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The finest regular coarsening and recursivelyregular subdivisions
"... We generalize the notion of regular polyhedral subdivision of a point (or vector) configuration in a new direction. This is done after studying some related objects, like the finest regular coarsening and the regularity tree of a subdivision. Properties of these two objects are derived, which confer ..."
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We generalize the notion of regular polyhedral subdivision of a point (or vector) configuration in a new direction. This is done after studying some related objects, like the finest regular coarsening and the regularity tree of a subdivision. Properties of these two objects are derived, which
A review of algebraic multigrid
, 2001
"... Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical onelevel methods had already reached their limits and new hierarchical algorithms had to b ..."
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Cited by 345 (11 self)
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Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical onelevel methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems. This paper gives a review of the first hierarchical and purely matrixbased approach, algebraic multigrid (AMG). AMG can directly be applied, for instance, to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes, both in 2D and 3D. Since AMG does not make use of any geometric information, it is a “plugin ” solver which can even be applied to problems without any geometric background, provided that the
An Algorithm for Coarsening Unstructured Meshes
 Numer. Math
, 1996
"... . We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the lin ..."
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Cited by 51 (5 self)
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. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...
Smooth ViewDependent LevelofDetail Control and Its Application to Terrain Rendering
"... The key to realtime rendering of largescale surfaces is to locally adapt surface geometric complexity to changing view parameters. Several schemes have been developed to address this problem of viewdependent levelofdetail control. Among these, the viewdependent progressive mesh (VDPM) framewor ..."
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Cited by 261 (1 self)
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The key to realtime rendering of largescale surfaces is to locally adapt surface geometric complexity to changing view parameters. Several schemes have been developed to address this problem of viewdependent levelofdetail control. Among these, the viewdependent progressive mesh (VDPM) framework represents an arbitrary triangle mesh as a hierarchy of geometrically optimized refinement transformations, from which accurate approximating meshes can be efficiently retrieved. In this paper we extend the general VDPM framework to provide temporal coherence through the runtime creation of geomorphs. These geomorphs eliminate "popping" artifacts by smoothly interpolating geometry. Their implementation requires new outputsensitive data structures, which have the added benefit of reducing memory use.
Selectively refinable subdivision meshes
"... We introduce RGB triangulations, an extension of redgreen triangulations that can support selective refinement over subdivision meshes generated through quadrisection of triangles. Our purpose is to define a mechanism based on local operators that act on subdivision meshes while supporting operatio ..."
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operations similar to those available in Continuous Level Of Detail models. Our mechanism permits to take an adaptive mesh at intermediate level of subdivision and process it through both refinement and coarsening operations, by remaining consistent with an underlying Loop subdivision scheme. Our method does
Survey of Polygonal Surface Simplification Algorithms
, 1997
"... This paper surveys methods for simplifying and approximating polygonal surfaces. A polygonal surface is a piecewiselinear surface in 3D defined by a set of polygons ..."
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Cited by 228 (3 self)
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This paper surveys methods for simplifying and approximating polygonal surfaces. A polygonal surface is a piecewiselinear surface in 3D defined by a set of polygons
Biorthogonal wavelets for subdivision volumes
 In: Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications
, 2002
"... Figure 1: Volume subdivision, manipulation, and fitting. A lattice (top left) is recursively subdivided and reshaped at the fourth subdivision level. This shape is lowpass filtered by removing fineresolution wavelet coefficients (bottom right). We present a biorthogonal wavelet construction based ..."
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Cited by 5 (0 self)
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Figure 1: Volume subdivision, manipulation, and fitting. A lattice (top left) is recursively subdivided and reshaped at the fourth subdivision level. This shape is lowpass filtered by removing fineresolution wavelet coefficients (bottom right). We present a biorthogonal wavelet construction
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 213 (7 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
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