Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Cortisol Awakening Response in Preschoolers and Depression Risk: Relations with Maternal History of Depression and Child Temperament

Increasing research suggests that elevations in the cortisol awakening response (CAR), the natural increase of cortisol 30 to 40 minutes after waking, may serve as a vulnerability marker for depression. However, existing studies have focused on adolescence and adulthood; very little is known about the CAR in early childhood and the factors that are associated with it. The current study aimed to examine the validity of the CAR as a potential early-emerging vulnerability marker for depression in a sample of preschool-age children. We examined associations between the CAR and two well-established risk factors for depression: maternal psychopathology and early child temperament (high negative emotionality (NE) and/or low positive emotionality (PE)). The sample consisted of 146 preschool-age children, of whom 71 (49.3%) had a biological mother with a history of depression and 65 (45.5%) had a biological mother with a history of anxiety. To assess the CAR, salivary cortisol samples were collected from the child upon waking, 30 and 45 minutes post-waking on two weekdays. Children’s CAR was examined as the total volume of cortisol secreted (AUCg) and the total increase in cortisol (AUCi) across waking. Evening cortisol was collected 30 minutes before bedtime. Child temperament was assessed using observational laboratory measures. Maternal depression and anxiety were assessed with clinical interviews. Associations with children’s CAR, as indicated by AUCg or AUCi, appeared to be specific to maternal current psychopathology and symptoms of anhedonia. Additionally, we observed significant interactions for both maternal lifetime and current depression and anxiety, in combination with child NE and PE, on elevated evening cortisol levels and flattened diurnal cortisol rhythms, indicating altered patterns of basal cortisol activity in offspring. Our study contributes to the limited but growing knowledge on the development of the CAR in preschool age children and as a marker of early risk. Findings suggest that there is a complex interplay between familial risk, affective vulnerability, and their joint effects on neuroendocrine dysfunction in young children, and highlight the need for future research to examine which aspects of the early diurnal rhythm predict the emergence of later depressive illness.

THE HEALTH OF YOUNG ADULTS WITH DISABILITIES IN THE UNITED STATES: THE COLLECTIVE ROLES OF RISK FACTORS IN SOCIAL ECOLOGIC FRAMEWORK

Background: Over the last few decades, the prevalence of young adults with disabilities (YAD) has steadily risen as a result of advances in medicine, clinical treatment, and biomedical technologythat enhanced their survival into adulthood. Despite investments in services, family supports, and insurance, they experience poor health status and barriers to successful transition into adulthood. Objectives: We investigated the collective roles of multi-faceted factors at intrapersonal, interpersonal and community levels within the social ecological framework on health related outcome including self-rated health (SRH) of YAD. The three specific aims are: 1) to examine sociodemographic differences and health insurance coverage in adolescence; 2) to investigate the role of social skills in relationships with family and peers developed in adolescence; and 3) to collectively explore the association of sociodemographic characteristics, social skills, and community participation in adolescence on SRH. Methods: Using longitudinal data (N=5,020) from the National Longitudinal Transition Study (NLTS2), we conducted multivariate logistic regression analyses to understand the association between insurance status as well as social skills in adolescence and YAD’s health related outcomes. Structural equation modeling (SEM) assessed the confluence of multi-faceted factors from the social ecological model that link to health in early adulthood. Results: Compared with YAD who had private insurance, YAD who had public health insurance in adolescence are at higher odds of experiencing poorer health related outcomes in self-rated health [adjusted odds ratio (aOR=2.89, 95% confidence interval (CI): 1.16, 7.23), problems with health (aOR=2.60, 95%CI: 1.26, 5.35), and missing social activities due to health problems (aOR=2.86, 95%CI: 1.39, 5.85). At the interpersonal level, overall social skills developed through relationship with family and peers in adolescence do not appear to have association with health related outcomes in early adulthood. Finally, at the community level, community participation in adolescence does not have an association with SRH in early adulthood. Conclusions: Having public health insurance coverage does not equate to good health. YAD need additional supports to achieve positive health outcomes. The findings in social skills and community participation suggest other potential factors may be at play for health related outcomes for YAD and the need for further investigation.

DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE MULTIPHASE FLOW WITH PHASE CHANGE

Phase change problems arise in many practical applications such as air-conditioning and refrigeration, thermal energy storage systems and thermal management of electronic devices. The physical phenomenon in such applications are complex and are often difficult to be studied in detail with the help of only experimental techniques. The efforts to improve computational techniques for analyzing two-phase flow problems with phase change are therefore gaining momentum. The development of numerical methods for multiphase flow has been motivated generally by the need to account more accurately for (a) large topological changes such as phase breakup and merging, (b) sharp representation of the interface and its discontinuous properties and (c) accurate and mass conserving motion of the interface. In addition to these considerations, numerical simulation of multiphase flow with phase change introduces additional challenges related to discontinuities in the velocity and the temperature fields. Moreover, the velocity field is no longer divergence free. For phase change problems, the focus of developmental efforts has thus been on numerically attaining a proper conservation of energy across the interface in addition to the accurate treatment of fluxes of mass and momentum conservation as well as the associated interface advection. Among the initial efforts related to the simulation of bubble growth in film boiling applications the work in \cite{Welch1995} was based on the interface tracking method using a moving unstructured mesh. That study considered moderate interfacial deformations. A similar problem was subsequently studied using moving, boundary fitted grids \cite{Son1997}, again for regimes of relatively small topological changes. A hybrid interface tracking method with a moving interface grid overlapping a static Eulerian grid was developed \cite{Juric1998} for the computation of a range of phase change problems including, three-dimensional film boiling \cite{esmaeeli2004computations}, multimode two-dimensional pool boiling \cite{Esmaeeli2004} and film boiling on horizontal cylinders \cite{Esmaeeli2004a}. The handling of interface merging and pinch off however remains a challenge with methods that explicitly track the interface. As large topological changes are crucial for phase change problems, attention has turned in recent years to front capturing methods utilizing implicit interfaces that are more effective in treating complex interface deformations. The VOF (Volume of Fluid) method was adopted in \cite{Welch2000} to simulate the one-dimensional Stefan problem and the two-dimensional film boiling problem. The approach employed a specific model for mass transfer across the interface involving a mass source term within cells containing the interface. This VOF based approach was further coupled with the level set method in \cite{Son1998}, employing a smeared-out Heaviside function to avoid the numerical instability related to the source term. The coupled level set, volume of fluid method and the diffused interface approach was used for film boiling with water and R134a at the near critical pressure condition \cite{Tomar2005}. The effect of superheat and saturation pressure on the frequency of bubble formation were analyzed with this approach. The work in \cite{Gibou2007} used the ghost fluid and the level set methods for phase change simulations. A similar approach was adopted in \cite{Son2008} to study various boiling problems including three-dimensional film boiling on a horizontal cylinder, nucleate boiling in microcavity \cite{lee2010numerical} and flow boiling in a finned microchannel \cite{lee2012direct}. The work in \cite{tanguy2007level} also used the ghost fluid method and proposed an improved algorithm based on enforcing continuity and divergence-free condition for the extended velocity field. The work in \cite{sato2013sharp} employed a multiphase model based on volume fraction with interface sharpening scheme and derived a phase change model based on local interface area and mass flux. Among the front capturing methods, sharp interface methods have been found to be particularly effective both for implementing sharp jumps and for resolving the interfacial velocity field. However, sharp velocity jumps render the solution susceptible to erroneous oscillations in pressure and also lead to spurious interface velocities. To implement phase change, the work in \cite{Hardt2008} employed point mass source terms derived from a physical basis for the evaporating mass flux. To avoid numerical instability, the authors smeared the mass source by solving a pseudo time-step diffusion equation. This measure however led to mass conservation issues due to non-symmetric integration over the distributed mass source region. The problem of spurious pressure oscillations related to point mass sources was also investigated by \cite{Schlottke2008}. Although their method is based on the VOF, the large pressure peaks associated with sharp mass source was observed to be similar to that for the interface tracking method. Such spurious fluctuation in pressure are essentially undesirable because the effect is globally transmitted in incompressible flow. Hence, the pressure field formation due to phase change need to be implemented with greater accuracy than is reported in current literature. The accuracy of interface advection in the presence of interfacial mass flux (mass flux conservation) has been discussed in \cite{tanguy2007level,tanguy2014benchmarks}. The authors found that the method of extending one phase velocity to entire domain suggested by Nguyen et al. in \cite{nguyen2001boundary} suffers from a lack of mass flux conservation when the density difference is high. To improve the solution, the authors impose a divergence-free condition for the extended velocity field by solving a constant coefficient Poisson equation. The approach has shown good results with enclosed bubble or droplet but is not general for more complex flow and requires additional solution of the linear system of equations. In current thesis, an improved approach that addresses both the numerical oscillation of pressure and the spurious interface velocity field is presented by featuring (i) continuous velocity and density fields within a thin interfacial region and (ii) temporal velocity correction steps to avoid unphysical pressure source term. Also I propose a general (iii) mass flux projection correction for improved mass flux conservation. The pressure and the temperature gradient jump condition are treated sharply. A series of one-dimensional and two-dimensional problems are solved to verify the performance of the new algorithm. Two-dimensional and cylindrical film boiling problems are also demonstrated and show good qualitative agreement with the experimental observations and heat transfer correlations. Finally, a study on Taylor bubble flow with heat transfer and phase change in a small vertical tube in axisymmetric coordinates is carried out using the new multiphase, phase change method.

Motion Planning and Controls with Safety and Temporal Constraints

Motion planning, or trajectory planning, commonly refers to a process of converting high-level task specifications into low-level control commands that can be executed on the system of interest. For different applications, the system will be different. It can be an autonomous vehicle, an Unmanned Aerial Vehicle(UAV), a humanoid robot, or an industrial robotic arm. As human machine interaction is essential in many of these systems, safety is fundamental and crucial. Many of the applications also involve performing a task in an optimal manner within a given time constraint. Therefore, in this thesis, we focus on two aspects of the motion planning problem. One is the verification and synthesis of the safe controls for autonomous ground and air vehicles in collision avoidance scenarios. The other part focuses on the high-level planning for the autonomous vehicles with the timed temporal constraints. In the first aspect of our work, we first propose a verification method to prove the safety and robustness of a path planner and the path following controls based on reachable sets. We demonstrate the method on quadrotor and automobile applications. Secondly, we propose a reachable set based collision avoidance algorithm for UAVs. Instead of the traditional approaches of collision avoidance between trajectories, we propose a collision avoidance scheme based on reachable sets and tubes. We then formulate the problem as a convex optimization problem seeking control set design for the aircraft to avoid collision. We apply our approach to collision avoidance scenarios of quadrotors and fixed-wing aircraft. In the second aspect of our work, we address the high level planning problems with timed temporal logic constraints. Firstly, we present an optimization based method for path planning of a mobile robot subject to timed temporal constraints, in a dynamic environment. Temporal logic (TL) can address very complex task specifications such as safety, coverage, motion sequencing etc. We use metric temporal logic (MTL) to encode the task specifications with timing constraints. We then translate the MTL formulae into mixed integer linear constraints and solve the associated optimization problem using a mixed integer linear program solver. We have applied our approach on several case studies in complex dynamical environments subjected to timed temporal specifications. Secondly, we also present a timed automaton based method for planning under the given timed temporal logic specifications. We use metric interval temporal logic (MITL), a member of the MTL family, to represent the task specification, and provide a constructive way to generate a timed automaton and methods to look for accepting runs on the automaton to find an optimal motion (or path) sequence for the robot to complete the task.