Centroidal power diagrams, Lloyd's algorithm and applications to optimal location problems

In this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems. The target energy can be written either in terms of atomic measures and the Wasserstein distance or in terms of weighted points and power diagrams (generalized Voronoi diagrams). The latter formulation is more suitable for computation. We show that critical points of the energy are centroidal power diagrams, which are generalizations of centroidal Voronoi tessellations, and that they can be approximated by a generalization of Lloyd's algorithm (Lloyd's algorithm is a common method for finding centroidal Voronoi tessellations). We prove that the algorithm is energy decreasing and prove a convergence theorem. Numerical experiments suggest that the algorithm converges linearly. We illustrate the algorithm in two and three dimensions using simple models of optimal location and crystallization (see online supplementary material)

Probabilistic and parallel algorithms for centroidal Voronoi tessellations with application to meshless computing and numerical analysis on surfaces

Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, and grid generation. A detailed review is given in chapter 1. In chapter 2, some probabilistic methods for determining centroidal Voronoi tessellations and their parallel implementation on distributed memory systems are presented. The results of computational experiments performed on a CRAY T3E-600 system are given for each algorithm. These demonstrate the superior sequential and parallel performance of a new algorithm we introduce. Then, new algorithms are presented in chapter 3 for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or superimposed meshes. Computational examples are provided that show, for both uniform and non-uniform point distributions that the algorithms result in high-quality point sets and high-quality support regions. The extensions of centroidal Voronoi tessellations to general spaces and sets are also available. For example, tessellations of surfaces in a Euclidean space may be considered. In chapter 4, a precise definition of such constrained centroidal Voronoi tessellations (CCVT\u27s) is given and a number of their properties are derived, including their characterization as minimizers of a kind of energy. Deterministic and probabilistic algorithms for the construction of CCVT\u27s are presented and some analytical results for one of the algorithms are given. Some computational examples are provided which serve to illustrate the high quality of CCVT point sets. CCVT point sets are also applied to polynomial interpolation and numerical integration on the sphere. Finally, some conclusions are given in chapter 5

Implementation of distributed partitioning algorithms using mobile Wheelphones

This thesis presents the implementation process of partitioning algorithms from the theorical ideas to sperimental result

Robust and parallel mesh reconstruction from unoriented noisy points.

Sheung, Hoi.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (p. 65-70).Abstract also in Chinese.Abstract --- p.vAcknowledgements --- p.ixList of Figures --- p.xiiiList of Tables --- p.xvChapter 1 --- Introduction --- p.1Chapter 1.1 --- Main Contributions --- p.3Chapter 1.2 --- Outline --- p.3Chapter 2 --- Related Work --- p.5Chapter 2.1 --- Volumetric reconstruction --- p.5Chapter 2.2 --- Combinatorial approaches --- p.6Chapter 2.3 --- Robust statistics in surface reconstruction --- p.6Chapter 2.4 --- Down-sampling of massive points --- p.7Chapter 2.5 --- Streaming and parallel computing --- p.7Chapter 3 --- Robust Normal Estimation and Point Projection --- p.9Chapter 3.1 --- Robust Estimator --- p.9Chapter 3.2 --- Mean Shift Method --- p.11Chapter 3.3 --- Normal Estimation and Projection --- p.11Chapter 3.4 --- Moving Least Squares Surfaces --- p.14Chapter 3.4.1 --- Step 1: local reference domain --- p.14Chapter 3.4.2 --- Step 2: local bivariate polynomial --- p.14Chapter 3.4.3 --- Simpler Implementation --- p.15Chapter 3.5 --- Robust Moving Least Squares by Forward Search --- p.16Chapter 3.6 --- Comparison with RMLS --- p.17Chapter 3.7 --- K-Nearest Neighborhoods --- p.18Chapter 3.7.1 --- Octree --- p.18Chapter 3.7.2 --- Kd-Tree --- p.19Chapter 3.7.3 --- Other Techniques --- p.19Chapter 3.8 --- Principal Component Analysis --- p.19Chapter 3.9 --- Polynomial Fitting --- p.21Chapter 3.10 --- Highly Parallel Implementation --- p.22Chapter 4 --- Error Controlled Subsampling --- p.23Chapter 4.1 --- Centroidal Voronoi Diagram --- p.23Chapter 4.2 --- Energy Function --- p.24Chapter 4.2.1 --- Distance Energy --- p.24Chapter 4.2.2 --- Shape Prior Energy --- p.24Chapter 4.2.3 --- Global Energy --- p.25Chapter 4.3 --- Lloyd´ةs Algorithm --- p.26Chapter 4.4 --- Clustering Optimization and Subsampling --- p.27Chapter 5 --- Mesh Generation --- p.29Chapter 5.1 --- Tight Cocone Triangulation --- p.29Chapter 5.2 --- Clustering Based Local Triangulation --- p.30Chapter 5.2.1 --- Initial Surface Reconstruction --- p.30Chapter 5.2.2 --- Cleaning Process --- p.32Chapter 5.2.3 --- Comparisons --- p.33Chapter 5.3 --- Computing Dual Graph --- p.34Chapter 6 --- Results and Discussion --- p.37Chapter 6.1 --- Results of Mesh Reconstruction form Noisy Point Cloud --- p.37Chapter 6.2 --- Results of Clustering Based Local Triangulation --- p.47Chapter 7 --- Conclusions --- p.55Chapter 7.1 --- Key Contributions --- p.55Chapter 7.2 --- Factors Affecting Our Algorithm --- p.55Chapter 7.3 --- Future Work --- p.56Chapter A --- Building Neighborhood Table --- p.59Chapter A.l --- Building Neighborhood Table in Streaming --- p.59Chapter B --- Publications --- p.63Bibliography --- p.6

Coverage problems in mobile sensing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2008.Includes bibliographical references (p. 177-183).Sensor-networks can today measure physical phenomena at spatial and temporal scales that were not achievable earlier, and have shown promise in monitoring the environment, structures, agricultural fields and so on. A key challenge in sensor-networks is the coordination of four actions across the network: measurement (sensing), communication, motion and computation. The term coverage is applied to the central question of how well a sensor-network senses some phenomenon to make inferences. More formally, a coverage problem involves finding an arrangement of sensors that optimizes a coverage metric. In this thesis we examine coverage in the context of three sensing modalities. The literature on the topic has thus far focused largely on coverage problems with the first modality: static event-detection sensors, which detect purely binary events in their immediate vicinity based on thresholds. However, coverage problems for sensors which measure physical quantities like temperature, pressure, chemical concentrations, light intensity and so on in a network configuration have received limited attention in the literature. We refer to this second modality of sensors as estimation sensors; local estimates from such sensors can be used to reconstruct a field. Third, there has been recent interest in deploying sensors on mobile platforms. Mobility has the effect of increasing the effectiveness of sensing actions. We further classify sensor mobility into incidental and intentional motion. Incidentally mobile sensors move passively under the influence of the environment, for instance, a floating sensor drifting in the sea. We define intentional mobility as the ability to control the location and trajectory of the sensor, for example by mounting it on a mobile robot. We build our analysis on a series of cases. We first analyze coverage and connectivity of a network of floating sensors in rivers using simulations and experimental data, and give guidelines for sensor-network design. Second, we examine intentional mobility and detection sensors.(cont.) We examine the problem of covering indoor and outdoor pathways with reconfigurable camera sensor-networks. We propose and validate an empirical model for detection behavior of cameras. We propose a distributed algorithm for reconfiguring locations of cameras to maximize detection performance. Finally, we examine more general strategies for the placement of estimation sensors and ask when and where to take samples in order to estimate an unknown spatiotemporal field with tolerable estimation errors. We discuss various classes of error-tolerant sensor arrangements for trigonometric polynomial fields.by Ajay A. Deshpande.Ph.D

Scaling laws for internally heated mantle convection

Diese Arbeit stellt eine neue Methode vor, um Mantelkonvektion terrestrischer Planeten in einer 3D sphärischen Kugelschale zu simulieren. Die erforderlichen Differentialgleichungen werden dabei erstmalig mittels der Finiten -Volumen-Methode für irreguläre Voronoi-Gitter diskretisiert. Diese Diskretisierung (D) ist zweite Ordnung und voll implizit. Für mehr als 1000 CPUs ist die Simulation effizient parallelisiert. Damit wurde das Spiralgitter, eine neuartige Gitterstruktur mit lateral variierenden Auflösungen, untersucht. Weiterhin wurde ein bikonjugiertes Gradientenverfahren angewandt um das resultierende Gleichungssystem zu lösen. Die D. des Spannungstensors erlaubt eine Zell-Zell Variation der Viskosität von 8 und systemweit von 45 Größenordnungen. Um die Anwendbarkeit dieser neuen Methode zu demonstrieren, wurde eine Parameterstudie mit 88 Fällen gewählt, wobei Skalierungsgesetze für Wärmetransport, Grenzschichtendicken und Strukturkomplexität für planetare Mäntel ermittelt wurden. This work presents a new method to simulate mantle convection in a 3D spherical shell with fully spatially varying viscosities. The formulation of the governing equations is based on the finite-volume (FV) method for fully irregular grids using Voronoi-cells. The simulation code is efficiently parallelized for more than 1000 CPUs. A new irregular grid with varying lateral resolution, the spiral grid, was investigated. The discretization method is second-order accurate in space and time. The Krylov-subspace solver BiCGS with a Jacobi preconditioner is employed to solve the resulting system of equations. The discretization of the stress tensor can handle viscosity variations of up to 8 orders of magnitude from cell-to-cell and up to 45 orders of magnitude system wide. As an application to purely internally heated mantle convection in a spherical shell, a parameter study of 88 cases is carried out to derive scaling laws for heat transport, stagnant-lid thickness and structural complexity

A gradient optimization approach to adaptive multi-robot control

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 181-190).This thesis proposes a unified approach for controlling a group of robots to reach a goal configuration in a decentralized fashion. As a motivating example, robots are controlled to spread out over an environment to provide sensor coverage. This example gives rise to a cost function that is shown to be of a surprisingly general nature. By changing a single free parameter, the cost function captures a variety of different multi-robot objectives which were previously seen as unrelated. Stable, distributed controllers are generated by taking the gradient of this cost function. Two fundamental classes of multi-robot behaviors are delineated based on the convexity of the underlying cost function. Convex cost functions lead to consensus (all robots move to the same position), while any other behavior requires a nonconvex cost function. The multi-robot controllers are then augmented with a stable on-line learning mechanism to adapt to unknown features in the environment. In a sensor coverage application, this allows robots to learn where in the environment they are most needed, and to aggregate in those areas. The learning mechanism uses communication between neighboring robots to enable distributed learning over the multi-robot system in a provably convergent way. Three multi-robot controllers are then implemented on three different robot platforms. Firstly, a controller for deploying robots in an environment to provide sensor coverage is implemented on a group of 16 mobile robots.(cont.) They learn to aggregate around a light source while covering the environment. Secondly, a controller is implemented for deploying a group of three flying robots with downward facing cameras to monitor an environment on the ground. Thirdly, the multi-robot model is used as a basis for modeling the behavior of a herd of cows using a system identification approach. The controllers in this thesis are distributed, theoretically proven, and implemented on multi-robot platforms.by Mac Schwager.Ph.D

Co-Optimization of Communication, Motion and Sensing in Mobile Robotic Operations

In recent years, there has been considerable interest in wireless sensor networks and networked robotic systems. In order to achieve the full potential of such systems, integrative approaches that design the communication, navigation and sensing aspects of the systems simultaneously are needed. However, most of the existing work in the control and robotic communities uses over-simplified disk models or path-loss-only models to characterize the communication in the network, while most of the work in networkingand communication communities does not fully explore the benefits of motion.This dissertation thus focuses on co-optimizing these three aspects simultaneously in realistic communication environments that experience path loss, shadowing and multi-path fading. We show how to integrate the probabilistic channel prediction framework, which allows the robots to predict the channel quality at unvisited locations, into the co-optimization design. In particular, we consider four different scenarios: 1) robotic routerformation, 2) communication and motion energy co-optimization along a pre-defined trajectory, 3) communication and motion energy co-optimization with trajectory planning, and 4) clustering and path planning strategies for robotic data collection. Our theoretical, simulation and experimental results show that the proposed framework considerably outperforms the cases where the communication, motion and sensing aspects of the system are optimized separately, indicating the necessity of co-optimization. They furthershow the significant benefits of using realistic channel models, as compared to the case of using over-simplified disk models

Floating production storage offloading unit structural fatigue analysis

This thesis examines the fatigue behaviour of FPSO structures. It has been compiled as a result of theoretical, analytical and experimental study. The Finite Element approach has been utilized to analyse the FPSO's structure. It is intended that this particular work will enable further computer simulations for fatigue assessment to be carried out. The thesis starts with the development of the general arrangement, structure and typical details of the City FPSO. The applied loads are then reviewed and this includes the so called static loads due to cargo loading and still water pressure, and the green loads due to dynamic loads induced by the vessel behaviour on waves at sea. Response to local loads such as, external sea pressure, internal pressure due to the cargo and ballast, wave slamming loads, etc. is then determined. The effect of the top structural loads on the FPSO is discussed with some practical calculation of typical topside processing palates loads. SCF evaluation methods are considered together with a discussion of the effect of structural dimensioning of local details, the use of specially performed test results conducted on ship structure. In particular, the structural stress concentration factor at the web-toe associated with the max loading conditions is developed. Confirmation of validity of the SCFs theory is provided from an extensive appraisal of the literature and from laboratory tests of the structure in question. The experimental technique developed in this thesis is based upon geometrical analogy to the simplified Peterson's Neuber notch theory, applied to the system parametric equations of SCFs and the geometric relations. The experimental results are in general accordance with published results. This research includes a calibration method for S-N Curves required for typical fatigue sensitive details in FPSOs. It also provides improved information on the important link between S-N data and finite element analysis for fatigue life assessment using a linear cumulative damage formulation