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Convergence of the Lloyd algorithm for computing centroidal Voronoi tessellations
 SIAM Journal on Numerical Analysis
"... Abstract. Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi tesse ..."
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Cited by 43 (4 self)
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Abstract. Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi tessellations may also be defined in more abstract and more general settings. Due to the natural optimization properties enjoyed by CVTs, they have many applications in diverse fields. The Lloyd algorithm is one of the most popular iterative schemes for computing the CVTs but its theoretical analysis is far from complete. In this paper, some new analytical results on the local and global convergence of the Lloyd algorithm are presented. These results are derived through careful utilization of the optimization properties shared by CVTs. Numerical experiments are also provided to substantiate the theoretical analysis.
Centroidal Voronoi tessellationbased reducedorder modeling of complex systems
, 2006
"... A reducedorder modeling methodology based on centroidal Voronoi tessellations (CVTs) is introduced. CVTs are special Voronoi tessellations for which the generators of the Voronoi diagram are also the centers of mass (means) of the corresponding Voronoi cells. For discrete data sets, CVTs are clos ..."
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Cited by 23 (4 self)
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A reducedorder modeling methodology based on centroidal Voronoi tessellations (CVTs) is introduced. CVTs are special Voronoi tessellations for which the generators of the Voronoi diagram are also the centers of mass (means) of the corresponding Voronoi cells. For discrete data sets, CVTs are closely related to the hmeans and kmeans clustering techniques. A discussion of reducedorder modeling for complex systems such as fluid flows is given to provide a context for the application of reducedorder bases. Then, detailed descriptions of CVTbased reducedorder bases and how they can be constructed from snapshot sets and how they can be applied to the lowcost simulation of complex systems are given. Subsequently, some concrete incompressible flow examples are used to illustrate the construction and use of CVTbased reducedorder bases. The CVTbased reducedorder modeling methodology is shown to be effective for these examples.
POD and CVTbased reducedorder modeling of Navier–Stokes flows
, 2006
"... A discussion of reducedorder modeling for complex systems such as fluid flows is given to provide a context for the construction and application of reducedorder bases. Reviews of the POD (proper orthogonal decomposition) and CVT (centroidal Voronoi tessellation) approaches to reducedorder modelin ..."
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Cited by 21 (4 self)
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A discussion of reducedorder modeling for complex systems such as fluid flows is given to provide a context for the construction and application of reducedorder bases. Reviews of the POD (proper orthogonal decomposition) and CVT (centroidal Voronoi tessellation) approaches to reducedorder modeling are provided, including descriptions of POD and CVT reducedorder bases, their construction from snapshot sets, and their application to the lowcost simulation of the Navier–Stokes system. Some concrete incompressible flow examples are used to illustrate the construction and use of POD and CVT reducedorder bases and to compare and contrast the two approaches to reducedorder modeling.
Advances in Studies and Applications of Centroidal Voronoi Tessellations
"... Centroidal Voronoi tessellations (CVTs) have become a useful tool in many applications ranging from geometric modeling, image and data analysis, and numerical partial differential equations, to problems in physics, astrophysics, chemistry, and biology. In this paper, we briefly review the CVT concep ..."
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Cited by 14 (4 self)
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Centroidal Voronoi tessellations (CVTs) have become a useful tool in many applications ranging from geometric modeling, image and data analysis, and numerical partial differential equations, to problems in physics, astrophysics, chemistry, and biology. In this paper, we briefly review the CVT concept and a few of its generalizations and wellknown properties. We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs. Whenever possible, we point out some outstanding issues that still need investigating.
Tessellation and Clustering by Mixture Models and Their Parallel Implementations
, 2004
"... Clustering and tessellations are basic tools in data mining. The kmeans and EM algorithms are two of the most important algorithms in the Mixture Modelbased clustering and tessellations. In this paper, we introduce a new clustering strategy which shares common features with both the EM and kmeans ..."
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Cited by 5 (3 self)
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Clustering and tessellations are basic tools in data mining. The kmeans and EM algorithms are two of the most important algorithms in the Mixture Modelbased clustering and tessellations. In this paper, we introduce a new clustering strategy which shares common features with both the EM and kmeans algorithms. Our methods also lead to more general tessellations of a spatial region with respect to a continuous and possibly anisotropic density distribution. Moreover, we propose some probabilistic methods for the construction of these clusterings and tessellations corresponding to a continuous density distribution. Some numerical examples are presented to demonstrate the effectiveness of our new approach. In addition, we also discuss the parallel implementation and performance of our algorithms on some distributed memory systems.
Multilevel and Adaptive Methods for Some Nonlinear Optimization Problems
, 2005
"... ∗ Signatures are on file in the Graduate School. In this thesis, we propose new multilevel and adaptive methods for solving nonlinear nonconvex optimization problems without relying on the linearization. We focus on two particular applications, that come from the fields of quantization and materia ..."
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Cited by 1 (1 self)
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∗ Signatures are on file in the Graduate School. In this thesis, we propose new multilevel and adaptive methods for solving nonlinear nonconvex optimization problems without relying on the linearization. We focus on two particular applications, that come from the fields of quantization and materials science. For the first problem, a multilevel quantization scheme is developed, that possesses a uniform convergence independent of the problem size. This is the first multilevel quantization scheme in the literature with a rigorous proof of uniform convergence with respect to the grid size and the number of grid levels for nonconstant densities. The proposed scheme can be generalized to higher dimensions, and both scalar and vector versions demonstrate significant speedup comparing to the traditional Lloyd method. We also provide some new characterizations for the convergence of the Lloyd iteration and other possible acceleration techniques including Newtonlike methods. For the second optimization problem, this thesis presents a novel algorithm aimed at automating phase diagram construc
Ideal Point Distributions, Best Mode Selections and Optimal Spatial Partitions via
"... There are many new applications of the centroidal Voronoi tessellations that come to life in recent years, along with more mathematical understandings and new algorithmic advances in their efficient computation. Some examples are presented in this paper as an illustration with an emphasis on the con ..."
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Cited by 1 (1 self)
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There are many new applications of the centroidal Voronoi tessellations that come to life in recent years, along with more mathematical understandings and new algorithmic advances in their efficient computation. Some examples are presented in this paper as an illustration with an emphasis on the construction of ideal point distributions, best mode selections and optimal spatial partitions. 1
Burkardt et al. NA03 Dundee 2003 Reduced order modeling of complex systems
"... 1 Reducedorder modeling Solutions of (nonlinear) complex systems are expensive with respect to both storage and CPU costs. As a result, it is difficult if not impossible to deal with a number of situations such as: continuation or homotopy methods for computing state solutions; parametric studies o ..."
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1 Reducedorder modeling Solutions of (nonlinear) complex systems are expensive with respect to both storage and CPU costs. As a result, it is difficult if not impossible to deal with a number of situations such as: continuation or homotopy methods for computing state solutions; parametric studies of state solutions; optimization and control problems (multiple state solutions); and feedback control settings (realtime state solutions). Not surprisingly, a lot of attention has been paid to reducing the costs of the nonlinear state solutions by using reducedorder models for the state; these are lowdimensional approximations to the state. Reducedorder modeling has been and remains a very active research direction in many seemingly disparate fields. We will focus on three approaches to reducedorder modeling: reduced basis methods; proper orthogonal decomposition (POD); andcentroidal Voronoi tessellations (CVT). Before describing the three approaches, we first discuss what we exactly mean by reducedordering modeling and make some general comments that apply to all reducedorder models. For a state simulation, a reducedorder method would proceed as follows. One first chooses a reduced basis ui, i = 1,..., n, where n is hopefully very
REDUCTION OF MODES FOR THE COMPUTATIONS OF NAVIERSTOKES EQUATIONS
"... Abstract. This article is a survey article for a reducedorder modeling approach for computation of timedependent NavierStokes flows. A reducedbasis method is introduced. The choices for the reduced basis method are the Lagrange subspace, the Hermite subspace and the Taylor subspace. We then int ..."
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Abstract. This article is a survey article for a reducedorder modeling approach for computation of timedependent NavierStokes flows. A reducedbasis method is introduced. The choices for the reduced basis method are the Lagrange subspace, the Hermite subspace and the Taylor subspace. We then introduce the PODbased method and CVTbased method. 1. Introducion Optimal control problems that involve partial differential equations as state equations are formidable problems to solve in real time. One such situation arises in control of fluid dynamical systems in which the state equations are the NavierStokes equations. We will discuss some reductiontype method which may help to overcome this difficulty. In order to illustrate the reducedbasis method, we consider the stationary NavierStokes equations −ν∆u+ (u · ∇)u+∇p = f(1.1) ∇ · u = 0(1.2) with appropriate boundary conditions for ν ∈ R and u ∈ X. The above problem is a parameterized one. The constant ν presents kinematic viscosity about which we choose to interpolate to obtain a reducedfinitedimensional set of basis elements. In standard finite element approximations, one approximate X with a piecewise polynomial space. However, the choices for the reduced basis method are different. The Lagrange subspace. In this case, the basis elements are solutions of the nonlinear problem under study at various parameter values νj. The reduced subspace is given by XR = span uj uj = u(νj), j = 1,...,M This kind of subspace was used to study structural problems in [1]. A possible advantage in this choice is that updating the basis elements can be done one basis vector at a time instead of generating the whole space.
Reducedorder Modeling of NavierStokes equations via Cenroidal Voronoi Tessellation
 CONTEMPORARY MATHEMATICS
"... A reducedbasis method based on centroidal Voronoi tessellations (CVT’s) is introduced. A discussion of reducedordering modeling for complex systems such as fluid flows is given to provide a context for the application of reducedorder bases. Then, detailed descriptions of CVTbased reducedorder ..."
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A reducedbasis method based on centroidal Voronoi tessellations (CVT’s) is introduced. A discussion of reducedordering modeling for complex systems such as fluid flows is given to provide a context for the application of reducedorder bases. Then, detailed descriptions of CVTbased reducedorder bases including their construction from snapshot sets and their application to the lowcost simulation of complex systems are given. Some concrete incompressible flow examples are used to illustrate the construction and use of CVTbased reducedorder bases.