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Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation

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Numerical Mathematics(Theory,methods and Applications) 2012年第2期5卷 242-259页

作者： Zichao Di Maria Emelianenko Stephen Nash Department of Mathematical Sciences George Mason UniversityFairfaxVA 22030USA. Systems Engineering and Operations Research Department George Mason UniversityFairfaxVA 22030USA

In a variety of modern applications there arises a need to tessellate the domain into representative regions,called Voronoi cells.A particular type of such tessellations,called centroidal Voronoi tessellations or CVTs,are in big demand due to their optimality properties important for many *** availability of fast and reliable algorithms for their construction is crucial for their successful use in practical *** paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization *** convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems,and O(k)complexity estimation is provided for a problem with k generators.

关键词： Centroidal Voronoi tessellation optimal quantization truncated Newton method Lloyd’s algorithm multilevel method uniform convergence

Fast multilevel CVT-Based Adaptive Data Visualization Algorithm

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Numerical Mathematics(Theory,methods and Applications) 2010年第2期3卷 195-211页

作者： M.Emelianenko Department of Mathematical Sciences George Mason University

Efficient data visualization techniques are critical for many scientific applications. Centroidal Voronoi tessellation(CVT) based algorithms offer a convenient vehicle for performing image analysis,segmentation and compression while allowing to optimize retained image quality with respect to a given *** experimental science with data counts following Poisson distributions,several CVT-based data tessellation algorithms have been recently *** they surpass their predecessors in robustness and quality of reconstructed data,time consumption remains to be an issue due to heavy utilization of the slowly converging Lloyd *** paper discusses one possible approach to accelerating data visualization *** relies on a multidimensional generalization of the optimization based multilevel algorithm for the numerical computation of the CVTs introduced in[1],where a rigorous proof of its uniform convergence has been presented in 1-dimensional *** multidimensional implementation employs barycentric coordinate based interpolation and maximal independent set coarsening *** is shown that when coupled with bin accretion algorithm accounting for the discrete nature of the data,the algorithm outperforms Lloyd-based schemes and preserves uniform convergence with respect to the problem *** numerical demonstrations provided are limited to spectroscopy data analysis,the method has a context-independent setup and can potentially deliver significant speedup to other scientific and engineering applications.

关键词： Centroidal Voronoi tessellations computational algorithms Lloyd's method acceleration schemes multilevel method binning image analysis visualization signal-to-noise ratio.

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

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Science China Mathematics 2012年第7期55卷 1513-1526页

作者： WANGFeng CHENJinRu HUANGpeiQi JiangsuKeyLaboratory 0rNSLSCSSchoolofMathematicalSciencesNanjingNormalUniversityNanjing210046China DepartmentofAppliedMathematics NanjingForestryUniversityNanjing210037China

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

关键词： multilevel method Crouzeix-Raviart element discontinuous coefficients

A UNIFIED FRAMEWORK FOR THE CONSTRUCTION OF VARIOUS ALGEBRAIC multilevel PRECONDITIONING methodS

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Acta Mathematicae Applicatae Sinica 1999年第2期15卷 132-143页

作者：白中治 Institute of Computational Mathematics and Scientific/Engineering Computing the Chinese Academy of Sciences Beijing China Faculty of Mathematics and Informatics the University of Nijmegen Nijmegen Netherlands

A framework for algebraic multilevel preconditioning methods is presented for solving largesparse systems of linear equations with symmetric positive definite coefficient matrices, whicharise in the discretization of second order elliptic boundary value problems by the finite elementmethod. This framework covers not only all known algebraic multilevel preconditioning methods,but yields also new ones. It is shown that all preconditioners within this framework have optimalorders of complexities for problems in two-dimensional (2-D) and three-dimensional(3-D) problemdomains, and their relative condition numbers are bounded uniformly with respect to the numbersof both the levels and the nodes.

关键词： multilevel method polynomial acceleration finite element method,optimal-order preconditioner