the predicted results. Hence, the proposed hybrid network methodology represents a first step towards a high-performance modeling and computational framework to solve highly complex networks of 1D-0D vessels, where the complexity does not only depend on the anatomical detail by which a network is described, but also on the level at which physiological mechanisms and mechanical characteristics of the cardiovascular system are modeled. Then, in the second part of the thesis, we focus on the modeling and simulation of cerebral blood flow, with emphasis on the venous side. We develop a methodology that, departing from the high-resolution MRI data obtained from a novel in-vivo microvascular imaging technique of the human brain, allows to reconstruct detailed subject-specific cerebral networks of specific vascular districts which are suitable to perform blood flow simulations. First, we extract segmentations of cerebral districts of interest in a way that the arterio-venous separation is addressed and the continuity and connectivity of the vascular structures is ensured. Equipped with these segmentations, we propose an algorithm to extract a network of vessels suitable and good enough, i.e. with the necessary properties, to perform blood flow simulations. Here, we focus on the reconstruction of detailed venous vascular networks, given that the anatomy and patho-physiology of the venous circulation is of great interest from both clinical and modeling points of view. Then, after calibration and parametrization of the MRI-reconstructed venous networks, blood flow simulations are performed to validate the proposed methodology and assess the ability of such networks to predict physiologically reasonable results in the corresponding vascular territories. From the results obtained we conclude that this work represents a proof-of-concept study that demonstrates that it is possible to extract subject-specific cerebral networks from the novel high-resolution MRI data employed, setting the basis towards the definition of an effective processing pipeline for detailed blood flow simulations from subject-specific data, to explore and quantify cerebral blood flow dynamics, with focus on venous blood drainage.

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This thesis establishes methods to quantify and explain uncertainty through high-order moments in time series data, along with first principal-based improvements on the standard autoencoder and variational autoencoder. While the first-principal improvements on the standard variational autoencoder provide additional means of explainability, we ultimately look to non-variational methods for quantifying uncertainty under the autoencoder framework. We utilize Shannon's differential entropy to accomplish the task of uncertainty quantification in a general nonlinear and non-Gaussian setting. Together with previously established connections between autoencoders and principal component analysis, we motivate the focus on differential entropy as a proper abstraction of principal component analysis to this more general framework, where nonlinear and non-Gaussian characteristics in the data are permitted. Furthermore, we are able to establish explicit connections between high-order moments in the data to those in the latent space, which induce a natural latent space decomposition, and by extension, an explanation of the estimated uncertainty. The proposed methods are intended to be utilized in economic and financial factor models in state space form, building on recent developments in the application of neural networks to factor models with applications to financial and economic time series analysis. Finally, we demonstrate the efficacy of the proposed methods on high frequency hourly foreign exchange rates, macroeconomic signals, and synthetically generated autoregressive data sets. / Master of Science / This thesis establishes methods to quantify and explain uncertainty in time series data, along with improvements on some latent variable neural networks called autoencoders and variational autoencoders. Autoencoders and varitational autoencodes are called latent variable neural networks since they can estimate a representation of the data that has less dimension than the original data. These neural network architectures have a fundamental connection to a classical latent variable method called principal component analysis, which performs a similar task of dimension reduction but under more restrictive assumptions than autoencoders and variational autoencoders. In contrast to principal component analysis, a common ailment of neural networks is the lack of explainability, which accounts for the colloquial term black-box models. While the improvements on the standard autoencoders and variational autoencoders help with the problem of explainability, we ultimately look to alternative probabilistic methods for quantifying uncertainty. To accomplish this task, we focus on Shannon's differential entropy, which is entropy applied to continuous domains such as time series data. Entropy is intricately connected to the notion of uncertainty, since it depends on the amount of randomness in the data. Together with previously established connections between autoencoders and principal component analysis, we motivate the focus on differential entropy as a proper abstraction of principal component analysis to a general framework that does not require the restrictive assumptions of principal component analysis. Furthermore, we are able to establish explicit connections between high-order moments in the data to the estimated latent variables (i.e., the reduced dimension representation of the data). Estimating high-order moments allows for a more accurate estimation of the true distribution of the data. By connecting the estimated high-order moments in the data to the latent variables, we obtain a natural decomposition of the uncertainty surrounding the latent variables, which allows for increased explainability of the proposed autoencoder. The methods introduced in this thesis are intended to be utilized in a class of economic and financial models called factor models, which are frequently used in policy and investment analysis. A factor model is another type of latent variable model, which in addition to estimating a reduced dimension representation of the data, provides a means to forecast future observations. Finally, we demonstrate the efficacy of the proposed methods on high frequency hourly foreign exchange rates, macroeconomic signals, and synthetically generated autoregressive data sets. The results support the superiority of the entropy-based autoencoder to the standard variational autoencoder both in capability and computational expense.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25099号 / 理博第5006号 / 新制||理||1714(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)講師 PETERSRobert, 教授 柳瀬 陽一, 教授 田中 耕一郎 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DGAM

This dissertation consists of three studies of immersed finite element (IFE) methods for inter- face problems related to partial differential equations (PDEs) with discontinuous coefficients. These three topics together form a continuation of the research in IFE method including the extension to elasticity systems, new breakthroughs to higher degree IFE methods, and its application to inverse problems. First, we extend the current construction and analysis approach of IFE methods in the literature for scalar elliptic equations to elasticity systems in the vector format. In particular, we construct a group of low-degree IFE functions formed by linear, bilinear, and rotated Q1 polynomials to weakly satisfy the jump conditions of elasticity interface problems. Then we analyze the trace inequalities of these IFE functions and the approximation capabilities of the resulted IFE spaces. Based on these preparations, we develop a partially penalized IFE (PPIFE) scheme and prove its optimal convergence rates. Secondly, we discuss the limitations of the current approaches of IFE methods when we try to extend them to higher degree IFE methods. Then we develop a new framework to construct and analyze arbitrary p-th degree IFE methods. In this framework, each IFE function is the extension of a p-th degree polynomial from one subelement to the whole interface element by solving a local Cauchy problem on interface elements in which the jump conditions across the interface are employed as the boundary conditions. All the components in the analysis, including existence of IFE functions, the optimal approximation capabilities and the trace inequalities, are all reduced to key properties of the related discrete extension operator. We employ these results to show the optimal convergence of a discontinuous Galerkin IFE (DGIFE) method. In the last part, we apply the linear IFE methods in the literature together with the shape optimization technique to solve a group of interface inverse problems. In this algorithm, both the governing PDEs and the objective functional for interface inverse problems are discretized optimally by the IFE method regardless of the location of the interface in a chosen mesh. We derive the formulas for the gradients of the objective function in the optimization problem which can be implemented efficiently in the IFE framework through a discrete adjoint method. We demonstrate the properties of the proposed algorithm by applying it to three representative applications. / Doctor of Philosophy / Interface problems arise from many science and engineering applications modeling the transmission of some physical quantities between multiple materials. Mathematically, these multiple materials in general are modeled by partial differential equations (PDEs) with discontinuous parameters, which poses challenges to developing efficient and reliable numerical methods and the related theoretical error analysis. The main contributions of this dissertation is on the development of a special finite element method, the so called immersed finite element (IFE) method, to solve the interface problems on a mesh independent of the interface geometry which can be advantageous especially when the interface is moving. Specifically, this dissertation consists of three projects of IFE methods: elasticity interface problems, higher-order IFE methods and interface inverse problems, including their design, analysis, and application.