Results 11  20
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80
On Generalizing the AMG Framework
 SIAM J. NUMER. ANAL
, 2003
"... We present a theory for algebraic multigrid (AMG) methods that allows for general smoothing processes and general coarsening approaches. The goal of the theory is to provide guidance in the development of new, more robust, AMG algorithms. In particular, we introduce several compatible relaxation met ..."
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Cited by 18 (3 self)
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We present a theory for algebraic multigrid (AMG) methods that allows for general smoothing processes and general coarsening approaches. The goal of the theory is to provide guidance in the development of new, more robust, AMG algorithms. In particular, we introduce several compatible relaxation methods and give theoretical justification for their use as tools for measuring the quality of coarse grids.
Using Approximate Inverses in Algebraic Multilevel Methods
, 1997
"... This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the twolevel type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related ..."
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Cited by 18 (6 self)
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This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the twolevel type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns, we analyze the effect of using an approximate inverse instead. We derive condition number estimates that are valid for any type of approximation of the Schur complementand that do not assume the use of the hierarchical basis. They show that the twolevel methods are stable when using approximate inverses based on modified ILU techniques, or explicit inverses that meet some rowsum criterion. On the other hand, we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate. These conclusions are numerically illustrated on some examples. keywords: iterative methods for linear...
Hierarchical Error Estimator for Eddy Current Computation
 In ENUMATH99: Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applictions
, 1999
"... We consider the quasimagnetostatic eddy current problem discretized by means of lowest order curlconforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in the finite element solution are desirable to control adaptive mesh refinement. We propose a loca ..."
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Cited by 15 (2 self)
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We consider the quasimagnetostatic eddy current problem discretized by means of lowest order curlconforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in the finite element solution are desirable to control adaptive mesh refinement. We propose a local aposteriori error estimator based on higher order edge elements: The residual equation is approximately solved in the space of phierarchical surpluses. Provided that a saturation assumption holds, we show that the estimator is both reliable and efficient.
Function, gradient, and Hessian recovery using quadratic edgebump functions
 SIAM J. Numer. Anal
"... Abstract. An approximate error function for the discretization error on a given mesh is obtained by projecting (via the energy inner product) the functional residual onto the space of continuous, piecewise quadratic functions which vanish on the vertices of the mesh. Conditions are given under which ..."
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Cited by 12 (7 self)
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Abstract. An approximate error function for the discretization error on a given mesh is obtained by projecting (via the energy inner product) the functional residual onto the space of continuous, piecewise quadratic functions which vanish on the vertices of the mesh. Conditions are given under which one can expect this hierarchical basis error estimator to give efficient and reliable function recovery, asymptotically exact gradient recovery and convergent Hessian recovery in the square norms. One does not find similar function recovery results in the literature. The analysis given here is based on a certain superconvergence result which has been used elsewhere in the analysis of gradient recovery methods. Numerical experiments are provided which demonstrate the effectivity of the approximate error function in practice.
A greedy strategy for coarsegrid selection
, 2006
"... Efficient solution of the very large linear systems that arise in numerical modelling of realworld applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considere ..."
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Cited by 10 (4 self)
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Efficient solution of the very large linear systems that arise in numerical modelling of realworld applications is often only possible through the use of multilevel techniques. While highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered, much recent interest has been in the development of purely algebraic approaches that may be applied in many situations, without problemspecific tuning. Here, we consider an algebraic approach to finding the fine/coarse partitions needed in multilevel approaches. The algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms, multilevel block factorization and algebraic multigrid. While no guarantee on the rate of coarsening is given, the splitting is shown to always yield an effective preconditioner in the twolevel sense. Numerical performance of twolevel and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations, and the advantages of each of these two algorithmic settings are explored. 1
ViewDependent Refinement of Multiresolution Meshes with Subdivision Connectivity
, 2001
"... We develop a new viewdependent levelofdetail algorithm for triangle meshes with subdivision connectivity. The algorithm is more suitable for textured meshes of arbitrary topology than existing progressive meshbased schemes. It begins with a wavelet decomposition of the mesh, and, per frame, find ..."
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Cited by 9 (1 self)
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We develop a new viewdependent levelofdetail algorithm for triangle meshes with subdivision connectivity. The algorithm is more suitable for textured meshes of arbitrary topology than existing progressive meshbased schemes. It begins with a wavelet decomposition of the mesh, and, per frame, finds a partial sum of wavelets necessary for highquality renderings from that frame's viewpoint. We present a new screenspace error metric that measures both geometric and texture deviation. In addition, wavelets that lie outside the view frustum or in backfacing areas are eliminated. The algorithm takes advantage of frametoframe coherence for improved performance, and supports geomorphs for smooth transitions between levels of detail.
POLYNOMIAL PRESERVING GRADIENT RECOVERY AND A POSTERIORI ESTIMATE FOR BILINEAR ELEMENT ON IRREGULAR QUADRILATERALS
, 2004
"... A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h 1+ρ) for ρ = min(α, 1), when the mesh is distorted O(h 1+α) (α > 0) from a regular one. Consequently, t ..."
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Cited by 8 (3 self)
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A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h 1+ρ) for ρ = min(α, 1), when the mesh is distorted O(h 1+α) (α > 0) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.
Strengthened Cauchy inequality in anisotropic meshes and application to an . . .
, 2001
"... In this document, we show an aposteriori error estimator which is efficient and reliable even on highly stretched meshes for the CrouzeixRaviart/P 0 pair. It relies on hierarchical space splitting whose main ingredient is the strengthened CauchySchwarz inequality. We demonstrate a method to enric ..."
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Cited by 8 (3 self)
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In this document, we show an aposteriori error estimator which is efficient and reliable even on highly stretched meshes for the CrouzeixRaviart/P 0 pair. It relies on hierarchical space splitting whose main ingredient is the strengthened CauchySchwarz inequality. We demonstrate a method to enrich the CrouzeixRaviart element so that the strengthened Cauchy constant is always bounded away from 1 independently of the aspect ratio. The performance of the aposteriori error estimator is confirmed by our numerical results.
Some Remarks on the Constant in the Strengthened C.B.S. Inequality: Application to h and pHierarchical Finite Element Discretizations of Elasticity Problems
, 1997
"... For a class of twodimensional boundary value problems including diffusion and elasticity problems it is proved that the constants in the corresponding strengthened CauchyBuniakowskiSchwarz (C.B.S.) inequality in the cases of hhierarchical and phierarchical finite element discretizations with t ..."
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Cited by 7 (1 self)
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For a class of twodimensional boundary value problems including diffusion and elasticity problems it is proved that the constants in the corresponding strengthened CauchyBuniakowskiSchwarz (C.B.S.) inequality in the cases of hhierarchical and phierarchical finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles triangles formulas are presented that show the dependence of the constant in the C.B.S. inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the C.B.S. inequality are given for threedimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly.
Deformable Surfaces for Feature Based Indirect Volume Rendering
 Computer Graphics International, IEEE Proceedings, 1998,752 – 760
, 1998
"... In this paper we present an indirect volume visualization method, based on the deformable surface model, which is a three dimensional extension of the snake segmentation method. In contrast to classical indirect volume visualization methods, this model is not based on isovalues but on boundary info ..."
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Cited by 7 (3 self)
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In this paper we present an indirect volume visualization method, based on the deformable surface model, which is a three dimensional extension of the snake segmentation method. In contrast to classical indirect volume visualization methods, this model is not based on isovalues but on boundary information. Physically speaking it simulates a combination of a thin plate and a rubber skin, that is influenced by forces implied by feature information extracted from the given data set. The approach proves to be appropriate for data sets that represent a collection of objects separated by distinct boundaries. These kind of data sets often occur in medical and technical tomography, as we will demonstrate by a few examples. We propose a multilevel adaptive finite difference solver, which generates a target surface minimizing an energy functional based on an internal energy of the surface and an outer energy induced by the gradient of the volume. This functional tends to produce very regular tr...