Riemannian Holonomy Groups of Statistical Manifolds

Normal distribution manifolds play essential roles in the theory of information geometry, so do holonomy groups in classification of Riemannian manifolds. After some necessary preliminaries on information geometry and holonomy groups, it is presented that the corresponding Riemannian holonomy group of the $d$-dimensional normal distribution is $SO\left(\frac{d\left(d+3\right)}{2}\right)$, for all $d\in\mathbb{N}$. As a generalization on exponential family, a list of holonomy groups follows.Comment: 11 page

MU-MIMO Communications with MIMO Radar: From Co-existence to Joint Transmission

Beamforming techniques are proposed for a joint multi-input-multi-output (MIMO) radar-communication (RadCom) system, where a single device acts both as a radar and a communication base station (BS) by simultaneously communicating with downlink users and detecting radar targets. Two operational options are considered, where we first split the antennas into two groups, one for radar and the other for communication. Under this deployment, the radar signal is designed to fall into the null-space of the downlink channel. The communication beamformer is optimized such that the beampattern obtained matches the radar's beampattern while satisfying the communication performance requirements. To reduce the optimizations' constraints, we consider a second operational option, where all the antennas transmit a joint waveform that is shared by both radar and communications. In this case, we formulate an appropriate probing beampattern, while guaranteeing the performance of the downlink communications. By incorporating the SINR constraints into objective functions as penalty terms, we further simplify the original beamforming designs to weighted optimizations, and solve them by efficient manifold algorithms. Numerical results show that the shared deployment outperforms the separated case significantly, and the proposed weighted optimizations achieve a similar performance to the original optimizations, despite their significantly lower computational complexity.Comment: 15 pages, 15 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

A robust simplex cut-cell method for adaptive high-order discretizations of aerodynamics and multi-physics problems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 189-199).Despite the wide use of partial differential equation (PDE) solvers, lack of automation still hinders realizing their full potential in assisting engineering analysis and design. In particular, the process of establishing a suitable mesh for a given problem often requires heavy person-in-the-loop involvement. This thesis presents work toward the development of a robust PDE solution framework that provides a reliable output prediction in a fully-automated manner. The framework consists of: a simplex cut-cell technique which allows the mesh generation process to be independent of the geometry of interest; a discontinuous Galerkin (DG) discretization which permits an easy extension to high-order accuracy; and an anisotropic output-based adaptation which improves the discretization mesh for an accurate output prediction in a fully-automated manner. Two issues are addressed that limit the automation and robustness of the existing simplex cut-cell technique in three dimensions. The first is the intersection ambiguity due to numerical precision. We introduce adaptive precision arithmetic that guarantees intersection correctness, and develop various techniques to improve the efficiency of using this arithmetic. The second is the poor quadrature quality for arbitrarily shaped elements. We propose a high-quality and efficient cut-cell quadrature rule that satisfies a quality measure we define, and demonstrate the improvement in nonlinear solver robustness using this quadrature rule. The robustness and automation of the solution framework is then demonstrated through a range of aerodynamics problems, including inviscid and laminar flows. We develop a high-order DG method with a dual-consistent output evaluation for elliptic interface problems, and extend the simplex cut-cell technique for these problems, together with a metric-optimization adaptation algorithm to handle cut elements. This solution strategy is further extended for multi-physics problems, governed by different PDEs across the interfaces. Through numerical examples, including elliptic interface problems and a conjugate heat transfer problem, high-order accuracy is demonstrated on non-interface-conforming meshes constructed by the cut-cell technique, and mesh element size and shape on each material are automatically adjusted for an accurate output prediction.by Huafei Sun.Ph. D

Impact of triangle shapes using high-order discretizations and direct mesh adaptation for output error

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 95-101).The impact of triangle shapes, including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for high-order discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and high-order discretizations. In addition, large angles are not problematic for the conditioning of the linear systems arising from discretization. They can be overcome through small increases in preconditioning costs. A direct adaptation scheme that controls the output error via mesh operations and mesh smoothing is also developed. The decision of mesh operations is solely based on output error distribution without any a priori assumption on error convergence rate. Anisotropy is introduced by evaluating the error changes due to potential edge split, and thus the anisotropies of both primal and dual solutions are taken into account. This scheme is demonstrated to produce grids with fewer degrees of freedom for a specified error level than the existing metric-based approach.by Huafei Sun.S.M

Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equation Q=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method

Logistic Model of Hit and Run Crashes in Calgary

Hit-and-run cashes refer to traffic collisions in which at least one driver flees from crash scene without reporting the crash. In the City of Calgary, for example, they accounted for 18 percent of total traffic collisions in 2005. The objective of this study is to identify the environment and road characteristics that contribute to the occurrence of hit-and-run crashes in the City of Calgary. A logistic regression model was developed to delineate the likelihood of hit-and-run crashes as opposed to non hit-and-run crashes. Our study showed that compared to weekday and daytime collisions, weekend and night time collisions have significantly higher likelihood of hit-and-run. In terms of weather condition, clear weather exhibited the greatest chance of hit-and-run when compared to any other weather conditions. Moreover, hit-and-run crashes are quite likely to occur on undivided one-way roads and the roads with artificial light. As for driver related factors, female drivers aged at 55 or above showed the greatest likelihood as compared to other age groups. Based on the findings from this study, a set of countermeasures will be proposed in this paper