In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.

The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in $\R^3$. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the two-dimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clément-type interpolation operator have to be introduced.

This paper presents a method to reduce time complexity of the computation of higher–order tensor lines. The method can be applied to higher–order tensors and the spherical harmonics representation, both widely used in medical imaging. It is based on a gradient descend technique and integrates well into fiber tracking algorithms. Furthermore, the method improves the angular resolution in contrast to discrete sampling methods which is especially important to tractography, since there, small errors accumulate fast and make the result unusable. Our implementation does not interpolate derived directions but works directly on the interpolated tensor information. The specific contribution of this paper is a fast algorithm for tracking lines tensor fields of arbitrary order that increases angular resolution compared to previous approaches.

In this thesis work the accuracy of the spherical harmonic synthesis and analysis are investigated, by simulated numerical studies.The main idea is to investigate the loss of accuracy, in the geopotential coeffcients, by the following testing method. We start with a synthesis calculation, using the coefficients(EGM2008), to calculate geoid heights on a regular grid. Those geoid heights are then used in an analysis calculation to obtain a new set of coeffcients, which are in turn used to derive a new set of geoid heights. The difference between those two sets of geoid heights will be analyzed to assess the accuracy of the synthesis and analysis calculations.The tests will be conducted with both point-values and area-means in the blocks in the grid. The area-means are constructed in some different ways and will also be compared to the mean value from 10000 point values as separate tests. Numerical results from this investigation show there are signifi…cant systematic errors in the geoid heights computed by spherical harmonic synthesis and analysis, sometimes reaching as high as several meters. Those big errors are most common at the polar regions and at the mid-latitude regions.

The Funk–Radon transform assigns to a function on the two-sphere its mean values along all great circles. We consider the following generalization: we replace the great circles by the small circles being the intersection of the sphere with planes containing a common point ζ inside the sphere. If ζ is the origin, this is just the classical Funk–Radon transform. We find two mappings from the sphere to itself that enable us to represent the generalized Radon transform in terms of the Funk–Radon transform. This representation is utilized to characterize the nullspace and range as well as to prove an inversion formula of the generalized Radon transform.

Profile analysis is a multivariate statistical method for comparing the mean vectors for different groups. It consists of three tests, they are the tests for parallelism, level and flatness. The results from each test give information about the behaviour of the groups and the variables in the groups. The test statistics used when there are more than two groups are likelihood-ratio tests. However, issues in the form indeterminate test statistics occur in the high-dimensional setting, that is when there are more variables than observations. This thesis investigates a method to approach this problem by reducing the dimensionality of the data using scores, that is linear combinations of the variables. Three different ways of choosing this score are compared: the eigendecomposition and two variations of the non-negative matrix factorization. The methods are compared using simulations for five different type of mean parameter settings. The results show that the eigendecomposition is the best technique for choosing the score, and that using more scores only slightly improves the results. Moreover, the results for the parallelism and the flatness tests are shown to be very good, but the results for the level hypothesis deviate from the expectation.

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This dissertation develops a computational method for simulating laser-induced cavitation and investigates the mechanism behind the formation of non-spherical bubbles induced by long-pulsed lasers. The proposed computational method accounts for the laser emission and absorption, phase transition, and the dynamics and thermodynamics of a two-phase fluid flow. In this new method, the model combines the Navier-Stokes (NS) equations for a compressible inviscid two-phase fluid flow, a new laser radiation equation, and a novel local thermodynamic model of phase transition. The Navier-Stokes equations are solved using the FInite Volume method with Exact two-phase Riemann solvers (FIVER). Following this method, numerical fluxes across phase boundaries are computed by constructing and solving one-dimensional bi-material Riemann problems. The new laser radiation equation is derived by customizing the radiative transfer equation (RTE) using the special properties of laser, including monochromaticity, directionality, high intensity, and a measurable focusing or diverging angle. An embedded boundary finite volume method is developed to solve the laser radiation equation on the same mesh created for the NS equations. The fluid mesh usually does not resolve the boundary and propagation directions of the laser beam, leading to the challenges of imposing the boundary conditions on the laser domain. To overcome this challenge, ghost nodes outside the laser domain are populated by mirroring and interpolation techniques. The existence and uniqueness of the solution are proved for the two-dimensional case, leveraging the special geometry of the laser domain. The method is up to second-order accuracy, which is also proved, and verified using numerical tests. A method of latent heat reservoir is developed to predict the onset of vaporization, which accounts for the accumulation and release of latent heat. In this work, the localized level set method is employed to track the bubble surface. Furthermore, the continuation of phase transition is possible in laser-induced cavitation problems, especially for long-pulsed lasers. A method of local correction and reinitialization is developed to account for continuous phase transitions. Several numerical tests are presented to verify the convergence of these methods. This multiphase laser-fluid coupled computational model is employed to simulate the formation and expansion of bubbles with different shapes induced by different long-pulsed lasers. The simulation results show that the computational method can capture the key phenomena in the laser-induced cavitation problems, including non-spherical bubble expansion, shock waves, and the ``Moses effect''. Additionally, the observed complex non-spherical shapes of vapor bubbles generated by long-pulsed laser reflect some characteristics (e.g., direction, width) of the laser beam. The dissertation also investigates the relation between bubble shapes and laser parameters and explores the transition between two commonly observed shapes -- namely, a rounded pear-like shape and an elongated conical shape -- using the proposed computational model. Two laboratory experiments are simulated, in which Holmium:YAG and Thulium fiber lasers are used respectively to generate bubbles of different shapes. In both cases, the predicted bubble nucleation and morphology agree reasonably well with the experimental observation. The full-field results of laser radiance, temperature, velocity, and pressure are analyzed to explain bubble dynamics and energy transmission. It is found that due to the lasting energy input, the vapor bubble's dynamics is driven not only by advection, but also by the continued vaporization at its surface. Vaporization lasts less than 1 microsecond in the case of the pear-shaped bubble, compared to over 50 microseconds for the elongated bubble. It is thus hypothesized that the bubble's morphology is determined by a competition between the speed of bubble growth due to advection and continuous vaporization. When the speed of advection is higher than that of vaporization, the bubble tends to grow spherically. Otherwise, it elongates along the laser beam direction. To test this hypothesis, the two speeds are defined analytically using a model problem and then estimated for the experiments using simulation results. The results support the hypothesis and also suggest that when the laser's power is fixed, a higher laser absorption coefficient and a narrower beam facilitate bubble elongation. / Doctor of Philosophy / Laser-induced cavitation is a process where laser beams create bubbles in a liquid. This phenomenon is widely applied in research and microfluidic applications for precise control of bubble dynamics. It also naturally occurs in various laser-based processes involving liquid environments. Understanding laser-induced cavitation is important for enhancing the effectiveness and safety of related technologies. However, experimental studies encounter limitations, highlighting the development of numerical methods to advance the understanding of laser-induced cavitation. The laser-induced cavitation can be roughly described as localized boiling through thermal radiation. The detailed physics involves the absorption of laser light by a liquid, the formation of vapor bubbles due to localized heating, and the dynamics of both the bubbles and the surrounding liquid. The first part of the dissertation introduces a new computational method for modeling these phenomena. The dynamics of the two-phase flow are modeled by the Navier-Stokes equations, which are solved using the FInite Volume method with Exact two-phase Riemann solvers (FIVER). The absorption of the laser light is modeled by a new laser radiation equation, which is derived from laser energy conservation and special properties of the laser. An embedded boundary finite volume method is developed to solve this equation on the same mesh created for the NS equations. Additionally, a method of latent heat reservoir is developed to predict the onset of vaporization. In this work, the level set method is employed to track the bubble surface, and a method of local correction and reinitialization is developed to account for possible continuous phase transitions. After developing this new method, several test cases are simulated. The simulation results show that the method can capture the key phenomena in the laser-induced cavitation problems, including the absorption of laser light, non-spherical bubble expansion, and shock waves. When the laser pulse is comparable to or longer than the acoustic time scale (long-pulsed laser), vapor bubbles generated often have complex non-spherical shapes. The bubble shapes reflect some characteristics (e.g., direction, width) of the laser beam. The second part of the dissertation investigates the relation between bubble shapes and laser parameters. Two laboratory experiments are simulated, in which two different lasers are used to generate bubbles of different shapes, namely, a rounded pear-like shape and an elongated conical shape. In both cases, the simulated bubbles exhibit shapes and sizes that reasonably match the experimental results. The simulation results of temperature, pressure, and velocity fields are analyzed to explain bubble dynamics and energy transmission. The analysis shows that the expansion of bubbles induced by long-pulsed lasers is determined not only by advection but also by the continued vaporization at its surface. Vaporization lasts less than $1$ microsecond in the case of the pear-shaped bubble, compared to over $50$ microseconds for the elongated bubble. It is thus hypothesized that the bubble expansion is determined by a competition between the speed of bubble growth due to advection and continuous vaporization. When the speed of advection is higher than that of vaporization, the bubble tends to grow spherically. Otherwise, it elongates along the laser beam direction. To test this hypothesis, the two speeds are defined analytically using a model problem and then estimated for the experiments using simulation results. The results support the hypothesis and also suggest that when the laser's power is fixed, a higher laser absorption coefficient and a narrower beam facilitate bubble elongation.

The first part of my thesis is concerned with estimation for longitudinal data using generalized semi-parametric mixed models and multilevel generalized linear mixed models for a binary response. Likelihood based inferences are hindered by the lack of a closed form representation. Consequently, various integration approaches have been proposed. We propose a spherical radial integration based approach that takes advantage of the hierarchical structure of the data, which we call the 2 SR method. Compared to Pinheiro and Chao's multilevel Adaptive Gaussian quadrature, our proposed method has an improved time complexity with the number of functional evaluations scaling linearly in the number of subjects and in the dimension of random effects per level. Simulation studies show that our approach has similar to better accuracy compared to Gauss Hermite Quadrature (GHQ) and has better accuracy compared to PQL especially in the variance components. The second part of my thesis is concerned with identifying differentially expressed gene pathways/gene sets. We propose a logistic kernel machine to model the gene pathway effect with a binary response. Kernel machines were chosen since they account for gene interactions and clinical covariates. Furthermore, we established a connection between our logistic kernel machine with GLMMs allowing us to use ideas from the GLMM literature. For estimation and testing, we adopted Clarkson's spherical radial approach to perform the high dimensional integrations. For estimation, our performance in simulation studies is comparable to better than Bayesian approaches at a much lower computational cost. As for testing of the genetic pathway effect, our REML likelihood ratio test has increased power compared to a score test for simulated non-linear pathways. Additionally, our approach has three main advantages over previous methodologies: 1) our testing approach is self-contained rather than competitive, 2) our kernel machine approach can model complex pathway effects and gene-gene interactions, and 3) we test for the pathway effect adjusting for clinical covariates. Motivation for our work is the analysis of an Acute Lymphocytic Leukemia data set where we test for the genetic pathway effect and provide confidence intervals for the fixed effects.

The knowledge of solid-state second phase heterogeneous nucleation process is limited due to the experimental difficulty, such as tiny length scale, short time period, and high temperature condition. In recent years, some significant breakthroughs in nucleation studies have been achieved by aid of computational techniques. In this study, we apply molecular dynamics (MD) simulations to perform with heterogeneous nucleation occurring at grain boundaries (GB) during the austenite (FCC) phase to ferrite (BCC) phase transformation in a pure Fe polycrystalline system. A neighbor vector analysis (NVA) method has been introduced and it is shown how the NVA can be used to determine the misorientation of grain or interphase boundaries, which allow a further investigation of the boundary structure correlated to interfacial energy and mobility during the nucleation and early grain growth stage. Meanwhile, benefited from the MD technique, the bulk energy, grain boundary energy, and interfacial energy can be individually captured during the simulations, which allow a detail analyze of the shape, critical size and nucleation energy of specific nuclei, through the classical nucleation theory (CNT) and according to a faceted-spherical cap geometric model (FSC). In addition, we also compared the results from the classical approach with a new algorithm that combination of the multi-phase field model (MPFM) and the nudged elastic band (NEB) method to demonstrate the CNT in the solid-state conduction. Finally, we extend our simulation method to a more complex triple GB junction nucleation event, and investigate the non-classical barrier-free nucleation behaviors. The results support the critical informations to clarify the initial state of austenite to ferrite transition, and improve our knowledge of the heterogeneous nucleation process, which help to bridge the gap between the experimental measurements and the theoretical calculations. The simulation method also provided a new approach for studying the complicate heterogeneous nucleation phenomenon in solid-state for a wide variety of polycrystalline material systems. / Thesis / Doctor of Philosophy (PhD)